Probability and Applications

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 function for the scaling limit and the behaviour of the spectral gaps the random graphs converge to their scaling limit.

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 first 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 Lévy 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.

tvkerneloverview_slides_140115 [PDF 720KB].pdf

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
------------
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/1706.10251.

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 L² 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 L² 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 obtain a quantitative lower bound on the entries of the time evolution operator associated to a general periodic operator in terms of its bandwidths, which allows us to obtain a quantitative lower bound on the irrationality of the frequency in terms of the minimum of the Lyapunov exponent for which the operator exhibits quasiballistic transport.

We study the concept of p-th variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. To start with we introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation is invariant with respect to the choice of the partition sequence. Finally, we introduce a notion called Horizontally rough which provides an invariance notion for p-th variation.

I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function on the critical line, especially in short intervals. The problems have interesting connections to statistical mechanics of disordered systems, both in their interpretations and in the techniques of proofs. These connections will be emphasized.

This is based in part on joint works with Emma Bailey, and with Paul Bourgade & Maksym Radziwill.

Rough volatility models have become quite popular recently, as they capture both the fractional scaling of the time series of the historic volatility (Gatheral et al. 2018) and the behaviour of the implied volatility surface (Fukasawa 2011, Bayer et al. 2016) remarkably well. In contrast to classical stochastic volatility models, the volatility process is neither a Markov process nor a semimartingale. Therefore, these models fall outside the scope of standard stochastic analysis and provide new mathematical challenges. In this talk, we present an overview of of this new paradigm in volatility modelling and consider the impact of rough volatility on portfolio optimisation. 

The talk is based on joint work with Johannes Muhle-Karbe and Denis Schelling.

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

Recently, there has been much progress in understanding stationary measures for coloured (also called multi-species or multi-type) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unifying approach to constructing stationary measures for most known such systems (including the classical multispecies Asymmetric Simple Exclusion Process) based on integrable stochastic vertex models and the Yang-Baxter equation. Joint work with Amol Aggarwal and Matthew Nicoletti.

We study an ensemble of random matrices called Elliptic Volatility Model. This consists of a product of independent matrices X = ΣZ where Z is a T by S matrix of i.i.d. light-tailed variables with mean 0 and variance 1 and Σ is a diagonal matrix of i.i.d. heavy tailed variables. The study of such ensembles first arose in financial mathematics as models of stock price log-returns. We obtain an explicit formula for the empirical spectral distribution of its covariance matrix when Σ_ii is distributed as the Student-t with parameter 3 and the distribution of its largest eigenvalue in a more general case. This is joint work with Svetlana Malysheva.

On \mathbb{Z}^d, consider \varphi, an \ell^2-normalized function that decays exponentially at infinity at a rate at least mu. One can define the onset length (of the exponential decay) of \varphi as the radius of the smallest ball, say, B, such that one has the following global bound \D |phi(x)| <= ||\varphi||_\infty e^(-mu dist(x,B)).

The present talk will describe the onset lengths of the localized eigenfunctions of random Schrödinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than l and localization center in a ball of radius L is smaller than C L^d exp(-c l), for l>0 large (for some constants C,c>0). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. The talk is mainly based on joint work with Jeff Schenker.

We present general conditions for the weak convergence of a discrete-time additive scheme
to a stochastic process with memory in the space D[0, T]. We investigate the convergence
of the related multiplicative scheme to a process that can be interpreted as an asset price with
memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process.

Domino tilings of the two-periodic Aztec diamond exhibit interesting statistical mechanical behaviors – a limit shape emerges separating three macroscopic interfaces, known as frozen, rough and smooth that depend on the local statistics. We survey some of the main results on this model, as well as recent progress on understanding the rough-smooth boundary.  This is based on joint work with Duncan Dauvergne and Thomas Finn. 

