Probability and Applications
Time: Wednesdays at 1pm
Location: Mathematical Sciences Building, Room: MB503
Organizer: Ilya Goldsheid and Linus Wunderlich

DateRoomSpeakerTitle

18/04/2012 5:00 AMM203Daniel Ueltschi (Warwick)Random permutations with cycle weightsSeminar series:We consider random permutations where cycles are weighted according to their lengths. I will discuss results about the lengths of typical cycles, the total number of cycles, and the number of finite cycles.

02/05/2012 5:00 PMM203Yee Whye Teh (UCL)Hierarchical Bayesian Models of Sequential DataSeminar series:In this talk I will present a novel approach to modelling sequence data called the sequence memoizer. As opposed to most other sequence models, our model does not make any Markovian assumptions. Instead, we use a hierarchical Bayesian approach which allows effective sharing of statistical strength across the different parts of the model. To make computations with the model efficient, and to better model the powerlaw statistics often observed in sequence data arising from datadriven linguistics applications, we use a Bayesian nonparametric prior called the PitmanYor process as building blocks in the hierarchical model. We show stateoftheart results on language modelling and text compression.

16/05/2012 5:00 PMM203Natanael Berestycki (Cambridge)Branching Brownian motion with selectionSeminar series:We consider random systems of particles which branch and move independently of one another, but are also subject to a selection mechanism that maintains the size of the population essentially constant. Models of this type were recently introduced by physicists Brunet, Derrida and collaborators. Using nonrigorous arguments they derived striking predictions for such systems: notably, the genealogy of the population is given by a universal object, the BolthausenSznitman coalescent. I will give an overview of some of these conjectures and some rigorous recent results in these directions.

16/05/2012 5:00 PMM203Natanael Berestycki (Cambridge)Branching Brownian motion with selectionSeminar series:Probability and ApplicationsWe consider random systems of particles which branch and move independently of one another, but are also subject to a selection mechanism that maintains the size of the population essentially constant. Models of this type were recently introduced by physicists Brunet, Derrida and collaborators. Using nonrigorous arguments they derived striking predictions for such systems: notably, the genealogy of the population is given by a universal object, the BolthausenSznitman coalescent. I will give an overview of some of these conjectures and some rigorous recent results in these directions.

06/06/2012 5:00 PMM 203Pierre Tarres (Oxford)Edge reinforced random walks, vertex reinforced jump process, and the SuSy hyperbolic sigma modelSeminar series:
Edgereinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertexreinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuoustime process favouring sites with more local time.
Then we prove that the VRJP is a mixture of timechanged Markov jump processes and calculate the mixing measure, which we interpret as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer.
This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010).

18/06/2012 5:00 PMM513Michael Kozdron (University of Regina)Using multiple SLE to explain a certain observable in the 2d Ising modelSeminar series:
The SchrammLoewner evolution (SLE) is a oneparameter family of random growth processes that has been successfully used to analyze a number of models from twodimensional statistical mechanics. Currently there is interest in trying to formalize our understanding of conformal field theory using SLE. Smirnov recently showed that the scaling limit of interfaces of the 2d critical Ising model can be described by SLE(3). The primary goal of this talk is to explain how a certain nonlocal observable of the 2d critical Ising model studied by Arguin and SaintAubin can be rigorously described using multiple SLE(3) and Smirnov's result. As an extension of this result, we explain how to compute the probability that a Brownian excursion and an SLE(k) curve, 0 < k < 4, do not intersect.

04/07/2012 5:00 AMM203Malwina Luczak (Sheffield)The supermarket model with arrival rate tending to 1Seminar series:
There are $n$ queues, each with a single server. Customers arrive in a Poisson process at rate $\lambda n$, where $0 < \lambda = \lambda (n) < 1$. Upon arrival each customer selects $d = d(n) \ge 1$ servers uniformly at random, and joins the queue at a leastloaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1.
We will review the literature, including results of Luczak and McDiarmid (2006), for the case where $\lambda$ and $d$ are constants independent of $n$.
We will then investigate the speed of convergence to equilibrium and the maximum length of a queue in the equilibrium distribution when $\lambda (n) \to 1$ and $d(n) \to \infty$ as $n \to \infty$. This is joint work with Graham Brightwell. 
26/09/2012 5:00 PMM203Leonid Pastur (Kharkov) Joint meeting with London Analysis and probability SeminarOn links between the random operator and random matrix theoriesSeminar series:
We present several families of selfadjoint ergodic operators for which we prove that if the parameter indexing operators of a given family tends to infinity then their Integrated Density of States converges weakly to the infinite size limit of the Normalized Counting Measure of eigenvalues of certain random matrices. We then give an informal discussion of these results as possible indications of the presence of the continuous spectrum of the random ergodic operators belonging to considered families for sufficiently large values of the indexing parameters.

03/10/2012 5:00 AMM203Matthias Winkel (University of Oxford)Hereditary properties, GaltonWatson real trees and Levy treesSeminar series:
Neveu studied leaflength erasure of GaltonWatson trees, Geiger and
Kauffmann the subtree spanned by vertices picked uniformly at random and
Duquesne and Winkel the subtree spanned by leaves picked uniformly at
random, each finding that the reduced tree is also a GaltonWatson tree.
We observe that the offspring distributions that occur in the threeHereditary properties, GaltonWatson real trees and Levy trees
examples are the same, and we introduce the notion of a hereditary
property to offer a unified approach. The notion of leaflength erasure
has recently been exploited by Evans, Winter and coauthors in a context
of real trees. We continue these developments and use results about
hereditary properties to obtain strong convergence results of
GaltonWatson real trees to Levy trees and characterisations and
properties of the limits. We also have an invariance principle for
GaltonWatson trees and decomposition results for GaltonWatson and Levy
trees. This is joint work with Thomas Duquesne. 
31/10/2012 4:00 AMM203Leonid Bogachev (University of Leeds)Gaussian fluctuations for Plancherel partitionsSeminar series:
The limit shape of Young diagrams under the Plancherel measure was found by
Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem
for fluctuations of Young diagrams in the bulk of the partition “spectrum”.
More specifically, under a suitable (logarithmic) normalization, the corresponding
random process converges (in the FDD sense) to a Gaussian process with independent
values. We also discuss a link with an earlier result by Kerov (1993) on the
convergence to a generalized Gaussian process. The proof is based on poissonization
of the Plancherel measure and an application of a general central limit theorem for
determinantal point processes. (Joint work with Zhonggen Su.) 
07/11/2012 1:30 PMM203Special EventOne Day Ergodic Theory Meeting

21/11/2012 4:00 AMM203Nadia Sidorova (UCL)A conditioning principle for GaltonWatson treesSeminar series:
We discuss the behaviour of a GaltonWatson tree conditioned on its
martingale limit being small. We prove that it converges to the smallest
possible tree, giving an example of entropic repulsion where the limit has
no entropy. We also discuss the first branching time of the conditioned
tree (which turns out to be almost deterministic) and the strength of the
first branching. This is a joint work with N. Berestycki (Cambridge), N.
Gantert (Munich), P. Moerters (Bath). 
12/12/2012 4:00 AMM203Malwina Luczak (QMUL)Law of large numbers for the SIR process on a random graph with given degree sequenceSeminar series:
We study the susceptibleinfectiverecovered (SIR) epidemic on a
random graph chosen uniformly among all the graphs with given vertex
degrees. In this model, infective vertices infect each of their
susceptible neighbours, and recover, at a constant rate. We show that
below a certain threshold in parameter values only a small number of
vertices get infected. Above the threshold, we prove that the fraction of
vertices that are infected if the epidemic becomes macroscopic is
approximately deterministic. In particular, we give a simple proof of
Volz's equations from biological literature.This is joint work with Svante Janson and Peter Windridge.

23/01/2013 4:00 AMM203Yan Fyodorov (QMUL)Fluctuations and extreme values in multifractal patternsSeminar series:
The goal is to understand sampletosample fluctuations in disordergenerated multifractal intensity patterns.
Arguably the simplest model of that sort is the exponential of an ideal periodic 1/f Gaussian noise.
The latter process can be looked at as a onedimensional "projection" of 2D Gaussian Free Field and inherits from it the logarithmic covariance structure. It most naturally emerges in the random matrix theory context, but attracted also an independent interest in statistical mechanics of disordered systems. We will determine the threshold of extreme values of 1/f noise and
provide a rather compelling explanation for the mechanism behind its universality.
Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disordergenerated multifractal fields. The presentation will be mainly based on the joint work with Pierre Le Doussal and Alberto Rosso, J Stat Phys: 149 (2012), 898920 as well as on some related earlier works by the speaker. 
06/02/2013 4:00 AMM203Svetlana Anulova (Inst Control Sci, Moscow)On ergodic properties of hybrid mechanical stochastic systemsSeminar series:
The model describes the dynamics of a point mass moving on a line in a force field. The force is disturbed by white noise, and depends on the random media. The random media has a finite number of states, switching in a markov chain regime. Under natural conditions on the force field we establish existence and uniqueness (in a weak sense) of solution of the equation, and the exponential rate of convergence to the stationary regime. The research stems from the investigation of F. Campillo and E. Pardoux into the issue of a controlled vehicle suspension device.

13/02/2013 4:00 AMM203Pierre Le Doussal (ENS, Paris)Statistics of the KardarParisiZhang growth equation from quantum mechanicsSeminar series:

06/03/2013 4:00 AMM203Andreas Kyprianou (Bath)Censored Stable ProcessesSeminar series:
We look at a general twosided jumping strictly alphastable process where alpha is in (0,2). By censoring its path each time it enters the negative half line we show that the resulting process is a positive selfsimilar Markov Process. Using Lamperti's transformation we uncover an underlying driving Lévy process and, moreover, we are able to describe in surprisingly explicit detail the WienerHopf factorization of the latter. Using this WienerHopf factorization together with a series of spatial path transformations, it is now possible to produce an explicit formula for the law of the original stable processes as it first ``enters'' a finite interval, thereby generalizing a result of Blumenthal, Getoor and Ray for symmetric stable processes from 1961.
This is joint work with Alex Watson (Bath) and JC Pardo (CIMAT)

13/03/2013 4:00 AMM203Ben Hambly (Oxford)Spectral properties for some random graphs and their scaling limits
I will consider the scaling limits of some random graphs such as the
continuum random tree and the critical random graph and discuss some
aspects of their spectra. In particular the high frequency asymptotics
of the eigenvalue counting functionfor the scaling limitand the
behaviour of the spectral gapas the random graphs converge to their
scaling limit. 
20/03/2013 4:00 AMM203Alexander Iksanov (National T. Shevchenko University, Kiev)Asymptotics of the number of empty boxes in the Bernoulli sieveSeminar series:
Suppose we are given a multiplicative random walk (a
stickbreaking set) generated by a random variable W taking values in the
interval (0,1) and a sample from the uniform [0,1] law which is
independent of the stickbreaking set. The Bernoulli sieve is a random
occupancy scheme in which 'balls' represented by the points of the uniform
sample are allocated over an infinite array of 'boxes' represented by the
gaps in the stickbreaking set. Assuming that the number of balls equals n
I am interested in the weak convergence of the number of empty boxes
within the occupancy range as n approaches infinity. Depending on the
behavior of the law of W near the endpoints 0 and 1 the number of empty
boxes can exhibit quite a wide range of different asymptotics. I will
discuss the most interesting cases with an emphasis on the methods
exploited. 
25/09/2013 5:00 AMM203Peter Windridge (QMUL)Critical SIR epidemic on a random graph with given degreesSeminar series:
The SIR epidemic is a simple Markovian model for disease spreading through
a finite graph. Each node is either susceptible, infective or recovered. An
infective node infects each neighbouring susceptible node, and
recovers, at a constant rate. We consider this process with the
underlying graph chosen uniformly at random, subject to having given vertex
degrees.The infection rate, recovery rate and vertex degrees determine a
parameter called the 'basic reproductive number' for the epidemic,
denoted R_0.It is known that R_0 \leq 1 implies only a few infections can occur w.h.p,
and that R_0 > 1 opens the possibility of a large outbreak. That this, there
is a threshold behaviour.In this talk we'll focus on the critical regime R_0 = 1 + \omega(n^{1/3}).
This is part of ongoing work with Svante Janson (Uppsala) and Malwina
Luczak (QMUL). 
30/10/2013 4:00 PMM203Olivier Henard (QMUL)The number of old families in Beta(2alpha,alpha)coalescents
The Lambdacoalescent is a partitionvalued process modelling the backward genealogy of a population, introduced independently by Pitman and Sagitov in 99. This genealogy may be represented as a tree. The definition of the Lambdacoalescent naturally builds the coalescent tree from the leaves.
In this talk, we present an attempt to construct the coalescent tree from the root. The first "branching event" from the root then corresponds to
the number of old families, and we determine its generating function explicitly in the special case of the Beta(2alpha,alpha)coalescent. 
06/11/2013 4:00 PMM203Albert FerreiroCastilla (QMUL)Multilevel Monte Carlo simulation for Lévy processes based on the WienerHopf factorization
In Kuznetsov et al. [2] a new Monte Carlo simulation technique was introduced for a
large family of Levy processes that is based on the WienerHopf decomposition. We pursue
this idea further by combining their technique with the recently introduced multilevel
Monte Carlo methodology. Moreover, we provide here for the rst time a theoretical
analysis of the new Monte Carlo simulation technique in [2] and of its multilevel variant
for computing expectations of functions depending on the historical trajectory of a Levy
process. We derive rates of convergence for both methods and show that they are uniform
with respect to the "jump activity" (e.g. characterised by the BlumenthalGetoor index).References
[1] FerreiroCastilla, A., Kyprianou, A.E., Scheichl, R. and Suryanarayana, G. (2013) Multi
level Monte Carlo simulation for Levy processes based on the WienerHopf factorization.
Stoch. Proc. Appl. (To appear).
[2] Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2011) A WienerHopf
Monte Carlo simulation technique for Levy process. Ann. App. Probab. 21(6), 21712190 
13/11/2013 4:00 PMM203Nicholas Simm (QMUL)fBm with Hurst index H=0 and statistics of GUE characteristic polynomials
We study the behaviour of the logmod of the characteristic polynomial \log\det(xM) as a function of the spectral parameter x, where M is a large GUE random matrix. We reveal that for x taken inside the bulk of the spectrum, that object is intimately related to various versions of the logarithmicallycorrelated random Gaussian processes, in particular, to the fractional Brownian motion (fBm) with Hurst exponent H=0. As the standard definitions always assume H>0, we provide a bona fide extension of fBm to the H=0 case in terms of a certain stochastic Fourier integral.

