Edge-reinforced 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 Vertex-reinforced 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 continuous-time process favouring sites with more local time.
Then we prove that the VRJP is a mixture of time-changed 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).
The Schramm-Loewner evolution (SLE) is a one-parameter family of random growth processes that has been successfully used to analyze a number of models from two-dimensional 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 non-local observable of the 2d critical Ising model studied by Arguin and Saint-Aubin 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.
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 least-loaded 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.
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.
Neveu studied leaf-length erasure of Galton-Watson 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 Galton-Watson tree.
We observe that the offspring distributions that occur in the threeHereditary properties, Galton-Watson 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 leaf-length erasure
has recently been exploited by Evans, Winter and co-authors in a context
of real trees. We continue these developments and use results about
hereditary properties to obtain strong convergence results of
Galton-Watson real trees to Levy trees and characterisations and
properties of the limits. We also have an invariance principle for
Galton-Watson trees and decomposition results for Galton-Watson and Levy
trees. This is joint work with Thomas Duquesne.
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.)
We discuss the behaviour of a Galton-Watson 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).
We study the susceptible-infective-recovered (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.
The goal is to understand sample-to-sample fluctuations in disorder-generated 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 one-dimensional "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 disorder-generated multifractal fields. The presentation will be mainly based on the joint work with Pierre Le Doussal and Alberto Rosso, J Stat Phys: 149 (2012), 898-920 as well as on some related earlier works by the speaker.
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.
We look at a general two-sided jumping strictly alpha-stable 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 self-similar 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 Wiener-Hopf factorization of the latter. Using this Wiener-Hopf 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)
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.
Suppose we are given a multiplicative random walk (a
stick-breaking 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 stick-breaking 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 stick-breaking 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.
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).
The Lambda-coalescent is a partition-valued 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 Lambda-coalescent 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(2-alpha,alpha)-coalescent.
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 Wiener-Hopf 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 Blumenthal-Getoor index).
References
[1] Ferreiro-Castilla, A., Kyprianou, A.E., Scheichl, R. and Suryanarayana, G. (2013) Multi-
level Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorization.
Stoch. Proc. Appl. (To appear).
[2] Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2011) A Wiener-Hopf
Monte Carlo simulation technique for Levy process. Ann. App. Probab. 21(6), 2171-2190
We study the behaviour of the log-mod of the characteristic polynomial \log|\det(x-M)| 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 logarithmically-correlated 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.
Recent progresses in Number Theory are due to the application of the Keating-Snaith 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 Keating-Snaith 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 L-functions. 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.
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 infinite-volume 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 shift-covariant 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 shift-covariance 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.
We present an estimate of the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables.
In a particular case of 0-1 random variables this yields the famous result by Barbour and Eagleson (1983). A generalisation to the case of dependent observations is given as well.
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 long-range 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.
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 semi-martingale 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 half-plane which we introduced a few years ago, and allows
to identify clearly the main terms in several limit theorems.
Abstract: The emphasis will be on some recent progress in the moment analysis of distributions and their characterization as being unique (M-determinate) or nonunique (M-indeterminate) in terms of the moments. Specific topics which will be discussed are:
(a) Stieltjes classes for M-indeterminate 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.
Optimal prediction of the ultimate maximum is a non-standard optimal stopping problem in the sense that the pay-off 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 one-sided 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 one-sided case.
This talk is based on joint work with Dr. Kees van Schaik which is due to appear in Acta Applicandae Mathematicae."
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 well-known 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. Gonzales-Casanova, N. Kurt, (Berlin).
Attachment: abstract [PDF 86KB]
The zero-range 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 zero-range process exhibits condensation.
We consider in this talk the non-homogeneous 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 non-homogeneous 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).
We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes 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 closed-form solutions to the equivalent free-boundary problems for the value functions with smooth-fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process.
In this talk, we will be interested in the asymptotics of random
trees built by linear preferential attachment (also known as
Barabasi-Albert trees or plane-oriented recursive tree). We will first try
to understand the influence of the initial tree (the seed) on the
long-term behavior of this process. Then we will see that this problem is
closely related to the existence of scaling limits of so-called 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."
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 size-biased empirical distribution
evolves in the limit according to a certain deterministic evolution equation. Although this equation involves a non-
local, non-linear operator, it can be studied thanks to a carefully chosen norm with respect to which this operator is
contractive.
In finite-dimensional 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 infinite-dimensional setting.
I will present a dynamic formulation of the mean-variance portfolio
selection problem and discuss possible ways of solving it.
Joint work with J. L. Pedersen (Copenhagen)
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 long-range dependencies on the model.
When G is the d-dimensional 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.
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 time-changed Brownian excursion.