The multispecies PushTASEP (where TASEP stands for totally
asymmetric simple exclusion process) is an interacting particle system
with multiple species of particles on a finite ring where the hopping
rates are site-dependent. (The homogeneous variant on Z is also known as
the Hammersley–Aldous–Diaconis process.) In its simplest variant with a
single species, a particle at a given site will hop, when that bell
rings, to the first available vacant site clockwise. We compute the
stationary distribution of this inhomogeneous process using a multiline
process. In particular, we show that the partition function of this
process is intimately related to the multispecies TASEP and to the
classical Macdonald polynomials. We also prove that large families of
events are symmetric under the interchange of these site-dependent
rates. This is joint work with James Martin.

Lambda quantiles have been introduced by Frittelli, Maggis and Peri (2014) in the context of risk measure theory with the name of Lambda Value at Risk. They are a generalization of quantiles that, instead of relying on a fixed confidence level, consider a function known as lambda. After a brief overview of the theory and the emerging literature on lambda quantiles, we will focus on their estimation. First, we will review their properties of robustness of the empirical estimator and elicitability under specific conditions on the distribution functions. Then, we will present a range of methods for estimating the lambda function and the lambda quantiles, including simulations, parametric models, and supervised learning approaches. Finally, we will provide an empirical application in risk measurement, comparing a parametric approach with a supervised learning method.

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

Abstract:

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. By determining the asymptotic growth rates of the expected Betti numbers, we discover that the graph undergoes higher-order phase transitions within the infinite-variance regime.

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

Abstract: We will consider regularly varying time series. The name comes from the marginal tails which are of power-law type. Davis and Hsing (1995) and Basrak and Segers (2009) started the analysis of such sequences. They found an accompanying sequence (spectral tail process) which contains the information about the influence of extreme values on the future behavior of the time series, in particular on extremal clusters. Using the spectral tail process, it is possible to derive limit theory for maxima, sums, point processes... of regularly varying sequences,  but also refined results like precise large deviation probabilities for these structures.

In this talk we will give a short introduction to regularly varying sequences and  and explain how the aforementioned limit results can be derived.     

Abstract: We will discuss the difference the matrix entries' number of finite moments makes to its eigenvalue distribution. 

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

We shall discuss the quantum dynamics associated with ergodic Schroedinger operators with singular continuous spectrum. Upper bounds on the transport moments have been obtained for several classes of one-dimensional operators, particularly, by Damanik--Tcheremchantsev, Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method which allows to recover most of the previous results and also to obtain new results in one and higher dimensions. The input required to apply the method is a large-deviation estimate on the Green function at a single energy. Based on joint work with S. Sodin.

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

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

Abstract: We will present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in the Euclidian space. The dynamics we study are those of a Markov birth-death process. We prove functional limit theorems in the so-called thermodynamic regime using the recent theory of Malliavin-Stein bounds. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs. (Joint work with Omer Bobrowski and Robert J. Adler)

Abstract:

In this talk we will discuss the deep parametric PDE method for parametric option pricing in high dimensions, underlying theoretical results for neural networks and an application in risk management.

The deep parametric PDE method uses deep neural networks to solve parametric partial differential equations, such as those arising in option pricing. Especially for a large number of risk factors, the efficiency of neural networks for high dimensional problems is beneficial. We investigate this efficiency theoretically by presenting approximation rates for networks with smooth activation functions. (joint work with Kathrin Glau)

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

We prove that Anderson localization near band edges of ergodic continuum
random Schroedinger operators with periodic background potential in in
dimension two and larger is universal.
In particular, Anderson localization holds without extra decay
assumptions on the random variables and independently of regularity or
degeneracy of the Floquet eigenvalues of the background operator.
Our approach is based on a robust initial scale estimate the proof of
which avoids Floquet theory altogether and uses instead an interplay
between quantitative unique continuation and large deviation estimates.
Furthermore, our reasoning is sufficiently flexible to prove this
initial scale estimate in a non-ergodic setting, which promises to be an
ingredient for understanding band edge localization also in these
situations.

Based on joint work with Albrecht Seelmann (TU Dortmund).