27/11/2013 4:00 PMM203YACINE BARHOUMI (Uni Zurich)KEATINGSNAITH PHILOSOPHY AND MOD* CONVERGENCE : SOME RECENT DEVELOPMENTS
Recent progresses in Number Theory are due to the application of the KeatingSnaith philosophy that consists in solving a surrogate problem in Random Matrix Theory where the computations are notably easier to achieve or to adapt results from Number Theory in the random matrix world. In this talk, we apply the KeatingSnaith philosophy to count the
number of zeroes of linear combinations of characteristic polynomials of independent random unitary matrices, a problem initially motivated by the study of Lfunctions. In particular, we explain why 100 % of the zeroes of such a combination lie on the unit circle. We then find
a probabilistic interpretation of mod* convergence, a particular type of convergence which is classical in Number Theory but unusual in Probability Theory and which is at the core of the celebrated moments conjecture. With such an interpretation, we are able to find approximations in distribution for sequences converging in the mod* sense with the use of Stein’s method, and to refine a probabilistic model about the number of prime divisors of a random uniform integer due to Erdös and Kac.This talk is partially based on a work with C. P. Hughes, J. Najnudel and A. Nikeghbali.

11/12/2013 4:00 PMM203Codina Cotar (UCL)Uniqueness of gradient Gibbs measures with disorder
We consider two versions of random gradient models. In model A) the interface feels a bulk term of random fields while in model B) the disorder enters though the potential acting on the gradients itself. It is well known that without disorder there are no Gibbs measures in infinite volume in dimension d = 2, while there are gradient Gibbs measures describing an infinitevolume distribution for the increments of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. Cotar and Kuelske proved the existence of shiftcovariant gradient Gibbs measures for model A) when d\ge 3 and the expectation with respect to the disorder is zero, and for model B) when d\ge 2. In the current work, we prove uniqueness of shiftcovariance gradient Gibbs measures with expected given tilt under the above assumptions. We also prove decay of covariances for both models. This is based on joint work with Christof Kuelske.

15/01/2014 4:00 PMM513Serguey Novak (Middlesex University London)Accuracy of Poisson approximation for sums of integervalued r.v.s.
We present an estimate of the accuracy of Poisson approximation to the distribution of a sum of independent integervalued random variables.
In a particular case of 01 random variables this yields the famous result by Barbour and Eagleson (1983). A generalisation to the case of dependent observations is given as well. 
29/01/2014 4:00 PMM513Zhian Liang (Shanghai)Evaluation of Geometric Asian Power Options under Fractional Brownian Motion
Modern option pricing techniques are often considered among the most mathematical complex of all applied areas of financial mathematics. In particular, the fractional Brownian motion is proper to model the stock dynamics for its longrange dependence. In this paper, we evaluate the price of geometric Asian options under fractional Brownian motion framework. Furthermore, the options are generalized to those with the added feature whose payoff is a power function. Based on the equivalent martingale theory, a closed form solution has been derived under the risk neutral probability.

05/02/2014 4:00 PMM513George Kapetanios (SEF, QMUL)Exponent of cross sectional dependence: estimation and inference.

12/02/2014 4:00 PMM513Jérémie Unterberger (Université de Lorraine)Analytic extension of fractional Brownian motion
Fractional Brownian motion (fBm) has emerged as the prominent model in the search for extensions of
stochastic calculus to random process which are not in the semimartingale class. This family of Gaussian processes
is mainly used as an ad hoc modelization of processes with power law correlations, but appears also at a more theoretical
level as the scaling limit of several natural discrete equilibrium models or processes. After a general introduction, we shall
discuss more specifically an analytic extension of fBm to the upper halfplane which we introduced a few years ago, and allows
to identify clearly the main terms in several limit theorems. 
19/02/2014 4:00 PMM513Jordan Stoyanov (Newcastle)Moment Determinacy of Probability Distributions
Abstract: The emphasis will be on some recent progress in the moment analysis of distributions and their characterization as being unique (Mdeterminate) or nonunique (Mindeterminate) in terms of the moments. Specific topics which will be discussed are:
(a) Stieltjes classes for Mindeterminate distributions. Index of dissimilarity.
(b) New Hardy’s criterion for uniqueness. Multidimensional moment problem.
(c) Nonlinear transformations of random data and their moment (in)determinacy.
(d) Moment determinacy of distributions of stochastic processes defined by SDEs.There will be new results, hints for their proof, examples and counterexamples, and also open questions and conjectures.

12/03/2014 4:00 PMM513Erik Baurdoux (LSE)Optimal prediction of the time of the ultimate maximum of a Lévy process
Optimal prediction of the ultimate maximum is a nonstandard optimal stopping problem in the sense that the payoff function depends on a process which is not adapted to the given filtration. Our aim is to approximate by stopping times as close as possible the (random) time of the ultimate maximum of a Lévy process. For a finite time horizon, this problem has been studied in various papers, including Du Toit, J. and Peskir, G. (AAP 2009) and Bernyk, V., Dalang, R.C. and Peskir, G. (2011 Ann. Probab.) for a Brownian motion and onesided stable process, respectively.
In this work we consider the infinite horizon case for a general Lévy process drifting to minus infinity. Using properties of the all time maximum of a L\'evy process and a reformulation of the problem as a standard optimal stopping problem, we find an optimal stopping time as a first passage time of the reflected process. The results are made more explicit in the spectrally onesided case.
This talk is based on joint work with Dr. Kees van Schaik which is due to appear in Acta Applicandae Mathematicae."

26/03/2014 4:00 PMM513Dario Spano (Oxford)On the ancestral process of longrange seed bank models
It has been observed that, in some bacterial species, spores may remain
dormant for a long time, to wake up much later, even up to "order of
population size" generations later. When they wake up, they can still
participate in the population's reproduction. This incredibly relaxed
attitude causes a relaxation of the population's Markov property,
forward in time. I will
describe some results about the genealogical process of seed bank models
which, in the scaling limit (as the population size tends to infinity),
may differ dramatically from the wellknown Kingman's Coalescent process.
The genealogy can be derived from the properties of a system of certain
types of Polya urns containing balls undergoing some sort of random
erosion. Joint work with J. Blath, A. GonzalesCasanova, N. Kurt, (Berlin). 
25/06/2014 5:00 PMM513Alexander Marynych (Kiev)Limit theorems for random processes with immigration at the epochs of a renewal process.
Attachment: abstract [PDF 86KB]

24/09/2014 5:00 PMCécile Mailler (Bath)Condensation and symmetrybreaking in the zerorange process with weak site disorder.
The zerorange process (ZRP) is described as follows: n sites contain respectively $(Q_1, \ldots, Q_n)$ particles, where the $(Q_1, \ldots, Q_n)$ are i.i.d. random variables. How does the ZRP behaves when we condition the system to have a fixed density (= average number of particles per site)? Under some conditions  described for example by Grosskinsky, Schütz and Spöhn (2003) and Janson (2012)  the zerorange process exhibits condensation.
We consider in this talk the nonhomogeneous ZRP, introduced by Godrèche and Luck (2012), in which we first sample random fitnesses in every site of the system, before running a ZRP, where the occupation numbers $(Q_1, \ldots, Q_n)$ are independent, but not identical. A site with a larger fitness will likely contain more particles.
I will describe how the nonhomogeneous ZRP behaves and under which conditions condensation occurs: this is an ongoing work, in collaboration with Peter Mörters (University of Bath) and Daniel Ueltschi (University of Warwick).

01/10/2014 5:00 PMNeofytos Rodosthenous (QML)Optimal stopping problems in diffusiontype models with running maxima and drawdowns
We study optimal stopping problems related to the pricing of perpetual American options in an extension of the BlackMertonScholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. We obtain closedform solutions to the equivalent freeboundary problems for the value functions with smoothfit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting threedimensional Markov process.

22/10/2014 5:00 PMM513Igor Kortchemski (Universität Zürich)Scaling limits and influence of the seed graph in preferential attachment trees
In this talk, we will be interested in the asymptotics of random
trees built by linear preferential attachment (also known as
BarabasiAlbert trees or planeoriented recursive tree). We will first try
to understand the influence of the initial tree (the seed) on the
longterm behavior of this process. Then we will see that this problem is
closely related to the existence of scaling limits of socalled looptrees
associated with these trees. Roughly speaking, a looptree of a tree
encodes the geometric structure of its nodes of large degree. This is
joint work with Nicolas Curien, Thomas Duquesne and Ioan Manolescu." 
29/10/2014 4:00 PMM513Pascal Maillard (Université ParisSud)Choices and Intervals
I will talk about a recent work with Elliot Paquette, the abstract of which reads as follows:
We consider a random interval splitting process, in which the splitting rule depends on the empirical
distribution of interval lengths. We show that this empirical distribution converges to a limit almost surely as the
number of intervals goes to infinity. We give a characterization of this limit as a solution of an ODE and use this to
derive precise tail estimates. The convergence is established by showing that the sizebiased empirical distribution
evolves in the limit according to a certain deterministic evolution equation. Although this equation involves a non
local, nonlinear operator, it can be studied thanks to a carefully chosen norm with respect to which this operator is
contractive.In finitedimensional settings, convergence results like this usually go under the name of stochastic approximation
and can be approached by a general method of Kushner and Clark. An important technical contribution of our work
is the extension of this method to an infinitedimensional setting. 
26/11/2014 4:00 PMM513Goran Peskir (Manchester)Optimal MeanVariance Portfolio Selection
I will present a dynamic formulation of the meanvariance portfolio
selection problem and discuss possible ways of solving it.Joint work with J. L. Pedersen (Copenhagen)

03/12/2014 4:00 PMM513Alexandre Stauffer (Bath)Random walk on dynamical percolation
We study the behavior of random walk on dynamical percolation.
In this model, the edges of a graph G are either open or closed, and refresh their status at rate \mu.
At the same time a random walker moves on G at rate 1 but only along edges which are open.
The regime of interest here is when \mu goes to zero as the number of vertices of G goes to infinity, since this creates longrange dependencies on the model.
When G is the ddimensional torus of side length n, we prove that in the subcritical regime,
the mixing times is of order n^2/\mu. We also obtain results concerning mean squared displacement and hitting times.
This is a joint work with Yuval Peres and Jeff Steif. 
14/01/2015 4:00 PMM513George Kapetanios (QM, SEF)Time Varying Estimation and Inference with Application to Large Dimensional Covariance Estimation and Portfolio Management

28/01/2015 4:00 PMM513Noah Forman (Oxford)The quantile rearrangement of random walk and Brownian motion increments
From a simple random walk one may obtain a random permutation of indices [1,n] via the lexicographic ordering first on the value of the walk at a given time, and second on the time itself. We demonstrate that by rearranging the increments of a random walk bridge according to this quantile permutation, we obtain a Dyck path. Passing to a Brownian limit gives a novel proof and a generalization of a theorem of Jeulin (1985) describing Brownian local times as a timechanged Brownian excursion.

11/02/2015 4:00 PMM513Nina Gantert (Technical University Munich)Random Walk among random conductances: Einstein relation and monotonicity of the speed
Many applications, such as porous media or composite materials, involve
heterogeneous media which are modeled by random fields. These media are
locally irregular but are “statistically homogeneous” in the sense that
their law has homogeneity properties. Considering random motions in such
a random medium, it turns out often that they can be described by their
effective behaviour. This means that there is a deterministic medium,
the effective medium, whose properties are close to the random
medium, when measured on long spacetime scales. In other words, the
local irregularities of the random medium average out over large
spacetime scales, and the random motion is characterized by the
“macroscopic” parameters of the effective medium. How do the macroscopic
parameters depend on the law of the random medium?As an example, we consider the effective diffusivity (i.e. the
covariance matrix in the central limit theorem) of a random walk among
random conductances. It is interesting
and nontrivial to describe this diffusivity in terms of the law of the
conductances. The Einstein relates this diffusivity with the derivative
of the speed of a biased random walk among random conductances. We
explain the Einstein relation and
we also discuss monotonicity questions for the speed of a biased random
walk among random conductances.The talk is based on joint work (in progress) with Noam Berger, Xiaoqin
Guo and and Jan Nagel. 
25/02/2015 4:00 PMM513HsienKuei Hwang (Academia Sinica, Taiwan)Nonlinear differential equations in applied probability
A brief survey, based largely on my research, is given
of nonlinear differential equations arising in applied probability.
The main focus will be on asymptotic and stochastic properties,
and their applications. 
11/03/2015 4:00 PMM513Louigi AddarioBerry (McGill University; Leverhulme Lecture)Slowdown of the front for branching Brownian motion with decay of mass.
Consider a branching Brownian motion particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical massm particles.
One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance Theta(t^{1/3}) behind the typical BBM front.
Many natural questions about the model remain open.

22/04/2015 5:00 PMM513Dan Crisan (Imperial CL)Limit theorems in stochastic filtering
Just as in any other area of probability theory, limit theorems are abundant in stochastic filtering. In this talk I will present a couple of examples. The first is an extension of the classical construction of the SuperBrownian Motion as a limit of systems branching Brownian particles. Variants of this construction lead to the theoretical and numerical approximations of the solution of the filtering problem. The second example is a propagation of chaos limit. The result is based on a representation of the solution of the filtering problem as the time marginal of a solution of a certain McKean–Vlasov type equation.
This is joint work with Jessica Gaines, Kari Heine, Terry Lyons and Jie Xiong.

06/05/2015 5:00 PMM513Stephen Connor (York)Perfect simulation of M/G/c queues
Unlike Markov chain Monte Carlo, perfect simulation algorithms produce a sample from the exact equilibrium distribution of a Markov chain, but at the expense of a random runtime. I'll give a short introduction to these algorithms for beginners, before talking about some recent work, jointly with Wilfrid Kendall (Warwick), on designing perfect simulation algorithms for M/G/c queues.

23/09/2015 5:00 PMM203Joaquin Miguez (QM)An overview of convergence results and some unconventional applications of particle filters
Particle filters are a class of recursive Monte Carlo algorithms that are used to approximate the sequences of posterior probability measures that arise in partially observed (dynamical) statespace systems. The approximations take the form of discrete random probability measures, consisting of samples in the state space with associated, properly computed weights. These random measures are typically used to approximate integrals with respect to the true probability distributions. In the talk, we will review the basic algorithm (often termed bootstrap filter) and standard convergence results. Then, we will proceed to discuss some notsousual applications of the methodology, namely the computation of maximum a posteriori estimators, the numerical solution of global optimisation problems and the estimation of probability density functions (pdf’s). Finally, we propose a new method for the online assessment of the convergence of particle filters that relies on the theory that we have developed for pdf estimation.