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 space-time scales. In other words, the
local irregularities of the random medium average out over large
space-time 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 non-trivial 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.
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.
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 mass-m 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.
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 Super-Brownian 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.
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 run-time. 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.
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) state-space 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 not-so-usual 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.
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).
Particle filters are very flexible algorithms for inferential computation in non-linear, non-Gaussian state-space 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 genetic-type selection, which is usually the bottleneck for parallelization. Can we do away with resampling, or at least re-structure it in such a way as to be more naturally suited to non-serial 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)
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.
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 hard-core model. In the undirected case, I'll describe connections to bootstrap percolation and to maximum-cardinality 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)).
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 non-trivial. In particular, as in
Bramson's results for the non-linear 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 Ebert-van Saarloos in the non-linear 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 Ebert-van Saarloos correction has to be modified.
The talk will discuss the combination of two classical ideas.
The first is the use of hierarchichal decompositions for test-functions 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 N-particle mean-field 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 N-particle empirical distribution could do so.
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 k-opt algorithms such as, now classical, Lin-Kernighan heuristic to reorder the vertices in order to construct a Hamiltonian cycle, although it is not restricted to sequential k-opt edge exchanges.
The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time, O(n4 log n) for this implementation. On-line demonstration will accompany presentation.
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.
In this talk I consider two-player nonzero-sum 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 two-player nonzero-sum games of optimal stopping and a certain class of two-player nonzero-sum games of singular control.
Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The eigenvalues of the upper-left 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 two-fold. 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.
The A-transform, if applied to a monomial $x^n$ results in a well-known 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 A-transform is especially useful for solving problems related to Lévy processes. For instance, it gives a straightforward formula for the calculation of European-type 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 A-transform method benefits from the absence of integro-differential equations, making the process of obtaining the solution much easier. We illustrate the method with some examples.
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 change-points detection in the Presence of Trends and Long-Range Dependence. To detect change-points 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...
We consider the problem of finding particular patterns in a realisation of a two-sided standard Brownian motion taking the value zero at time zero. Examples include two-sided 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.
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.
Sabot and Tarres showed that a discrete time version of vertex-reinforced
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
edge-reinforced random walk has the same law as a mixture of the discrete
time version of vertex-reinforced 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.
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).
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 anti-preferential 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 anti-preferential attachment) are special cases of random mappings with exchangeable in-degrees. 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 in-degree 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 in-degrees 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 Intra-European Fellowship
No. 236845 (RANDOMAPP) within the 7th European Community Frame-
work Programme.
Abstract
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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 2-dimensional 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 non-Gaussian.
A SIRSN obeys axioms proposed by Aldous [1] and provides random (almost surely) unique routes between specified locations in a statistically scale-invariant 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). Scale-Invariant Random Spatial Networks. Electronic J. Prob., 21, no. 19, 1-41.
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).
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 Feynman-Kac’ formula on the infinite horizon with variable signs of the potential.
In dimensions higher than two it is expected that a disordered quantum system undergoes a metal-insulator transition from a region of localization to delocalization. For the one-particle 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 two-particle Anderson model with short-range 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 metal-insulator transport transition.
This is joint work with A. Klein and S. T. Nguyen.
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.
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.
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.
The Robinson-Schensted-Knuth (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.
The Wright-Fisher 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.
It is well-known 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 3-dimensional 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 self-interacting 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.
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.
Consider the problem of a government that wants to control the debt-to-GDP (gross domestic product) ratio of a country, while taking into consideration the evolution of the inflation rate. The uncontrolled inflation rate follows an Ornstein-Uhlenbeck 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 two-dimensional singular stochastic control problem over an infinite time-horizon. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation-dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.
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 Ca-induced-Ca-release, generating high-amplitude 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.
In 2005, Schramm considered the random interchange model on the complete graph and he proved
that the lengths of long cycles have Poisson-Dirichlet 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.
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 so-called s-holomorphic observables (aka lattice fermions). Surprisingly and embarrassingly, despite the facts that the rigid structure of s-holomorphic 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/super-harmonicity 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 half-plane (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.
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 two-periodic 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 liquid-gas boundary. This is
based on joint works with Vincent Beffara (Grenoble), Kurt Johansson
(Stockholm) and Benjamin Young (Oregon).
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.
For a given Markov process X and survival function H on R_+, the inverse first-passage time problem (IFPT) is to find a barrier function b : R_+ → [−∞,+∞] such that the survival function of the first-passage 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 time-change I such that for the time-changed 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 quasi-invariant distributions of the process X killed at the epoch of first entrance into the negative half-axis. 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.
We discuss the timing of trades under mean-reverting 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 Ornstein-Uhlenbeck (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.
