Probability and Applications
Time: Wednesday at 1pm
Location: W316, Queen's Building
Organizer: Mira Shamis

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. 
07/03/2018 1:00 PMW316, Queens' BuildingBen Green (Oxford)TBA

07/03/2018 4:00 PMQueens' Building, Room W316Prof. 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.

21/03/2018 1:00 PMW316, Queens' BuildingYan Fyodorov (KCL)TBA

28/03/2018 1:00 PMW316, Queens' BuildingBenjamin Schlein (Zurich)TBA