30/09/2015 5:00 PMM203John Moriarty (QM)A geometric answer to an open question of singular control with stopping
We study a family of optimal stopping problems with a parameter, with respect to which the value function is continuous but the boundary of the stopping set is discontinuous. This solves a certain open problem of singular stochastic control with discretionary stopping suggested by Karatzas, Ocone, Wang and Zervos (2000) by providing suitable candidates for the moving boundaries in an unsolved parameter range. The discontinuity, which would not be considered in the original solution method, is found by inspecting the geometry of obstacle problems in a sense going back to Dynkin and Yushkievich (1969).

07/10/2015 5:00 PMBancroft Rd 3.02Nick Whiteley (Bristol)Particle filtering subject to interaction constraints
Particle filters are very flexible algorithms for inferential computation in nonlinear, nonGaussian statespace models. The potential benefits of parallel and distributed implementation of particle filters motivates study of their interaction structure, especially the "resampling" step, in which particles interact through a genetictype selection, which is usually the bottleneck for parallelization. Can we do away with resampling, or at least restructure it in such a way as to be more naturally suited to nonserial implementation? What role does resampling really play in endowing these algorithms with attractive properties? This talk will introduce some new algorithms and discuss properties of existing ones, in this context.
Joint work with Kari Heine (UCL) and Anthony Lee (Warwick)

13/10/2015 5:00 PMM103Rosemary Harris (QM)Memory effects in complex systems (joint meeting with Complex Systems seminar)

28/10/2015 4:00 PMM203Christopher Joyner (QM)Random Walk approach to spectral statistics in random Bernoulli matrices
Abstract:
Random Bernoulli matrices (in which the matrix elements are chosen independently from plus or minus 1 with equal probability) are intimately connected to the adjacency matrices of random graphs and share many spectral properties. In the limit of large matrix dimension the distribution of eigenvalues from such matrices resembles that from matrices in which the elements are chosen randomly from a Gaussian distribution  the question is why? We take a dynamical approach to this problem, which is achieved by initiating a discrete random walk process over the space of matrices. Previously we have used this idea to analyse the corresponding eigenvalue motion but I will discuss some recent developments which involve the adaptation of Stein’s method to this context. 
25/11/2015 4:00 PMM203James Martin (Oxford)Percolation games
Let G be a graph (directed or undirected), and let v be some vertex of G. Two players play the following game. A token starts at v. The players take turns to move, and each move of the game consists of moving the token along an edge of the graph, to a vertex that has not yet been visited. A player who is unable to move loses the game. If the graph is finite, then one player or the other must have a winning strategy. In the case of an infinite graph, it may be that, with optimal play, the game continues for ever.
I'll focus in particular on games played on the lattice Z^d, directed or undirected, with each vertex deleted independently with some probability p. In the directed case, the question of whether draws occur is closely related to ergodicity for certain probabilistic cellular automata, and to phase transitions for the hardcore model. In the undirected case, I'll describe connections to bootstrap percolation and to maximumcardinality matchings and independent sets.
This is based on joint work with Alexander Holroyd and Irène Marcovici (http://arxiv.org/abs/1503.05614(link is external)) and with Riddhipratim Basu, Alexander Holroyd and Johan Wästlund (http://arxiv.org/abs/1505.07485(link is external)).

09/12/2015 4:00 PMM203Julien Berestycki (Oxford)Vanishing corrections for the position in a linear model of FKPP fronts
Based on joint work with E. Brunet S. Harris and M. Roberts
Take the linearised FKPP equation
\[
\partial_t h =\partial^2_x h +h
\]
with boundary condition $h(m(t),t)=0$. Depending on the behaviour of the
initial condition $h_0(x)=h(x,0)$ we obtain the asymptotics ~up to a
$o(1)$ term $r(t)$~ of the absorbing boundary $m(t)$ such that $\omega(x) :=
\lim_{t\to\infty} h(x+m(t) ,t)$ exists and is nontrivial. In particular, as in
Bramson's results for the nonlinear FKPP equation, we recover the
celebrated $3/2\log t$ correction for initial conditions decaying faster
than $x^{\nu}e^{x}$ for some $\nu<2$.Furthermore, when we are in this regime, the main result I will present is the identification (to first order) of the $r(t)$ term which ensures the fastest convergence to $\omega(x)$. When $h_0(x)$ decays faster than $x^{\nu}e^{x}$ for some $\nu<3$, we show that $r(t)$ must be
chosen to be $3\sqrt{\pi/ t}$ which is precisely the term predicted heuristically by Ebertvan Saarloos in the nonlinear case. When the initial condition decays as $x^{\nu}e^{x}$ for some $\nu\in[3,2)$, we show that even though we are still in the regime where Bramson's correction is $3/2\log t$, the Ebertvan Saarloos correction has to be modified. 
13/01/2016 4:00 PMM103James Norris (Cambridge)Fluid limits for population processes with a continuum of types: an example from gas kinetics
The talk will discuss the combination of two classical ideas.
The first is the use of hierarchichal decompositions for testfunctions in estimating the Wasserstein distance
of a probability measure from its sample empirical distribution.
The second is the use of martingale estimates to show convergence of Markov chains to solutions of differential equations.
The ideas can be combined because the techniques used for sample empirical distributions extend naturally to martingale
measures associated to a Markov chain.The driving example is Kac's Nparticle meanfield model for velocity exchange by elastic collision in a dilute gas of spherical particles.
We will show that, for large N, the empirical distribution of particles converges in Wasserstein distance to the solution of the spatially homogeneous
Boltzmann equation, as fast as any Nparticle empirical distribution could do so. 
20/01/2016 4:00 PMM103Vladimir Ejov (Flinders University of South Australia/ MPI Bonn): joint meeting of Probability and Combinatorics Research Groups"Snakes and Ladders" Heuristic for the Hamiltonian Cycle Problem and Flinders HCP Challenge
We present a polynomial complexity heuristic for solving the Hamiltonian Cycle Prob
lem in an undirected graph of order n. Although finding a Hamiltonian cycle is not theoretically
guaranteed, we have observed that this heuristic is successful even in cases where such cycles are
extremely rare. It has not yet failed on a single graph under 2000 vertices. It uses transformations, inspired by kopt algorithms such as, now classical, LinKernighan heuristic to reorder the vertices in order to construct a Hamiltonian cycle, although it is not restricted to sequential kopt edge exchanges.
The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time, O(n4 log n) for this implementation. Online demonstration will accompany presentation. 
27/01/2016 4:00 PMM103Chris Rogers (Cambridge): talk in the framework of the London Probability SeminarCombining different models
In a situation where a large number of assets are available to a fund, the
question of how to allocate capital to those assets is a perennial and
important one. The issues of estimating the means and covariances of
returns are very well known, and there seems still no good solution if we
take all the assets at once. If we choose to decompose the set of all
assets into smaller subsets, we expect to find it much easier to estimate
means and covariances, but then the question remains how to combine these
smaller studies. The smaller models will typically be talking about sets of
assets that overlap but do not coincide, and the question we would really
like to understand is how we might go about combining the wisdom gained
from studying small subsamples of the assets into some useable statement
about all the assets. This talk offers a few very preliminary ideas about
how this could be approached. 
10/02/2016 4:00 PMM103Pavel Gapeev (LSE)Risk sensitive utility indifference pricing of perpetual American options under fixed transaction costs.

24/02/2016 4:00 PMM103Giorgio Ferrari (Bielefeld)Nash equilibria of threshold type for twoplayer nonzerosum games of stopping
In this talk I consider twoplayer nonzerosum games of optimal stopping on a class of regular diffusions with singular boundary behaviour (in the sense of Itô and McKean, p. 108). I show that Nash equilibria are realised by stopping the diffusion at the first exit time from suitable intervals whose boundaries solve a system of algebraic equations. Under mild additional assumptions we also prove uniqueness of the equilibrium. Finally, I discuss some recent results on the connection between twoplayer nonzerosum games of optimal stopping and a certain class of twoplayer nonzerosum games of singular control.

02/03/2016 4:00 PMM103Elliott PaquetteThe law of fractional logarithm in the GUE minor process
Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The eigenvalues of the upperleft NxN submatrices, form what is called the GUE minor process. We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk. This GUE minor process is determinantal, and our proof is twofold. First, we reduce the law of fractional logarithm to a set of correlation and decorrelation estimates that must be made about the largest eigenvalues of pairs of GUE matrices. We then make these estimates using the explicit form of the GUE minor kernel. We also pose an open problem related to this kernel.
This is joint work with Ofer Zeitouni.

09/03/2016 4:00 PMM103Elena Boguslavskaya (Brunel)Atransforms and Lévy processes
The Atransform, if applied to a monomial $x^n$ results in a wellknown Appell polynomial $Q_n^\eta(x)$. Not surprisingly, the transformed function has properties similar to an Appell polynomial. For example, the transformed function is a martingale if the transform is built on a Lévy process. As a consequence of the above, the Atransform is especially useful for solving problems related to Lévy processes. For instance, it gives a straightforward formula for the calculation of Europeantype functionals of Lévy processes. In the context of optimal stopping, one can obtain an optimal stopping rule by studying the geometrical properties of the transformed payoff. If compared to the standard approach, the Atransform method benefits from the absence of integrodifferential equations, making the process of obtaining the solution much easier. We illustrate the method with some examples.

16/03/2016 4:00 PMM103Evgeny Burnaev (Institute of Information Transmission Problems, Moscow)On some Machine Learning Applications in Computational Finance
The talk consists of two parts. The purpose of the first part is to give a broad introduction to the techniques of machine learning, and to place those techniques within the context of computational finance. The purpose of the second part is to present some new methodology for changepoints detection in the Presence of Trends and LongRange Dependence. To detect changepoints and anomalies, we develop a machine learning approach based on the ensembles of “weak” statistical detectors. We demonstrate the performance of the proposed methodology using an artificial dataset, the publicly available Abilene dataset as well as the proprietary geoinformation system dataset.
This talk is mostly based on the following publications:
https://www.researchgate.net/publication/294580553_Optimal_Estimation_of...(link is external)
https://www.researchgate.net/publication/290440067_Ensembles_of_Detector...(link is external)
https://www.researchgate.net/publication/290440053_Nonparametric_Decompo...

04/05/2016 4:00 PMM103Hermann Þórisson (Uni Iceland)Finding patterns in twosided Brownian motion
We consider the problem of finding particular patterns in a realisation of a twosided standard Brownian motion taking the value zero at time zero. Examples include twosided Skorohod imbedding, the Brownian bridge and several other patterns, also in planar Brownian motion. The key tool here are recent allocation results in Palm theory.

21/09/2016 4:00 PMM103Fabrizio Leisen (Kent)Some recent results on Exchangeable Occupancy Models
This talk focuses on Exchangeable Occupancy Models (EOMs) and their relations with
the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As
our main purpose, we show how definitions and results presented in Shaked, Spizzichino,
and Suter (2004) can be unified and generalized in the frame of occupancy models. We first
show some general facts about EOMs. Then we introduce a class of EOMs, called M(a)
models, and a concept of generalized Uniform Order Statistics Property in discrete time.
For processes with this property, we prove a general characterization result in terms of
M(a)models. Finally, we will investigate some closure properties of Exchangeable occupancy models w.r.t. some natural transformations of EOMs. In particular, a new transformation of occupancy distributions, called merging, is introduced and studied when M(a)
models are considered. This talk resumes two joint works with Francesca Collet, Fabio Spizzichino and Florentina Suter. 
28/09/2016 4:00 PMM103Silke Rolles (Munich)Vertexreinforced jump processes
Sabot and Tarres showed that a discrete time version of vertexreinforced
jump processes has the same law as a random walk in a random environment,
where the environment can be described in terms of a supersymmetric sigma
model introduced by Zirnbauer. Furthermore, they showed that linearly
edgereinforced random walk has the same law as a mixture of the discrete
time version of vertexreinforced jump processes. In the talk I will
describe these connections and indicate how they can be used to prove properties
of the reinforced processes.The talk is based on joint papers with Margherita Disertori and Franz
Merkl. 
19/10/2016 4:00 PMM103Ashkan Nikeghbali (Zurich)Some remarkable applications of coupling and strong convergence for the circular unitary ensemble
It is standard in random matrix theory to study weak convergence of the eigenvalue point process, but how about almost sure convergence? In this talk we introduce a way to couple all dimensions of random unitary matrices together to prove a quantitative strong convergence for eigenvalues for random unitary matrices. Then we show how this can give some remarkable simple answers to important questions related to moments and ratios of characteristic polynomials of random unitary matrices (and insight in some conjectures related to the Riemann zeta function).

09/11/2016 4:00 PMM103Jennie Hansen (HeriotWatt University, Edinburgh)Random mappings with exchangeable indegrees
Random mapping models have been studied by various authors since the
1950's and have applications in modelling epidemic processes, the analysis of
cryptographic systems (e.g. DES) and of Pollard's algorithm, and random
number generation. In this talk I consider random mappings from a perspective which is inspired, in part, by results for preferential and antipreferential attachment in other random graph models. It turns out that both the usual uniform random mapping model and other models (e.g. random mappings
with preferential and antipreferential attachment) are special cases of random mappings with exchangeable indegrees. By viewing random mappings from this perspective, questions related to their asymptotic structure can be tackled by using a calculus that is based on the moments of the joint
distribution of the exchangeable indegree sequence of the vertices in the (directed) graphical representation of the random mapping. This calculus gives us tools to tackle questions about the component structure of a random mapping which would be more dicult to attack using classical combinatorial approaches such as generating function arguments. In this talk I give an overview of the development of this calculus and of results which can be obtained using it. In addition, I explore some natural and attractive connections between random mappings with exchangeable indegrees and various urn schemes.This talk is based on joint work with Jerzy Jaworski (Adam Mickiewicz Uni
versity), who was supported by the Marie Curie IntraEuropean Fellowship
No. 236845 (RANDOMAPP) within the 7th European Community Frame
work Programme. 
23/11/2016 4:00 PMM103Igor Wigman (KCL)Nodal intersections of random toral eigenfunctions with a test curve
Abstract

This talk is based on joint works with Zeev Rudnick, and Maurizia Rossi.We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard 2dimensional flat torus ("arithmetic random waves") with a fixed reference curve. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry.
Our first result prescribes the asymptotic behaviour of the nodal intersections variance for generic smooth curves in the high energy limit; remarkably, it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber, and the geometry of the given curve. For these curves we can prove the Central Limit Theorem. In a work in progress we construct some exceptional examples of curves where the variance is of smaller order of magnitude, and the limit distribution is nonGaussian.