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 $\alpha \le 1/2$, $P(A_{n,[\alpha n]} \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, 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).
The talk will present an overview of results about the sequential selection from posets and related optimality problems. The problem of choosing on-line 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 log-concavity 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).
We shall discuss the properties of the logarithmic derivative
of the Riemann zeta-function, rescaled about a point chosen at random
point on the critical line. The talk will be mostly self-contained.
Lambda-coalescents 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.
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).
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 power-law approximation by rationals, we show that the central gaps of H are open and provide a lower bound for their widths.
We consider two sharp next-order asymptotics problems, namely the asymptotics for the minimum energy for optimal point con figurations and the asymptotics for the many-marginals Optimal Transport, in both cases with Coulomb and Riesz costs with inverse power-law long-range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, while the second appears as a semi-classical limit of the Density Functional Theory energy modelling a quantum version of the same system. Recently the second-order 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 inverse-power-law interactions with power d-2\le s.
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.
Abstract. In this talk I will provide an overview of some recent results on proba-bilistic proofs of continuity and Lipschitz continuity of optimal stopping boundaries in multi-dimensional 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).
Dobrushin and Sethuuraman-Varadhan have proved sharp Central Limit Theorem for additive functionals of finite non-stationary Markov chains. We discuss the Local Limit Theorem in the same setting and give some extensions and applications. Joint work with Omri Sarig.
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 non-trivial extension of Nash's moment bound to this context and on down-to-earth concrete functional analytic arguments.
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 one-cut regular beta-ensembles and general linear statistics.
This is joint work with Michel Ledoux and Christian Webb, available at https://arxiv.org/abs/1706.
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.
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 Marchenko-Pastur law for sample covariance matrices. In the non-Hermitian 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.
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 one-dimensional Anderson model. Comm. Math. Phys. 101 (1985), no. 1, 1–19.
In this talk I will focus on the hard sphere model in R^d, in which a random set of non-intersecting 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).
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).
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 Rasch-type 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.
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 Finetti-type 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 non-singleton 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: erased-interval 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 one-to-one 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 tree-like 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.
I will present a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the condition number) of the left and right eigenvectors for non-selfadjoint Gaussian random matrices. First I derive the exact finite-N expression in the case of real eigenvalues and the associated non-orthogonality factors in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distributions are maximally heavy-tailed, so that all integer moments beyond normalization are divergent. Then I present results for a complex eigenvalue and the associated non-orthogonality 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.
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 white-noise representation of Gaussian fields and (ii) the randomised algorithm approach to noise sensitivity. Joint work with Hugo Vanneuville
Given a smooth compact Riemannian manifold an important problem in Quantum Chaos studies the distribution of L^{2} mass of eigenfunctions of the Laplace-Beltrami 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 half-integral weight automorphic forms. This is based on joint work Maksym Radziwill.
I will not assume any background knowledge of automorphic forms.
The spectral statistics of large random matrices exhibit a new type of universality as postulated by Eugene Wigner in the 1950’s. This celebrated Wigner-Dyson-Mehta 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.
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 large-matrix limit the joint moments are related to a solution of Painlevé III equation.
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.
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.
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 infinite-dimensional groups, and the other appears in combinatorial and probabilistic questions. In the talk I will discuss differences and similarities between these two settings.
We study a general class of log-gas 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 q-boxed 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 one-dimensional section of a pullback of the two-dimensional Gaussian free field.
Our approach is based on a q-analogue of the Schwinger-Dyson (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.
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.
The Erdös-Rényi 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.
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.
In this joint work with Gerónimo Uribe-Bravo, 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
jump-times, 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 non-Markovian, since the walker needs to
remember all its past.
Under moment conditions on the inter-jump-times, 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.
There are several conditions either sufficient or necessary for uniqueness or for non-uniqueness 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 Orleans-Paris), P. Kopanov (Plovdiv) and E. Yarovaya (Moscow).
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 2-dimensional 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 state-dependent rate of the `local-time' of an auxiliary 2-dimensional reflecting diffusion.
I will discuss the asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.
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 self-intersection local times and describe its asymptotics.
References:
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 non-integer $\beta>1$ 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)
We derive a multiscale generalisation of the Bakry--Emery
criterion for a measure to satisfy a Log-Sobolev inequality. Our
criterion relies on the control of an associated PDE well known in
renormalisation theory: the Polchinski equation. It implies the usual
Bakry--Emery criterion, but we show that it remains effective for
measures which are far from log-concave. Indeed, using our criterion,
we prove that the massive continuum Sine-Gordon model with $\beta<6\pi$
satisfies asymptotically optimal Log-Sobolev 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.
We will first give a general introduction on the topic with a special focus on the non-interacting 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.