Random walk in random environment (RWRE) is a model for random movement of a particle in a disordered medium, which is intrinsically related to random difference equations. For these I discuss a parabolic Harnack inequality in a not necessarily elliptic balanced i.i.d. setting. The talk is based on joint work with Noam Berger (TUM).

We review several results on the geometry and the asymptotic shape of
random polytopes generated by N independent vectors distributed according
to a log-concave probability measure on the n-dimensional Euclidean space.
We also discuss the case of the uniform measure on the discrete cube and
applications to combinatorial questions about 0/1 polytopes.

Consider an n x n random unitary matrix U taken with respect to normalized Haar measure. It is a well known consequence of the strong Szego limit theorem that the traces of powers of U converge to independent (complex) normal random variables as n grows. I will discuss a recent result together with
Gaultier Lambert where we obtain a super-exponential rate of convergence in total variation between the traces of the first m powers of an n × n random unitary matrices and a 2m-dimensional Gaussian random variable. This generalizes previous results in the scalar case, which answered a conjecture by
Diaconis, to the multivariate setting.  We are especially interested in the regime where m grows with n. The problem on how the rate of convergence changes as m grows with n was raised recently by Sarnak. The result we obtain gives the precise dependence on the dimensions m and n in the estimate with explicit constants for m almost up to the square root of n.

We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using a message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and similarly in the expected size of small clusters containing a given node. Results can be obtained for single large instances of finite networks and in the limit of infinite system size, byderiving self-consistency equations for the limiting distributions that emerge from the single instance formulation as the infinite system size limit is taken.  Distributions of node dependent probabilities to belong to the giant cluster in each instance of a number of repeated random edge removal experiments are also briefly discussed.

We consider a semimartingale reflected Brownian motion in a 
two-dimensional cone. The main goal of the talk is to study the 
algebraic nature of the Laplace transform of its stationary 
distribution. We derive necessary and sufficient conditions for the 
Laplace transform to be differentially algebraic, D-finite, algebraic 
or rational. These conditions are algebraic dependencies among the 
parameters of the model (drift, opening of the wedge, angles of the 
reflections on the axes). As a consequence we obtain new derivations 
of the Laplace transform in several well known cases, namely the 
skew-symmetric case, the orthogonal reflections case and the 
sum-of-exponential densities case. The third of these occurs exactly 
when the so-called Dieker-Moriarty condition holds. Joint work with M. 
Bousquet-Mélou, A. Elvey Price, S. Franceschi and C. Hardouin.

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.

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:

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 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.

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.

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 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)

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.

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.   

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. 

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 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.

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.

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.

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.

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi_{n+1}$, where $a$ is a constant, stays non-negative for a long time. Assuming that the i.i.d. innovations $\xi_n$ take only two values $\pm 1$ and $a \le \frac23$, we find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\frac12< a \le \frac23$, except for the case $a=\frac23$ and $\pr(\xi_n=1)=\frac12$, where this distribution is uniform on the interval $[0,3]$. These properties are similar to those of the Bernoulli convolutions. To solve our problem, we employ a dynamical system defined by a certain linear mod 1 transform. Such mappings are well studied due to their use in expansions of numbers in non-integer bases, the so-called generalised $\beta$-expansions. This is a joint work with V. Wachtel.

We investigate reinforced non-linear urns with interacting types, and
show that where there are three interacting types there are phenomena
which do not occur with two types. In a model with three types where the
interactions between the types are symmetric, we show the existence of a
double phase transition with three phases: as well as a phase with an
almost sure limit where each of the three colours is equally represented
and a phase with almost sure convergence to an asymmetric limit, which
both occur with two types, there is also an intermediate phase where
both symmetric and asymmetric limits are possible.  In a model with
anti-symmetric interactions between the types, we show the existence of
a phase where the proportions of the three colours cycle and do not
converge to a limit, alongside a phase where the proportions of the
three colours can converge to a limit where each of the three is equally
represented.  This is joint work with Marcelo Costa.

TBA