30/11/2016 4:00 PMM103Wilfrid Kendall (Warwick)Random Walks in Scaleinvariant Random Spatial Networks (SIRSN)
A SIRSN obeys axioms proposed by Aldous [1] and provides random (almost surely) unique routes between specified locations in a statistically scaleinvariant manner. A planar construction based on a randomized dyadic rectilinear network is established in [1]: a further construction based on Poisson line processes has now been established in [2,3] and even delivers SIRSN in dimensions 3 and higher. I will describe recent planar results concerning random walks (actually Rayleigh random flights) in such SIRSN, aimed at providing better insight into the behaviour of SIRSN routes.
1. Aldous, D.J. (2014). ScaleInvariant Random Spatial Networks. Electronic J. Prob., 21, no. 19, 141.
2. WSK (2016). From Random Lines to Metric Spaces. Ann. Prob. (to appear).
3. Kahn, J. (2016). Improper poisson line process as SIRSN in any dimension. Ann. Prob. (to appear). 
07/12/2016 4:00 PMM103Alexandre Veretennikov (Leeds)Poisson equations with a potential in the whole space for "ergodic" generators
In several recent papers Poisson equation ``in the whole space’’ was studied for so called ergodic generators $L$ corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for the method of diffusion approximation in the theory of stochastic averaging and homogenisation. In his talk a similar equation with a potential is considered, firstly because it is natural for PDEs, and secondly with a hope that it may be also useful for some extensions related to homogenization and averaging. The title could have also used the term FeynmanKac’ formula on the infinite horizon with variable signs of the potential.

18/01/2017 1:00 PMM103Constanza RojasMolina (Uni Bonn)Characterization of the metalinsulator transport transition for the twoparticle Anderson model
In dimensions higher than two it is expected that a disordered quantum system undergoes a metalinsulator transition from a region of localization to delocalization. For the oneparticle Anderson model, F. Germinet and A. Klein showed that the transport exponent in these regions can be related to the applicability of the multiscale analysis method used in the proof of localization. In this talk we present a recent generalization of this characterization to the twoparticle Anderson model with shortrange interactions. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the Bootstrap Multiscale Analysis. In the case of two particles, this gives the desired characterization of the metalinsulator transport transition.
This is joint work with A. Klein and S. T. Nguyen. 
08/02/2017 1:00 PMM203Ilya Goldsheid (QM)Products of nonidentically distributed independent matrices.
The theory concerned with the study of the asymptotic behaviour of independent identically distributed (iid) random matrices can probably be perceived as a basically complete  at least as far as the matrices of a fixed dimension are concerned. However, not much is known when the matrices can be drawn from several (even just two) distinct distributions. Perturbations of iid matrices provide just one example of such products.
In should be emphasized that the methods used in the iid case don't work for the non stationary sequences of matrices.
I shall discuss several result addressing this problem.

15/02/2017 1:00 PMM203Nick Simm (Warwick)Gaussian multiplicative chaos and random matrix theory
I will describe recent results on the relation between the subject of Gaussian multiplicative chaos (GMC) and random matrix theory (RMT). This relation has been the subject of continued interest lately, and touches on various parts of mathematics including the Riemann zeta function, Gaussian free fields, and branching processes. Our basic object of study is the number of eigenvalues of a random unitary matrix lying in a small arc of the unit circle. After an appropriate regularization, we prove that the exponential of this stochastic process converges to a limiting GMC measure as the size of the matrix becomes large. A key advance is that our results hold in the entire subcritical regime of GMC. Our technique is likely to apply to a wide range of random matrix models and beyond. This is joint work with Gaultier Lambert and Dmitry Ostrovsky.

01/03/2017 1:00 PMW316Roland Bauerschmidt (Cambridge)Local KestenMcKay law for random regular graphs
I will discuss results on the delocalisation of eigenvectors and the
spectral measure of random regular graphs with large but fixed degree. Our
approach combines the almost deterministic structure of random regular graphs
at small distances with random matrix like behaviour at large distances. 
08/03/2017 1:00 PMW316Neil O'Connell (Bristol)From longest increasing subsequences to Whittaker functions and random polymers
The RobinsonSchenstedKnuth (RSK) correspondence is a combinatorial bijection which plays an important role in the theory of Young tableaux and provides a natural framework for the study of longest increasing subsequences in random permutations and related percolation problems. I will give some background on this and then explain how a birational version of the RSK correspondence provides a similar framework for the study of GL(n)Whittaker functions and random polymers.

15/03/2017 1:00 PMW316Robert Griffiths (Oxford)A coalescent dual process for a WrightFisher diffusion with recombination and its application to haplotype partitioning
The WrightFisher diffusion process with recombination models the haplotype frequencies in a population where a length of DNA contains $L$ loci, or in a continuous model where the length of DNA is regarded as an interval $[0,1]$. Recombination may occur at any point in the interval and split the length of DNA. A typed dual process to the diffusion, backwards in time, is related to the ancestral recombination graph, which is a random branching coalescing graph. Transition densities in the diffusion have a series expansion in terms of the transition functions in the dual process. The history of a single haplotype back in time describes the partitioning of the haplotype into fragments by recombination. The stationary distribution of the fragments is of particular interest and we show an efficient way of computing this distribution. This is joint research with
Paul A. Jenkins, University of Warwick, and
Sabin Lessard, Universit{\'e } de Montr{\'e}al. 
22/03/2017 1:00 PMW316Christophe Sabot (Lyon)Vertex Reinforced Jump Process, random Schrödinger operator and hitting time of Brownian motion.
It is wellknown that the first hitting time of 0 by a negatively drifted Brownian motion starting at $a>0$ has the inverse Gaussian law. Moreover, conditionally on this first hitting time, the BM up to that time has the law of a 3dimensional Bessel bridge. In this talk, we will give a generalization of this result to a familly of Brownian motions with interacting drifts. The law of the hitting times will be given by the inverse of the random potential that appears in the context of the selfinteracting process called the Vertex Reinforced Jump Process (VRJP). The spectral properties of the associated random Schrödinger operator at ground state are intimately related to the recurrence/transience properties of the VRJP.
We will also explain some "commutativity" property of these BM and its relation with the martingale that appeared in previous work on the VRJP.
Work in progress with Xiaolin Zeng. 
05/04/2017 1:00 PMW316Nadav Yesha (King's)Nodal intersections for random waves on the threedimensional torus.
We study the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard flat torus ("arithmetic random waves") with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Rudnick and Wigman found the asymptotic behaviour of the nodal intersections variance on the two dimensional torus; we discuss the three dimensional case and give an upper bound for the variance. These results in particular imply that the nodal intersections number admits a universal asymptotic law with arbitrarily high probability.
This is a joint work with Zeev Rudnick and Igor Wigman. 
24/05/2017 1:00 PMW316Giorgio Ferrari (Uni Bielefeld)On the Optimal Management of Public Debt: a Solvable TwoDimensional Singular Stochastic Control Problem
Consider the problem of a government that wants to control the debttoGDP (gross domestic product) ratio of a country, while taking into consideration the evolution of the inflation rate. The uncontrolled inflation rate follows an OrnsteinUhlenbeck dynamics and affects the growth rate of the debt ratio. The level of the latter can be reduced by the government through fiscal interventions. The government aims at choosing a debt reduction policy which minimises the total expected cost of having debt, plus the total expected cost of interventions on debt ratio. We model such problem as a twodimensional singular stochastic control problem over an infinite timehorizon. We show that it is optimal for the government to adopt a policy that keeps the debttoGDP ratio under an inflationdependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully twodimensional optimal stopping problem.

04/10/2017 1:00 PMW316, Queen's BuildingAnna Maltsev (QM)Intracellular calcium signalling and the Ising model
Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal to noise ratio is achieved by a cooperative action of Ca release channels arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via CainducedCarelease, generating highamplitude local Ca signals known as puffs in neurons and sparks in muscle cells. Despite positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. We construct an exact mapping of such molecular clusters to an Ising model and demonstrate that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field. This is joint work with Prof. Stern's laboratory at the National Institutes of Health.

11/10/2017 1:00 PMQueen's Building, W316Daniel Ueltschi (Warwick)Random interchange model on the complete graph and the PoissonDirichlet distribution
In 2005, Schramm considered the random interchange model on the complete graph and he proved
that the lengths of long cycles have PoissonDirichlet distribution PD(1). If one adds the weight
2^{#cycles}, one gets Tóth’s representation of the quantum Heisenberg model. In this case, we
prove (essentially) that long cycles have distribution PD(2). In a related model of random loops,
that involves “double bars” as well as “crosses”, we prove that long loops have distribution PD(1).
Joint work with J. Björnberg and J. Fröhlich. 
18/10/2017 1:00 PMQueen's Building, W316Dmitry Chelkak (ENS, Paris)Ising model on planar graphs: sholomorphic functions and embeddings.
During the last decade, a significant progress in the understanding of the critical Ising model on nice 2D lattices has been achieved, basing on the careful analysis of the socalled sholomorphic observables (aka lattice fermions). Surprisingly and embarrassingly, despite the facts that the rigid structure of sholomorphic functions exists on every weighted planar graph and that the conformally invariant behavior arising in the scaling limit should be very universal, the existing proofs of convergence results highly rely on some particular trick (sub/superharmonicity of the primitives of f^2), which works only for the special case of isoradial graphs, with prescribed Ising weights. The main purpose of this talk is to discuss what can be done in more general settings: from some explicit computations for the "layered" model in the halfplane (unpublished work with Clement Hongler (Lausanne)) to a new embedding of generic weighted planar graphs into the plane which might pave a way to true universality results for the critical Ising model.

08/11/2017 1:00 PMQueen's Building, W316Sunil Chhita (Durham)The twoperiodic Aztec diamond
Simulations of uniformly random domino tilings of large Aztec
diamonds give striking pictures due to the emergence of two macroscopic
regions. These regions are often referred to as solid and liquid phases.
A limiting curve separates these regions and interesting probabilistic
features occur around this curve, which are related to random matrix
theory. The twoperiodic Aztec diamond features a third phase, often
called the gas phase. In this talk, we introduce the model and discuss
some of the asymptotic behavior at the liquidgas boundary. This is
based on joint works with Vincent Beffara (Grenoble), Kurt Johansson
(Stockholm) and Benjamin Young (Oregon). 
23/10/2013 5:00 PMM203Shaun McKinlay (Uni Melbourne)A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits
We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n−1)...X(1) of i.i.d. copies X(k) of X converge a.s. as n→∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

26/02/2014 4:00 PMM513Martijn Pistorius (Imperial)ON EXPLICIT SOLUTION OF AN INVERSE FIRSTPASSAGE TIME PROBLEM FOR LEVY PROCESSES AND COUNTERPARTY CREDIT RISK
For a given Markov process X and survival function H on R_+, the inverse firstpassage time problem (IFPT) is to find a barrier function b : R_+ → [−∞,+∞] such that the survival function of the firstpassage time τ_b = inf{t ≥ 0 : X(t) ≤ b(t)} is given by H. In this paper we consider a version of the IFPT problem where the barrier is fixed at zero and the problem is to find an entrance law μ and a timechange I such that for the timechanged process X ◦ I the IFPT problem is solved by a constant barrier at the level zero. For any Levy process X satisfying an exponential moment condition, we identify explicitly the solution of this problem
in terms of quasiinvariant distributions of the process X killed at the epoch of first entrance into the negative halfaxis. For a given multivariate survival function H of generalised frailty type we construct subsequently an explicit solution to the corresponding IFPT with the barrier level fixed at zero. We apply these results to the valuation of financial contracts that are subject to counterparty credit risk. 
18/03/2015 4:00 PMM513Tim Leung (Columbia University)Optimal Mean Reversion Trading with Transaction Costs
We discuss the timing of trades under meanreverting price dynamics subject to fixed transaction costs. We consider an optimal double stopping approach to determine the optimal times to enter and subsequently exit the market, when prices are driven by an OrnsteinUhlenbeck (OU), exponential OU, or CIR process. In addition, we analyze a related optimal switching problem that involves an infinite sequence of trades, and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Among our results, we find that the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero, leading to a disconnected continuation (waiting) region for entry.

29/03/2017 1:00 PMW316Christina Goldschmidt (Oxford)Parking on a tree
Consider the following particle system. We are given a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. Each node of the tree has space for a single particle (we think of them as cars). A number $m \le n$ of cars arrives one by one, and car $i$ wishes to park at node $S_i$, $1 \le i \le m$, where $S_1, S_2, \ldots, S_m$ are i.i.d. uniform random variables on $[n]$. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until the first time it encounters an empty space, in which case it parks there; otherwise, it leaves the tree. Let $A_{n,m}$ denote the event that all $m$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Set $m = [\alpha n]$. Then if $\alpha \le 1/2$, $P(A_{n,[\alpha n]} \to \frac{\sqrt{12\alpha}}{1\alpha}$, whereas if $\alpha > 1/2$ we have $P(A_{n,[\alpha n]}) \to 0$. (In fact, they proved more precise asymptotics in $n$ for $\alpha \ge 1/2$.) In this talk, I will give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Time permitting, I will also discuss some generalisations.
Joint work with Michał Przykucki (Oxford).

18/11/2015 4:00 PMM203Michał Morayne (Wroclaw)Sequential selection from posets
The talk will present an overview of results about the sequential selection from posets and related optimality problems. The problem of choosing online a maximal element from a poset with the greatest possible probability during an examination of a random permutation of its elements is a direct generalization of the classical secretary problem. Several new algorithms either optimal on a given natural poset or simply eﬃcient which are universal for certain families of posets have recently been obtained. We shall also state a logconcavity type inequality that is a criterion for a (general) process to fall into the so called monotone case where ﬁnding an optimal stopping algorithm is especially simple. We will apply this inequality to sequential selections from posets. (These last results were obtained jointly with Malgorzata Kuchta).

25/01/2017 1:00 PMM203Sasha Sodin (QM)The distribution of ζ'/ζ about a random point on the critical line
We shall discuss the properties of the logarithmic derivative
of the Riemann zetafunction, rescaled about a point chosen at random
point on the critical line. The talk will be mostly selfcontained. 
01/11/2017 1:00 PMW316, Queen's BuildingAlexander Gnedin (QMUL)The collision spectrum of Lambdacoalescents
Lambdacoalescents model the evolution of a coalescing system in which any number of blocks
randomly sampled from the whole may merge into a larger block. There is a variety of quantitatively
different behaviours of this process depending on the concentration of the directing measure near zero.
In particular, different limiting distributions appear for the total number of collisions in the coalescent starting with $n$ singleton blocks. In this talk we survey available results on the number of collisions and
then focus on recent findings on more delicate collision spectrum $(X_{n,k} : 2 \leq k\leq\leq n)$, where $X_{n,k}$ is the number of $k$fold collisions. This is a joint work with A. Iksanov, A. Marynych and M. Moehle. 
08/11/2017 2:00 PMW316, Queen's BuildingSaid Hamadene (Universite du Maine)A new existence result for reflected BSDEs with interconnected obstacles
In this talk we prove existence of a solution to a system of Markovian BSDEs with interconnected obstacles. A key feature of our system, and the main novelty of this paper, is that we allow for the driver $f_i$ of the $i$th component of the $Y$process to depend on all components of the $Z$process. This extends the existing theory on reflected BSDEs, which only addresses problems where $f_i$ depends on $Z^i$.
This is a joint work with De Angelis T. (University of Leeds, UK) and G.Ferrari (Univ. of Bielfeld, Germany).

15/11/2017 1:00 PMW316 Queen's BuildingIgor Krasovsky (ICL)Central spectral gaps of the almost Mathieu operator
We consider the spectrum of the almost Mathieu operator H with an
irrational frequency and in the case of the critical coupling.
For frequencies admitting a powerlaw approximation by rationals, we show that the central gaps of H are open and provide a lower bound for their widths. 
22/11/2017 1:00 PMW316, Queen's BuildingCodina Cotar (UCL)Equality of the Jellium and Uniform Electron Gas nextorder asymptotic terms for Coulomb and Riesz potentials
We consider two sharp nextorder asymptotics problems, namely the asymptotics for the minimum energy for optimal point con figurations and the asymptotics for the manymarginals Optimal Transport, in both cases with Coulomb and Riesz costs with inverse powerlaw longrange interactions. The first problem describes the ground state of a Coulomb or Riesz gas, while the second appears as a semiclassical limit of the Density Functional Theory energy modelling a quantum version of the same system. Recently the secondorder term in these expansions was precisely described, and corresponds respectively to a Jellium and to a Uniform Electron Gas model. The present work shows that for inversepowerlaw interactions with power d2\le s.

29/11/2017 1:00 PMW316, Queen's BuildingJason Miller (Cambridge)Convergence of percolation on uniform quadrangulations
Let Q be a uniformly random quadrangulation with simple boundary decorated by a critical (p=3/4) face percolation configuration. We prove that the chordal percolation exploration path on Q between two marked boundary edges converges in the scaling limit to SLE(6) on the Brownian disk. Our method of proof is robust and, up to certain technical steps, extends to any percolation model on a random planar map which can be explored via peeling. Based on joint work with E. Gwynne.

06/12/2017 1:00 PMW316, Queen's BuildingTiziano de Angelis (Leeds)PROBABILISTIC RESULTS ON REGULARITY OF OPTIMAL STOPPING BOUNDARIES
Abstract. In this talk I will provide an overview of some recent results on probabilistic proofs of continuity and Lipschitz continuity of optimal stopping boundaries in multidimensional problems. The probabilistic argument complements some similar results known from the PDE literature concerning free boundary problems, and oers an alternative point of view on the topic. In some instances the methods presented in this talk allow to relax standard assumptions made in the PDE approach, as for example uniform ellipticity of the underlying diusion. Some applications to models for irreversible investment and actuarial sciences will be illustrated. If time allows I will also connect the regularity of the boundary to questions of smoothness of the value function. This talk draws from joint work with G. Stabile (Sapienza University of Rome) and ongoing work with G. Peskir (University of Manchester).

13/12/2017 1:00 PMW316, Queen's BuildingDmitry Dolgopyat (University of Maryland)Local Limit Theorem for Nonstationary Markov chains
Dobrushin and SethuuramanVaradhan have proved sharp Central Limit Theorem for additive functionals of finite nonstationary Markov chains. We discuss the Local Limit Theorem in the same setting and give some extensions and applications. Joint work with Omri Sarig.

10/01/2018 1:00 PMW316, Queens' BuildingMartin Hairer (ICL)TBA

17/01/2018 1:00 PMW316, Queens' BuildingBalint Toth (Bristol)Quenched CLT for random walk in doubly stochastic random environment
I will present the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{1}$condition. The proof relies on nontrivial extension of Nash's moment bound to this context and on downtoearth concrete functional analytic arguments.

24/01/2018 1:00 PMW316, Queens' BuildingGaultier Lambert (Zurich)Stein's method for normal approximation of linear statistics of betaensembles
In the first part, I will review the basic ideas of Stein’s method for normal approximation and present a new application which is valid for statistics which are approximate eigenfunctions of the infinitesimal generator of a Markov process. In the second part, I will report on some applications to random matrix theory. We will prove a CLT for polynomial linear statistics of the Gaussian Unitary Ensemble and discuss the generalizations to onecut regular betaensembles and general linear statistics.
This is joint work with Michel Ledoux and Christian Webb, available at https://arxiv.org/abs/1706.10251 . 
31/01/2018 1:00 PMW316, Queens' BuildingImre Barany (UCL and Renyi Inst. Budapest)Convex cones, integral zonotopes, and their limit shape
Given a convex cone $C$ in $R^d$, an integral zonotope $T$ is the sum of segments $[0,v_i]$ ($i=1,\ldots,m$) where each $v_i \in C$ is a vector with integer coordinates. The endpoint of $T$ is $k=\sum_1^m v_i$. Let $F(C,k)$ be the family of all integral zonotopes in $C$ whose endpointis $k \in C$. We prove that, for large $k$, the zonotopes in $F(C,k)$ have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in $F(C,k)$ are very close to a
fixed convex set which is actually a zonoid. We also establish several combinatorial properties of a typical zonotope in $F(C,k)$. This is joint work with Julien Bureaux and Ben Lund.

07/02/2018 1:00 PMW316, Queens' BuildingJohannes Alt (IST Austria, Vienna)Local inhomogeneous circular law
The density of eigenvalues of large random matrices typically converges to a deterministic limit as the dimension of the matrix tends to infinity. In the Hermitian case, the best known examples are the Wigner semicircle law for Wigner ensembles and the MarchenkoPastur law for sample covariance matrices. In the nonHermitian case, the most prominent result is Girko’s circular law: The eigenvalue distribution of a matrix X with centered, independent entries converges to a limiting density supported on a disk. Although inhomogeneous in general, the density is uniform for identical variances. In this special case, the local circular law by Bourgade et al. shows this convergence even locally on scales slightly above the typical eigenvalue spacing. In the general case, the density is obtained via solving a system of deterministic equations. In my talk, I explain how a detailed stability analysis of these equations yields the local inhomogeneous circular law in the bulk spectrum for a general variance profile of the entries of X. This result was obtained in joint work with László Erdos and Torben Krüger.

21/02/2018 1:00 PMW316, Queens' BuildingPedro Miguel Duarte (Lisbon)Regularity of Lyapunov exponents of random linear cocycles (joint seminar with dynamical systems)
In [1] Émile Le Page established the Holder continuity of the top Lyapynov exponent for irreducible random linear cocycles with a gap between its first and second Lyapunov exponents. An example of B. Halperin (see Appendix 3 in [2]) suggests that in general, uniformly hyperbolic cocycles apart, this is the best regularity that one can hope for. We will survey on recent results and limitations on the regularity of the Lyapunov exponents for random GL(2)cocycles.
[1] Émile Le Page, Régularité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes et applications. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 2, 109–142.
[2] Barry Simon and Michael Taylor, Harmonic analysis on SL(2,R) and smoothness of the density of states in the onedimensional Anderson model. Comm. Math. Phys. 101 (1985), no. 1, 1–19. 
28/02/2018 1:00 PMW316, Queens' BuildingAlexander Magazinov (TAU)On percolation in the hard sphere model.
In this talk I will focus on the hard sphere model in R^d, in which a random set of nonintersecting unit balls is sampled with an intensity parameter λ.
Consider the graph in which the vertex set is the set of balls, and two balls are adjacent if they are at distance ≤ε from each other. We will discuss the connectivity of this graph for large λ in dimensions d = 2 and 3. I will sketch the proof that the graph is highly connected when λ is greater than a certain threshold depending on ε. Namely, a cube annulus with inner radius L_1 and outer radius L_2 is crossed by this graph with probability at least 1 − C exp(−c L_1^{d  1}). This answers (a variant of) a question by Bowen, Lyons, Radin and Winkler (2006) and strengthens a result by Aristoff (2014). 
07/03/2018 1:00 PMW316, Queens' BuildingBen Green (Oxford)The probability of fixing a set of size k
Fix k, and let n be large. What is the probability that a random permutation on {1,...,n} has a fixed set of size k? As n tends to infinity, this tends to a limit which we call p(k). For example p(1) = 1  1/e, since a permutation fixes some set of size 1 if and only if it is not a derangement. I will discuss joint work with Eberhard and Ford in which we estimate p(k).

07/03/2018 4:00 PMW316, Queens' BuildingProf. Dr. Claudia KlüppelbergMultivariate boundary crossings in a bipartite network
We study boundary crossings for single and multivariate components of a compound Poisson process. The dependence structure between the components is induced by a random bipartite graph. The focus of our analysis lies in the study of the influence of the random graph on boundary crossings, where we consider the Bernoulli graph and a Raschtype graph as examples. We investigate the influence of the random graph on subsets of components. In particular, we contrast the influence of the network on single components and on multivariate vectors. As applications, risk balancing networks in ruin theory and load balancing networks in queueing theory are presented.

14/03/2018 1:00 PMW316Julian Gerstenberg (Hannover)Exchangeable interval hypergraphs and limits of ordered discrete structures
A hypergraph (V, E) is called an interval hypergraph if there exists a linear order l on V such that every edge e ∈ E is an interval w.r.t. l; we also assume that {j} ∈ E for every j ∈ V . Our main result is a de Finettitype representation of random exchangeable interval hypergraphs on N (EIHs): the law of every EIH can be obtained by sampling from some random compact subset K of the triangle {(x, y) : 0 ≤ x ≤ y ≤ 1} at iid uniform positions U1, U2, . . . , in the sense that, restricted to the node set [n] := {1, . . . , n} every nonsingleton edge is of the form e = {i ∈ [n] : x < Ui < y} for some (x, y) ∈ K. We obtain this result via the study of a related class of stochastic objects: erasedinterval processes (EIPs). These are certain transient Markov chains (In, ηn)n∈N such that In is an interval hypergraph on V = [n] w.r.t. the usual linear order (called interval system). We present an almost sure representation result for EIPs. Attached to each transient Markov chain is the notion of Martin boundary. The points in the boundary attached to EIPs can be seen as limits of growing interval systems. We obtain a onetoone correspondence between these limits and compact subsets K of the triangle with (x, x) ∈ K for all x ∈ [0, 1].
Interval hypergraphs are a generalizations of hierarchies and as a consequence we obtain a representation result for exchangeable hierarchies, which is close to a result of Forman, Haulk and Pitman. Several ordered discrete structures can be seen as interval systems with additional properties, i.e. Schröder trees (rooted, ordered, no node has outdegree one) or even more special: binary trees. We describe limits of Schröder trees as certain treelike compact sets. These can be seen as an ordered counterpart to real trees, which are widely used to describe limits of discrete unordered trees. Considering binary trees we thus obtain a homeomorphic description of the Martin boundary of Remy’s tree growth chain, which has been analyzed by Evans, Gröbel and Wakolbinger.

21/03/2018 1:00 PMW316, Queens' BuildingYan Fyodorov (KCL)On statistics of biorthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry.
I will present a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'nonorthogonality overlap factor' (also known as the condition number) of the left and right eigenvectors for nonselfadjoint Gaussian random matrices. First I derive the exact finiteN expression in the case of real eigenvalues and the associated nonorthogonality factors in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distributions are maximally heavytailed, so that all integer moments beyond normalization are divergent. Then I present results for a complex eigenvalue and the associated nonorthogonality factor in the complex Ginibre ensemble complementing recent studies by P. Bourgade & G. Dubach. The presentation will be mainly based on the paper arXiv:1710.04699 and a joint work with Jacek Grela and Eugene Strahov arXiv:1711.07061.

28/03/2018 1:00 PMRoom W316, Queens' BuildingBenjamin Schlein (Zurich)Bogoliubov excitation spectra for BoseEinstein condensatesWe consider systems of N interacting bosons trapped in a box with volume one and interacting through a potential with effective range of the order 1/N (GrossPiteavskii regime). We show that lowenergy states exhibits complete BoseEinstein condensation, with optimal rate. Furthermore, we determine the lowenergy spectrum up to errors that vanishes in the limit of large N. As a result, we rigorously confirm the validity of Bogoliubov’s 1947 predictions. This talk is based on joint work with C. Boccato, C. Brennecke and S. Cenatiempo.

03/10/2018 1:00 PMQueens' Building, Room: W316Konstantin MatveevGibbs measures on the Young graph with Macdonald multiplicities and Kerov's conjecture.I will talk about the recent proof of the Kerov's conjecture (1992) classifying the homomorphisms from the algebra of symmetric functions to reals with nonnegative values on Macdonald functions. This allows to describe the set of extreme Gibbs measures on the Young graph with Macdonald multiplicities. For the special case of Schur functions this is equivalent to classifying totally nonnegative infinite Toeplitz matrices, and the result was first proved by Schoenberg, Edrei, and others in the beginning of the 1950s. Their motivation came from Analysis, but in the 1960s Thoma has discovered a connection with the representation theory of the infinite symmetric group. Some other special cases of the Kerov's conjecture are also connected to asymptotic representation theory. Our proof is a combination of two methods.1). Developing in the Macdonald generality the "pole elimination" argument developed for the Schur case by Schoenberg.2). A new method based on showing certain diffusivity in the branching graph of the Macdonald functions.I will explain the history of the problem and all the relevant notions.

10/10/2018 1:00 PMQueens' Building, Room W316Gordon Slade (UBC)Longrange O(n) models below the upper critical dimensionThe critical behaviour of spin systems and selfavoiding walk is well understood in dimensions above the upper critical dimension d=4, where meanfield behaviour applies.
In the physics literature, critical behaviour for dimensions d=4epsilon is studied by expansion methods. These methods can be made rigorous for longrange O(n) models, where positive n refers to the ncomponent phi^4 model and n=0 refers to the weakly selfavoiding walk. We will discuss recent work in this direction, using a rigorous renormalisation group method. 
17/10/2018 1:00 PMW316, Queens' BuildingAntti Kupiainen, HelsinkiRenormalization Group and Stochastic PDE’s

24/10/2018 1:00 PMQueens' Building, Room: W316Stephen Muirhead (QMUL)The phase transition for level sets of smooth planar Gaussian fields
In recent years the strong links between the geometry of smooth planar Gaussian fields and percolation have become increasingly apparent, and it is now believed that the connectivity of the level sets of a wide class of smooth, stationary planar Gaussian fields exhibits a sharp phase transition that is analogous to the phase transition in, for instance, Bernoulli percolation. In recent work we prove this conjecture under the assumptions that the field is (i) symmetric, (ii) positively correlated, and (iii) the covariance kernel decays sufficiently rapidly at infinity (roughly speaking, the integrability of the kernel is enough). Key to our proofs are (i) the whitenoise representation of Gaussian fields and (ii) the randomised algorithm approach to noise sensitivity. Joint work with Hugo Vanneuville

31/10/2018 1:00 PMQueens' Building, Room: W316Steve Lester (QMUL)Mass equidistribution for halfintegral weight automorphic forms
Given a smooth compact Riemannian manifold an important problem in Quantum Chaos studies the distribution of L^{2} mass of eigenfunctions of the LaplaceBeltrami operator in the limit as the eigenvalue tends to infinity. For manifolds with negative curvature Rudnick and Sarnak have conjectured that the L^{2} mass of the eigenfunctions equidistributes with respect to the Riemannian volume form. This is known as the Quantum Unique Ergodicity (QUE) Conjecture. In certain arithmetic settings QUE is now known. In this talk I will discuss the analogue of QUE in the context of halfintegral weight automorphic forms. This is based on joint work Maksym Radziwill.
I will not assume any background knowledge of automorphic forms.

14/11/2018 1:00 PMQueens' Building, Room W316Laszlo Erdős (IST)Тhe matrix Dyson equation in random matrix theory
The spectral statistics of large random matrices exhibit a new type of universality as postulated by Eugene Wigner in the 1950’s. This celebrated WignerDysonMehta conjecture has recently been proved for hermitian matrices with independent, identically distributed entries. Wigner’s original vision, however, extends well beyond this class of matrix
ensembles and it predicts universal behavior for any random operator with “sufficient complexity”. One of main mathematical tools is the matrix Dyson equation (MDE), a deterministic quadratic equation for large matrices that computes the density of states. We will discuss new classes of matrix ensembles that have become accessible by a systematic analysis of the MDE. 
21/11/2018 1:00 PMQueens' Building, Room W316Reda ChhaibiTBA

28/11/2018 1:00 PMQueens' Building, Room: W316Jon Keating, BristolJoint Moments of Characteristic Polynomials of Random Unitary Matrices
I shall describe recent results (obtained with E. Basor, R. Buckingham, A. Its, E. Its and T. Grava) relating the joint moments of the characteristic polynomial of a CUE random matrix and its derivative to a solution of the Painlevé V equation. This connection can be used to derive explicit formulae and to show that in the largematrix limit the joint moments are related to a solution of Painlevé III equation.

05/12/2018 1:00 PMQueens' Building, Room: W316Geoffrey Grimmett, CambridgeSelfavoiding walks on graphs and groupsThe problem of selfavoiding walks (SAWs) arose in statistical mechanics in the 1940s, and has connections to probability, combinatorics,
and the geometry of groups. The basic question is to count SAWs. The socalled 'connective constant' is the exponential growth rate of the number of nstep SAWs. We summarise joint work with Zhongyang Li concerned how the connective constant depends on the choice of graph. This work includes equalities and inequalities for connective constants, and a partial answer to the socalled 'locality problem' for graphs and particularly Cayley graphs. A number of open problems remain. 
23/01/2019 1:00 PMQueens' Building, Room: W316Yuri YakubovichOrderings of Gibbs random samples

03/04/2024 1:00 PMMB 503Purba Das (KCL)TBA
TBA

28/02/2024 1:00 PMMB503Christoph Czichowsky (LSE)Portfolio optimisation in rough volatility models
Rough volatility models have become quite popular recently, as they capture both the fractional scaling of the time series of the historic volatility (Gatheral et al. 2018) and the behaviour of the implied volatility surface (Fukasawa 2011, Bayer et al. 2016) remarkably well. In contrast to classical stochastic volatility models, the volatility process is neither a Markov process nor a semimartingale. Therefore, these models fall outside the scope of standard stochastic analysis and provide new mathematical challenges. In this talk, we present an overview of of this new paradigm in volatility modelling and consider the impact of rough volatility on portfolio optimisation.
The talk is based on joint work with Johannes MuhleKarbe and Denis Schelling. 
13/03/2024 1:00 PMMB503Amanda TurnerTBA
TBA

27/03/2024 1:00 PMMB 503LouisPierre Arguin (Oxford)TBA
TBA

21/02/2024 1:00 PMMB503Tommaso Rosati (Warwick)The AllenCahn equation with weakly critical initial datum.
Inspired by questions concerning the evolution of phase fields, we study the AllenCahn equation in dimension 2 with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.

14/02/2024 1:00 PMMB503Samuel Johnston (KCL)Free probability via entropic optimal transportThe basic operations of free probability  additive free convolution, multiplicative free convolution, and free compression  describe respectively how the spectra of large random matrices interact under the basic matrix operations: addition, multiplication, and taking minors.In this talk we discuss how these operations can be formulated in terms of an “entropic optimal transport” problem – an optimal transport problem but with an entropy penalty for the coupling measure.Our proof of this formulation uses the quadrature formulas of Marcus, Spielman and Srivastava, which relate the expected characteristic polynomial of matrices under random unitary vs symmetric conjugation. Our approach involves an asymptotic analysis of the quadrature formulas using a large deviation principle on the symmetric group.This is joint work with Octavio Arizmendi (CIMAT).

20/03/2024 1:00 PMMB503Igor Wigman (KCL)Around Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
This talk is based on a joint work with Steve Lester.
We review the Gauss circle problem, and Hardy's conjecture regarding the order of magnitude of the remainder term. It is attempted to rigorously formulate the folklore heuristics behind Hardy's conjecture. Some weaker forms of the likely statement are proved to support it.

24/01/2024 1:00 PMMB503Leonid Petrov (University of Virginia)Coloured Interacting Particle Systems on the Ring
Recently, there has been much progress in understanding stationary measures for coloured (also called multispecies or multitype) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unifying approach to constructing stationary measures for most known such systems (including the classical multispecies Asymmetric Simple Exclusion Process) based on integrable stochastic vertex models and the YangBaxter equation. Joint work with Amol Aggarwal and Matthew Nicoletti.

06/12/2023 1:00 PMMB503Anna Maltsev (QMUL)Spectral properties of the heavytailed elliptic volatility model
We study an ensemble of random matrices called Elliptic Volatility Model. This consists of a product of independent matrices X = ΣZ where Z is a T by S matrix of i.i.d. lighttailed variables with mean 0 and variance 1 and Σ is a diagonal matrix of i.i.d. heavy tailed variables. The study of such ensembles first arose in financial mathematics as models of stock price logreturns. We obtain an explicit formula for the empirical spectral distribution of its covariance matrix when Σ_ii is distributed as the Studentt with parameter 3 and the distribution of its largest eigenvalue in a more general case. This is joint work with Svetlana Malysheva.

29/11/2023 1:00 PMMB503Frédéric Klopp (Sorbonne Université)On the spatial extent of localized eigenfunctions for random Schrödinger operators
On \mathbb{Z}^d, consider \varphi, an \ell^2normalized function that decays exponentially at infinity at a rate at least mu. One can define the onset length (of the exponential decay) of \varphi as the radius of the smallest ball, say, B, such that one has the following global bound \D phi(x) <= \varphi_\infty e^(mu dist(x,B)).
The present talk will describe the onset lengths of the localized eigenfunctions of random Schrödinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than l and localization center in a ball of radius L is smaller than C L^d exp(c l), for l>0 large (for some constants C,c>0). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. The talk is mainly based on joint work with Jeff Schenker.

22/11/2023 1:00 PMMB503Alexander Gnedin (QMUL)Infinite SizeBiased Orders: Construction, Algorithms and ClassificationSizebiased permutation of a finite set is a random arrangement, with distribution depending onthe sizes (or weights) of elements. The concept has numerous roots in random algorithms, social choice, theory of records and species sampling models. Extension for infinite (countable) ground set is straightforward if the weights are summable, in which case the permutation is a well defined selfbijection of N. If the weights are not summable, the counterpart of a finite sizebiased permutation is a random infinite order, whose settheoretic type (e.g. Z, Q) is determined by the sequence of weights. For the general case we collect basic properties and constructions of the infinite sizebiased order, some of which belong to the folklore, and give a complete classification of the order types.

15/11/2023 1:00 PMMB503Yuliya Mishura (Taras Shevchenko National University of Kyiv)General Conditions of Weak Convergence of DiscreteTime Multiplicative Scheme to Asset Price with Memory
We present general conditions for the weak convergence of a discretetime additive scheme
to a stochastic process with memory in the space D[0, T]. We investigate the convergence
of the related multiplicative scheme to a process that can be interpreted as an asset price with
memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. 
01/11/2023 12:00 PMMB503Oleg Zaboronski (Warwick)Asymptotic expansions for Fredholm Pfaffians and interacting particle systems
Motivated by the phenomenon of duality for interacting particle systems
we introduce two classes of Pfaffian kernels describing a number of Pfaffian
point processes in the ‘bulk’ and at the ‘edge’. Using the probabilistic
method due to Mark Kac, we prove two Szegotype asymptotic expansion
theorems for the corresponding Fredholm Pfaffians. The idea of the proof is
to introduce an effective random walk with transition density determined by
the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through
expectations with respect to the random walk, and analyse the expectations
using general results on random walks. We demonstrate the utility of the theorems
by calculating asymptotics for the empty interval and noncrossing
probabilities for a number of examples of Pfaffian point processes: coalescing/
annihilating Brownian motions, massive coalescing Brownian motions,
real zeros of Gaussian power series and Kac polynomials, and real eigenvalues
for the real Ginibre ensemble. This is a joint work with Roger Tribe and Will FitzGerald.

25/10/2023 12:00 PMMB503Sunil Chhita (Durham)The TwoPeriodic Aztec diamond
Domino tilings of the twoperiodic Aztec diamond exhibit interesting statistical mechanical behaviors – a limit shape emerges separating three macroscopic interfaces, known as frozen, rough and smooth that depend on the local statistics. We survey some of the main results on this model, as well as recent progress on understanding the roughsmooth boundary. This is based on joint work with Duncan Dauvergne and Thomas Finn.

18/10/2023 12:00 PMMB503Arvind Ayyer (IISC)The multispecies PushTASEP
The multispecies PushTASEP (where TASEP stands for totally
asymmetric simple exclusion process) is an interacting particle system
with multiple species of particles on a finite ring where the hopping
rates are sitedependent. (The homogeneous variant on Z is also known as
the Hammersley–Aldous–Diaconis process.) In its simplest variant with a
single species, a particle at a given site will hop, when that bell
rings, to the first available vacant site clockwise. We compute the
stationary distribution of this inhomogeneous process using a multiline
process. In particular, we show that the partition function of this
process is intimately related to the multispecies TASEP and to the
classical Macdonald polynomials. We also prove that large families of
events are symmetric under the interchange of these sitedependent
rates. This is joint work with James Martin. 
11/10/2023 1:00 PMMB503Ilaria Peri (Birkbeck)Estimating lambda quantiles. An application to risk management
Lambda quantiles have been introduced by Frittelli, Maggis and Peri (2014) in the context of risk measure theory with the name of Lambda Value at Risk. They are a generalization of quantiles that, instead of relying on a fixed confidence level, consider a function known as lambda. After a brief overview of the theory and the emerging literature on lambda quantiles, we will focus on their estimation. First, we will review their properties of robustness of the empirical estimator and elicitability under specific conditions on the distribution functions. Then, we will present a range of methods for estimating the lambda function and the lambda quantiles, including simulations, parametric models, and supervised learning approaches. Finally, we will provide an empirical application in risk measurement, comparing a parametric approach with a supervised learning method.

27/09/2023 1:00 PMMB503Alexander Marynych (Taras Shevchenko National University of Kyiv)Generalised convexity with respect to families of affine maps
The standard convex closed hull of a subset of $\mathbb{R}^d$ is defined as the intersection of all images, under the action of a group of rigid motions, of a halfspace containing the given set. We propose a generalisation of this classical notion, that we call a $(K,\mathbb{H})$hull, and which is obtained from the above construction by replacing a halfspace with some other convex closed subset $K$ of the Euclidean space, and a group of rigid motions by a subset $\mathbb{H}$ of the group of invertible affine transformations. The above construction encompasses and generalises several known models in convex stochastic geometry and allows us to gather them under a single umbrella. The talk is based on recent works by Kalbuchko, Marynych, Temesvari, Thäle (2019), Marynych, Molchanov (2022) and Kabluchko, Marynych, Molchanov (2023+).

14/09/2023 1:00 PMMB503Chun Yin Siu (Cornell University)The Topology of Preferential Attachment Graphs
Abstract:
The preferential attachment model is a natural and popular random graph model for a growing network that contains very wellconnected ``hubs''. We study the higherorder connectivity of such a network by investigating the topological properties of its clique complex. By determining the asymptotic growth rates of the expected Betti numbers, we discover that the graph undergoes higherorder phase transitions within the infinitevariance regime.

12/04/2023 2:00 PMMB503Igor Wigman (KCL)Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genusThis talk is based on a joint work with Zeev Rudnick.We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the WeilPetersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. We show that in this same limit, the energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature.

29/03/2023 2:00 PMMB503Leonid Pastur (KCL and ILTP (Ukraine)On random matrices arising in deep neural networksTBC

05/04/2023 2:00 PMMB503Thomas Bothner (Bristol)Bulk spacings in nonHermitian matrix models
Random matrix eigenvalue spacings tend to show up in problems not directly related to random matrices: for instance, bumper to bumper distances of parked cars in a number of roads in central London are well represented by the socalled eigenvalue bulk spacing distribution of a suitable Hermitian matrix model. In this talk we will first survey several occurrences of these Hermitian spacing distributions and afterwards try to generalise them to nonHermitian models. As it turns out, the theory of integrable systems, especially Painlev\'e special function theory, plays a crucial role in this field. Based on arXiv:2212.00525, joint work with Alex Little (Bristol)

15/03/2023 2:00 PMMB503Thomas Mikosch (University of Copenhagen)Extreme value theory for heavytailed time series
Abstract: We will consider regularly varying time series. The name comes from the marginal tails which are of powerlaw type. Davis and Hsing (1995) and Basrak and Segers (2009) started the analysis of such sequences. They found an accompanying sequence (spectral tail process) which contains the information about the influence of extreme values on the future behavior of the time series, in particular on extremal clusters. Using the spectral tail process, it is possible to derive limit theory for maxima, sums, point processes... of regularly varying sequences, but also refined results like precise large deviation probabilities for these structures.
In this talk we will give a short introduction to regularly varying sequences and and explain how the aforementioned limit results can be derived. 
01/03/2023 2:00 PMMB503Svetlana Malysheva (QMUL)Heavytailed random matrices
Abstract: We will discuss the difference the matrix entries' number of finite moments makes to its eigenvalue distribution.

08/02/2023 2:00 PMMB503Natasha Blitvic (QMUL)(More on) Combinatorial Moment SequencesAbstract: We will revisit a topic discussed in the Internal Colloquium (November 2022), namely that of enumerative sequences which play a privileged role in probability. Starting from the same premise (which we will briefly summarize), we will branch off in a different direction, looking at another motivation for these types of positivity questions. We will discuss different ways in which combinatorial properties translate into probabilistic ones. Finally, we will look beyond finite n at how some of the constructions of interest to us behave asymptotically and at how universal are some of these behaviors. Based on joint works with Einar Steingrímsson and Slim Kammoun.

15/02/2023 2:00 PMMB503Martin Barlow (Vancouver)Stability of the elliptic Harnack InequalityAbstract. A manifold has the Liouville property if every bounded harmonic function is constant. A theorem of T. Lyons is that the Liouville property is not preserved under mild perturbations of the space. Stronger conditions on a space, which imply the Liouville property, are the parabolic and elliptic Harnack inequalities (PHI and EHI). In the early 1990s Grigor'yan and SaloffCoste gave a characterisation of the parabolic Harnack inequality (PHI), which immediately gives its stability under mild perturbations. In this talk we prove the stability of the EHI. The proof uses the concept of a quasi symmetric transformation of a metric space, and the introduction of these ideas to Markov processes suggests a number of new problems. (Joint work with Mathav Murugan)

01/02/2023 2:00 PMMB503Alexander Gnedin (QMUL)A Random Graph Growth Model
Abstract: A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edgecounting at times when the graph has $n\leq N$ occupied vertices. Two different Poisson limits are identified for $n\asymp N^{1/3}$ and $Nn\asymp 1$. For the bulk of the process, when $n\asymp N$, the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of $N^{3/2}$ and approximable by a Gaussian bridge. (Joint work with Michael Farber and Wajid Wajid Mannan arXiv:2301.07809)

22/02/2023 2:00 PMMB503Mira Shamis (QMUL)Upper bounds on quantum dynamics
We shall discuss the quantum dynamics associated with ergodic Schroedinger operators with singular continuous spectrum. Upper bounds on the transport moments have been obtained for several classes of onedimensional operators, particularly, by DamanikTcheremchantsev, JitomirskayaLiu, JitomirskayaPowell. We shall present a new method which allows to recover most of the previous results and also to obtain new results in one and higher dimensions. The input required to apply the method is a largedeviation estimate on the Green function at a single energy. Based on joint work with S. Sodin.

22/03/2023 2:00 PMMB503Antal Jarai (Bath)The Abelian sandpile in two dimensions
Abstract: The Abelian sandpile model describes the discretetime evolution of a system of particles on a finite graph. At each step, a new particle is added to the system, following which the particles are redistributed according to certain local rules, and some particles may leave the system. Interest in the model comes from the fact that the stationary distribution of the dynamics is characterised by power laws (often referred to as selforganised criticality).The model on 2D lattices is special in that certain observables have been shown to haveconformally covariant scaling limits (Durre (2008) and Kassel & Wu (2015)). I will discuss recent progress on some aspects of the scaling limit, as well as conjectures.(includes joint work with Miles Elvidge)

05/10/2022 2:00 PMMB503Alexander Gnedin (QMUL)Random Permutations and Queues
Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discretetime Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infiniteserver queues. We use this connection to show how results from the queuing theory can be interpreted in terms of the evolution of the cycle counts of permutation. (joint work with Dudley Stark (QMUL))

07/12/2022 2:00 PMMB503Efe Onaran (Technion)Functional CLTs for Local Statistics of Spatial BirthDeath Processes in the Thermodynamic Regime
Abstract: We will present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in the Euclidian space. The dynamics we study are those of a Markov birthdeath process. We prove functional limit theorems in the socalled thermodynamic regime using the recent theory of MalliavinStein bounds. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs. (Joint work with Omer Bobrowski and Robert J. Adler)

19/10/2022 2:00 PMMB503Alexander Veretennikov (Leeds  UoL, visitor)Recurrence for SDEs with switching and applications
Second & possibly higher order recurrence of a $d$dimensional diffusion with an additive Wiener process with switching and with one recurrent and one transient regime under suitable conditions will be discussed. The approach is based on embedded Markov chains and on a priori bounds for the moments of $X_t$ at times of jumps of the discrete component. An easier part of the problem is positive recurrence; higher order ones do require  a bit unexpectedly  more assumptions and some new hints.
Positive recurrence is discussed in a recent paper https://rdcu.be/cRkgO; https://doi.org/10.1007/s40072022002657 and is closely related to the existence of an invariant measure of the process. Time permitting, some links to the asymptotic behaviour of solutions of PDE systems will be shown be discussed. 
16/11/2022 2:00 PMMB503Mohammed Osman (QMUL)Local universality for complex nonHermitian matricesWe will discuss the universality of the correlation functions for complex nonHermitian matrices. In the Hermitian case, an important step for the proof of universality for general matrices is the proof for the special case of matrices with a small Gaussian perturbation. We show how this step can be adapted to the complex nonHermitian case via the asymptotic analysis of an explicit formula for the correlation functions of the perturbed matrices.

02/11/2022 2:00 PMMB503Jan Palczewski (Leeds)Nonzero sum game of exit from a stochastic marketThe timing of strategic exit is one of the most important but difficult business decisions, especially under competition and uncertainty. Motivated by this problem, we examine a stochastic game of exit in which players are uncertain about their competitor's exit payoff. It is a nonzero sum stopping game with asymmetric information as each player does not know their competitor's exit payoff. The market uncertainty, observed by both players, is represented by a general onedimensional diffusion. Under the condition that a single player exit problem has a solution of a threshold type, we construct a symmetric equilibrium in pure strategies. This equilibrium is further shown to be unique in a wide subclass of symmetric perfect Bayesian equilibria. Our arguments are mainly probabilistic with occasional use of PDE methods.

14/12/2022 2:00 PMMB503Linus Wunderlich (QMUL)Neural networks for highdimensional parametric option pricing
Abstract:
In this talk we will discuss the deep parametric PDE method for parametric option pricing in high dimensions, underlying theoretical results for neural networks and an application in risk management.
The deep parametric PDE method uses deep neural networks to solve parametric partial differential equations, such as those arising in option pricing. Especially for a large number of risk factors, the efficiency of neural networks for high dimensional problems is beneficial. We investigate this efficiency theoretically by presenting approximation rates for networks with smooth activation functions. (joint work with Kathrin Glau)

30/11/2022 2:00 PMMB503Nick Simm (Sussex)Eigenvalue statistics of real asymmetric random matrices
Abstract: Consider a real N x N random matrix, all of whose entries are i.i.d. standard normals with no symmetry assumptions imposed. Although the definition is simple and appealing, the study of eigenvalue statistics for such asymmetric matrices is quite involved. The eigenvalues are mainly complex, but a certain fraction lie precisely on the real line with positive probability. Their statistics have been studied for both finite N and in scaling limits when N tends to infinity, and seem to form a distinct universality class. I will review some of its important properties. Then I will present recent work related to this on products of random matrices. The latter is joint work with Will FitzGerald (Manchester).

11/03/2020 1:00 PMMathematical Sciences Building, Room: MB503Matthias Täufer (QMUL)A robust initial scale estimate and localization at band edges of the continuum Anderson model
We prove that Anderson localization near band edges of ergodic continuum
random Schroedinger operators with periodic background potential in in
dimension two and larger is universal.
In particular, Anderson localization holds without extra decay
assumptions on the random variables and independently of regularity or
degeneracy of the Floquet eigenvalues of the background operator.
Our approach is based on a robust initial scale estimate the proof of
which avoids Floquet theory altogether and uses instead an interplay
between quantitative unique continuation and large deviation estimates.
Furthermore, our reasoning is sufficiently flexible to prove this
initial scale estimate in a nonergodic setting, which promises to be an
ingredient for understanding band edge localization also in these
situations.
Based on joint work with Albrecht Seelmann (TU Dortmund). 
01/04/2020 1:00 PMMathematical Sciences Building, Room: MB503David Criens (Munich)A parabolic Harnack inequality for RWRE in balanced environments
Random walk in random environment (RWRE) is a model for random movement of a particle in a disordered medium, which is intrinsically related to random difference equations. For these I discuss a parabolic Harnack inequality in a not necessarily elliptic balanced i.i.d. setting. The talk is based on joint work with Noam Berger (TUM).

04/03/2020 1:00 PMMathematical Sciences Building, Room: MB503Simone Warzel (Munich)Spectral Gaps, Incompressibility and Fragmented MatrixProduct States in a $\nu = 1/3$ Fractional Quantum Hall SystemIn the thin cylinder regime Haldane’s pseudopotential corresponding to onethird filling results in a frustrationfree fermionic lattice Hamiltonian which is dipoleconserving with an added electrostatic interaction. Its zeroenergy eigenspace is exponentially large. Nevertheless, it admits a a rather simple, full description in terms of a certain class of fragmented matrixproduct states, which I will introduce and discuss in this talk.As I will sketch, the complete classification of zeroenergy states can be taken as a basis for a proof of a uniform spectral gap in the excitation spectrum of these Hamiltonians. The latter is vital for the theoretical explanation of the incompressibility of the FQH system at maximal filling. (Based on a joint work with B. Nachtergaele and A. Young)

19/02/2020 1:00 PMMathematical Sciences Building, Room: MB503Raphael LachiezeRey (Paris)Percolation of shot noise excursionsStationary shot noise fields are a special class of infinitely divisible random fields, they can be seen as a spatial moving average based on a homogeneous Poisson measure. We consider the excursion sets of a planar symmetric shot noise field. In particular, we consider wether the excursion above some level u percolates, under assumptions involving the smoothness of the field and the decay of its correlation function at infinity. We find results which are similar to the Gaussian case, under analogous hypotheses: it percolates only at u < 0, and there is a sharp phase transition at 0.

12/02/2020 1:00 PMMathematical Sciences Building, Room: MB503Alon Nishry (Tel Aviv)Zeros of Gaussian Taylor series  Fluctuations and rigidityThe zero set of a random analytic function can be considered as a point process in the complex plane. When the function is represented by a Taylor Series with independent Gaussian coefficients, many statistical properties of the zero set are known.I will describe the asymptotics of the variance for both the number of zeros as well as smooth statistics of the zeros. The latter have a surprising connection to rigidity of the zero set, first introduced by Ghosh and Peres. In particular, it is possible to construct a random function, such that if the locations of its zeros in the complement of a compact set are given, then its zeros inside that set can be determined uniquely.Based on a joint work with A. Kiro (WIS).

05/02/2020 1:00 PMMathematical Sciences Building, Room: MB503Apostolos Giannopoulos (Athens)Asymptotic shape of random polytopes
We review several results on the geometry and the asymptotic shape of
random polytopes generated by N independent vectors distributed according
to a logconcave probability measure on the ndimensional Euclidean space.
We also discuss the case of the uniform measure on the discrete cube and
applications to combinatorial questions about 0/1 polytopes. 
26/02/2020 1:00 PMMathematical Sciences Building, Room: MB503Kurt Johansson (KTH)Multivariate normal approximation for traces of random unitary matrices
Consider an n x n random unitary matrix U taken with respect to normalized Haar measure. It is a well known consequence of the strong Szego limit theorem that the traces of powers of U converge to independent (complex) normal random variables as n grows. I will discuss a recent result together with
Gaultier Lambert where we obtain a superexponential rate of convergence in total variation between the traces of the first m powers of an n × n random unitary matrices and a 2mdimensional Gaussian random variable. This generalizes previous results in the scalar case, which answered a conjecture by
Diaconis, to the multivariate setting. We are especially interested in the regime where m grows with n. The problem on how the rate of convergence changes as m grows with n was raised recently by Sarnak. The result we obtain gives the precise dependence on the dimensions m and n in the estimate with explicit constants for m almost up to the square root of n. 
22/01/2020 1:00 PMMathematical Sciences Building, Room: MB503Reimer Kuhn (KCL)Heterogeneous MicroStructure of Percolation in Complex Networks
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using a messagepassing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and similarly in the expected size of small clusters containing a given node. Results can be obtained for single large instances of finite networks and in the limit of infinite system size, byderiving selfconsistency equations for the limiting distributions that emerge from the single instance formulation as the infinite system size limit is taken. Distributions of node dependent probabilities to belong to the giant cluster in each instance of a number of repeated random edge removal experiments are also briefly discussed.

29/01/2020 1:00 PMMathematical Sciences Building, Room: MB503Kilian Raschel (Tours)The stationary distribution of the reflected Brownian motion in a cone: differential properties of the Laplace transform
We consider a semimartingale reflected Brownian motion in a
twodimensional cone. The main goal of the talk is to study the
algebraic nature of the Laplace transform of its stationary
distribution. We derive necessary and sufficient conditions for the
Laplace transform to be differentially algebraic, Dfinite, algebraic
or rational. These conditions are algebraic dependencies among the
parameters of the model (drift, opening of the wedge, angles of the
reflections on the axes). As a consequence we obtain new derivations
of the Laplace transform in several well known cases, namely the
skewsymmetric case, the orthogonal reflections case and the
sumofexponential densities case. The third of these occurs exactly
when the socalled DiekerMoriarty condition holds. Joint work with M.
BousquetMélou, A. Elvey Price, S. Franceschi and C. Hardouin. 
04/12/2019 1:00 PMMathematical Sciences Building, Room: MB503Antti Knowles (Geneva)Spectral analysis of critical ErdösRényi graphs
The ErdösRényi graph G(N,p) is the simplest model of a random graph, where each edge of the complete graph on N vertices is open with probability p, independently of the others. If p = p_N is not too small then the degrees of the graph concentrate with high probability and the graph is homogeneous. On the other hand, for p of order (log N) / N and smaller, the degrees cease to concentrate and the graph is with high probability inhomogeneous, containing isolated vertices, leaves, hubs, etc. I present results on the eigenvalues and eigenvectors of the adjacency matrix of G(N,p) at and below this critical scale. I show a rigidity estimate for the locations of the eigenvalues and explain a transition from localized to delocalized eigenvectors at a specific location in the spectrum.

27/11/2019 1:00 PMMathematical Sciences Building, Room: MB503Giovanni Peccati (Luxembourg)Extensions of a theorem by de JongIn a wellknown contribution from 1990, P. de Jong proved a general invariance principle for degenerate $U$statistics, based on a drastic simplification of the method of moments and cumulants.My aim in this talk is to present two recent extensions of such a result to an infinitedimensional setting: the first concerns nonlinear functionals of Poisson point processes, and the second yields a general invariance principle for $U$processes based on symmetric $U$statistics.Among the techniques acting behind the scenes, I will evoke Stein's method, Malliavin calculus and the "Gammatype" calculus associated with (nondiffusive) Markov semigroups.Essentially based on joint works with Ch. Döbler and M. Kasprzak (Luxembourg).

20/11/2019 1:00 PMMathematical Sciences Building, Room: MB503Olga Izyumtseva (QMUL)Selfintersection local times of random fields in stochastic flows.
The present talk is devoted to the evolution of Gaussian field in the flow of interacting particles. We present completely new approach for the description of evolution based on the equation with interaction introduced by A.A. Dorogovtsev in 2003. It allows to describe the motion of field taking into account its shape. For defined random field we prove the existence of selfintersection local times and describe its asymptotics.
References:
1. A.A. Dorogovtsev, O.L. Izyumtseva, Hilbertvalued selfintersection local times for planar Brownian motion, Stochastics, 91, no.1, 2019, 1431542. A.A. Dorogovtsev, Stochastic flows with interaction and measurevalued processes, International Journal of Mathematics and Mathematical Sciences 63 (2003), 396339773. A.A. Dorogovtsev, O.L. Izyumtseva, Local times of selfintersection, Ukrainian Mathematical Journal, 68, no. 3, 2016, 325379(joint work with Andrey Dorogovtsev and Alexander Gnedin) 
13/11/2019 1:00 PMMathematical Sciences Building, Room: MB503Roland Bauerschmidt (Cambridge)LogSobolev inequality for the continuum SineGordon model
We derive a multiscale generalisation of the BakryEmery
criterion for a measure to satisfy a LogSobolev inequality. Our
criterion relies on the control of an associated PDE well known in
renormalisation theory: the Polchinski equation. It implies the usual
BakryEmery criterion, but we show that it remains effective for
measures which are far from logconcave. Indeed, using our criterion,
we prove that the massive continuum SineGordon model with $\beta<6\pi$
satisfies asymptotically optimal LogSobolev inequalities for Glauber
and Kawasaki dynamics. These dynamics can be seen as singular SPDEs
recently constructed via regularity structures, but our results are
independent of this theory. 
23/10/2019 1:00 PMMathematical Sciences Building, Room: MB503Tiziano de Angelis (Leeds)Optimal dividends with partial information and stopping of a degenerate reflecting diffusion
We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a 2dimensional degenerate diffusion, whose first component is singularly controlled and it is absorbed as it hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with `creation'. One key feature of the stopping problem is that creation occurs at a statedependent rate of the `localtime' of an auxiliary 2dimensional reflecting diffusion.

06/11/2019 1:00 PMMathematical Sciences Building, Room: MB503Sarah Pennington (Bath)Branching Brownian motion with selection and a free boundary problem
Consider a system of N particles moving according to Brownian motions and branching at rate one. Each time a particle branches, the particle in the system furthest from the origin is killed. It turns out that we can use results about a related partial differential equation known as a free boundary problem to control the long term behaviour of this particle system for large N.
This is joint work with Julien Berestycki, Eric Brunet and James Nolen.

09/10/2019 1:00 PMMathematics Building, Room: MB503Robert Seiringer (IST)The polaron at strong coupling
We review old and new results on the Froehlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.

30/10/2019 1:00 PMMB503Diana Conache (Munich)A model for dislocation lines in 3D solids at low temperatureWe propose a model for threedimensional solids on a mesoscopic scale with a statistical mechanical description of dislocation lines in thermal equilibrium. The model has a linearized rotational symmetry, which is broken by boundary conditions. We show that this symmetry is spontaneously broken in the thermodynamic limit at small positive temperatures. In particular, we will focus on the statistical mechanical properties of a random Burgers vector configuration. The talk is intended for a general audience. This is joint work with Roland Bauerschmidt, Markus Heydenreich, Franz Merkl and Silke Rolles and is based on https://link.springer.com/
article/10.1007/s00023019 008299 
02/10/2019 1:00 PMMathematical Sciences Building, Room: MB503Igor Krasovsky (Imperial College)Hausdorff dimension of the spectrum of the almost Mathieu operatorWe will discuss the wellknown quasiperiodic operator: the almost Mathieuoperator in the critical case. We give a new and elementary proof (the first proof was completed in 2006 by Avila and Krikorian by a different method) of the fact that its spectrum is a zero measure Cantor set. We furthermore prove a conjecture going back to the work of David Thouless in 1980s, that the Hausdorff dimension of the spectrum is not larger than 1/2. This is a joint work with Svetlana Jitomirskaya.

25/09/2019 1:00 PMMathematical Sciences Building, Room: MB503Folkmar Bornemann (TU München)Finite size corrections in random matrices, random permutations, and last passage percolation

16/10/2019 1:00 PMMathematical Sciences Building, Room: MB503Misha Sodin (Tel Aviv)The Wiener spectrum and Taylor series pseudorandom coefficients
The theme of my talk will be the influence of the multipliers $\xi (n) $ on the angular distribution of zeroes of the Taylor series
\[ F_\xi (z) = \sum_{n\ge 0} \xi (n) \frac{z^n}{n!}.\]
This is a classical topic initiated by Littlewood together with his pupils and collaborators Chen, Nassif, and Offord.
Our main finding is that the leading term in the asymptotic behaviour of $ \log F_\xi (z) $ (and hence, the distribution of zeroes of $F_\xi$) is governed by the Wiener spectrum of the sequence $ \xi $, that is, by the support of spectral measure of $\xi$.
It applies to random stationary sequences, to the sequences $\xi(n)=\exp(n^\beta)$ with noninteger $\beta>1$ and $\xi (n) = \exp(Q(n))$, where $Q$ is a Weyl polynomial, to Besicovitch almost periodic sequences, to multiplicative random sequences, and to the Möbius function (assuming ``the binary Chowla conjecture'').
The talk will be based on the joint works with Jacques Benatar, Alexander Borichev, and Alon Nishry (arXiv:1409.2736, 1908.09161)

25/03/2020 1:00 PMTBAMichael Farber (QMUL)TBA

04/07/2019 1:00 PMQueens Building, Room: W316Renato Soares dos Santos (NYU Shanghai)Brownian motion in Poisson potential

24/06/2019 1:00 PMScape 2.01Asad Lodhia (University of Michigan)Harmonic Means of Wishart Random Matrices

27/03/2019 1:00 PMQueens' Building, Room: W316Cécile Mailler (Bath)The monkey walk: a random walk with random reinforced relocations and fading memory.
In this joint work with Gerónimo UribeBravo, we prove and extend
results from the physics literature about a random walk with random
reinforced relocations. The "walker" evolves in $\mathbb Z^d$ or
$\mathbb R^d$ according to a Markov process, except at some random
jumptimes, where it chooses a time uniformly at random in its past,
and instatnly jumps to the position it was at that random time. This
walk is by definition nonMarkovian, since the walker needs to
remember all its past.
Under moment conditions on the interjumptimes, and provided that the
underlying Markov process verifies a distributional limit theorem, we
show a distributional limit theorem for the position of the walker at
large time. The proof relies on exploiting the branching structure of
this random walk with random relocations; we are able to extend the
model further by allowing the memory of the walker to decay with time. 
20/03/2019 1:00 PMQueens’ Building, Room: W316Leonid Parnovski (UCL)Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons
I will discuss the asymptotic behaviour of the eigenvalues of the Steklov problem (aka DirichlettoNeumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.

13/03/2019 1:00 PMBancroft Road Teaching Rooms, Room: 4.02 *Please note change to roomRoman Kotecky (Warwick)Metastability for a liquid/gas model on continuumA big open problem of mathematical statistical physics is a proof of existence of a phase transition for any realistic model of interacting particles on continuum.After introducing the WidomRowlinson model—known as quermassinteraction process to probabilists—one of the few for which the phase transition is well understood,I will proceed to a discussion of a transition from a metastable supercooled gas phase to the liquid phase for a system subjected to a stochastic dynamics.In particular, the theorem about Arrhenius law with a new nonstandard entropic correction will be formulated.The relevant terms stem from large and moderate deviations of the shape of a critical droplet.(Based on a work in progress, joint with F. den Hollander, S. Jensen, and E. Pulvirenti.)

06/03/2019 1:00 PMQueens' Building, Room: W316Alexey Bufetov (Bonn)Representations of classical Lie groups: two growth regimes
Asymptotic representation theory deals with representations of groups of growing size. For classical Lie groups there are two distinguished regimes of growth. One of them is related to representations of infinitedimensional groups, and the other appears in combinatorial and probabilistic questions. In the talk I will discuss differences and similarities between these two settings.

27/02/2019 1:00 PMQueens' Building, Room: W316Jordan Stoyanov (Sofia)New Look at Checkable Conditions for MDeterminacy of Probability Distributions
There are several conditions either sufficient or necessary for uniqueness or for nonuniqueness of a probability distribution in terms of its moments (assume that all moments are finite): Cramer, Carleman, Hardy, Krein, rate of growth of moments, etc. Besides the moments, the cumulants/semiinvariants will also be involved. Any of these conditions is of interest by itself, each can be checked, hence checkable! However, it is a challenging problem to make a complete picture of all possible relationships between different conditions leading to the same property of a probability distribution. Some new recent results will be reported, hints for their proof will be given. Both discrete and continuous distributions will be treated. There will be illustrative examples and counterexamples, and also open questions and conjectures.
The talk is partly based on joint work with G.D. Lin (Taipei), Ch. Vignat (New OrleansParis), P. Kopanov (Plovdiv) and E. Yarovaya (Moscow).

06/02/2019 1:00 PMQueens' Building, Room: W316Alisa Knizel (Columbia University)Loggases on a quadratic lattice via discrete loop equations
We study a general class of loggas ensembles on a quadratic lattice. Using a variational principle we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a qboxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a onedimensional section of a pullback of the twodimensional Gaussian free field.
Our approach is based on a qanalogue of the SchwingerDyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to a quadratic lattice. Based on joint work with Evgeni Dimitrov. 
30/01/2019 1:00 PMQueens’ Building, Room: W316Markus Riedle (KCL)Cylindrical Lévy processes
Cylindrical Lévy processes are a natural extension of cylindrical Brownian motion which has been the standard model of random perturbations of partial differential equations for the last 50 years. In this talk, we introduce cylindrical Lévy processes, present some specific examples, and discuss their relations to other models of random perturbations in the literature. The talk continues with presenting a theory of stochastic integration for random integrands with respect to cylindrical Lévy processes, which requires a completely new approach. We finish the talk by discussing the challenges of studying stochastic partial differential equations driven by cylindrical Lévy processes.

13/02/2019 1:00 PMQueens' Building, Room: W316Andrey Dorogovtsev (Kiev)Comparison theorems and their applicationIn the talk different kinds of comparison theorems will be discussed. We begin with Slepian's inequality which was the key for Sudakov's estimations and Dudley's continuity condition for Gaussian processes. It has a lot of generalizations. We present one related to the families of martingales. It has surprising consequences about the behavior of onedimensional flows of Brownian particles. Among them we present the asymptotics of maximal deviation for particles starting from the interval and asymptotics of the number of the clusters in the flow.
Next part of the lecture is devoted to the limit behavior of Gaussian twodimensional integrators, one class of processes which can be used as base for stochastic calculus.Here the application of Slepian's inequality is based on the properties of compact operators in the Hilbert space.
Comparison between the solutions to SDE leads to the heat kernel estimations for the transition
density of the diffusion processes. Basing on such estimates we present asymptotics of the intersection local time for two diffusion processes on the plane.
The results have connection to the different branches of applied mathematics and mathematical
modeling such as turbulence theory and polymer structure. In the talk we shall give examples of applications in these areas.
The talk is partly based on the joint work with Olga Izyumtseva. 
16/01/2019 1:00 PMQueens' Building, Room: W316Mylène Maïda (Lille)A statistical physics approach to the sine beta processThe Sine process (corresponding to inverse temperature beta equal to 2) is a well known determinantal point process. It appears as the bulk limit of some particle systems in various contexts (random matrix ensembles, zeros of Lfuntions, growth models etc.) Its universality properties are fascinating. More recently, Valko and Virag introduced a family of point processes as the bulk limit of Gaussian beta ensembles, for any positive beta. As soon as beta is different from 2, much less is known.In a work with David Dereudre, Adrien Hardy (Université de Lille) and Thomas Leblé (Courant Institute, New York), we use tools from classical statistical mechanics based on DLR equations to understand better the Sine beta process and in particular show that it is rigid.

12/12/2018 1:00 PMQueens' Building, Room: W316Frédéric Klopp (Paris Rive Gauche)Interacting one dimensional quantum particles in a random background.
We will first give a general introduction on the topic with a special focus on the noninteracting case. We then will present a toy model in dimension one. For this model, one can prove the exponential decay of correlation both at 0 and positive temperature.

07/11/2018 1:00 PMQueens' Building, Room: W316Fabio Cunden (UCD)Some new perspectives on moments of random matrices
The study of ‘moments’ of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, mainly due to its connections to enumerative geometry. I will give some background on this and then describe some recent work which offers some new perspectives (and new results).
This talk is based on joint works with Antoine Dahlqvist, Francesco Mezzadri, Neil O'Connell and Nick Simm.