Recurrence/transience of random affine recursions and a fluctuation-theoretic duality

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Iterated-logarithm laws for convex hulls of random walks with drift

In the talk we will present laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan. Our first result is a version of Strassen's functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we will briefly show how to compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero-one law for functionals of convex hulls of walks with drift, of some independent interest.

This is a joint project with Nikola Sandrić, Stjepan Šebek and Andrew Wade 

Ordered random walks

We study a d-dimensional random walk with independent and identically  distributed increments in a Weyl chamber.  We discuss existence of the harmonic function for the random walks killed on leaving the Weyl chamber . We then construct an ordered process using Doob's h-transform. For this process we discuss limit theorems when d is fixed and when d slowly increases.

Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations $u_t+u=Qu$, where $Q$ is an inhomogeneous smoothing transform. We show that under mild assumptions on $Q$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$. The talk is based on a joint work with Dariusz Buraczweski and Alexander Marynych

Biased Random walk on dynamical percolation or: should we be decisive or flexible?

We consider biased random walks on dynamical percolation on $\Z^d$. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for $d=1$ the speed is increasing, we show that in general this fails in dimension $d \geq 2$. As our main result, we establish two regimes of parameters, separated by an explicit critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero. Based on joint work with Sebastian Andres, Perla Sousi and Dominik Schmid

Tail asymptotics for exponential functionals of subordinators and extinction times of self-similar fragmentations

Exponential functionals of subordinators have been thoroughly investigated since they are involved in the description of various processes ranging from the analysis of algorithms to coagulation or fragmentation processes. In this talk we will provide the exact large-time equivalents of the density and upper tail distribution of the exponential functional of a subordinator in terms of its Laplace exponent. This improves previous results on the logarithmic asymptotic behaviour of this tail. We will then see how this result can be used to determine the large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity. Indeed, the extinction time of a typical fragment in such a process is an exponential functional of a subordinator. However the tail of the extinction time of the whole fragmentation process decreases much more slowly in general. We will quantify this difference by determining the asymptotic ratio of the two tails.

Local limit of Preferential Attachment model

In this talk I will present the local limit of preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014). The degree distribution of this limiting random graph, which we call the random Pólya point tree, has a surprising size-biasing phenomenon. Our models incorporate negative values of the preferential attachment fitness parameter, which allows us to consider preferential attachment models with infinite-variance degrees. We shall use the local limit to identify the percolation phase transition on preferential attachment models. We identify the critical percolation threshold explicitly using the branching process limit. This is based on joint works with Alessandro Garavaglia, Remco van der Hofstad and Rounak Ray.

Scaling limits of uniform spanning trees and forests

In two and three dimensions, the scaling limit of the uniform spanning tree of Z^d is a measured, rooted spatial tree, and its embedding in R^d fills the space. For these integer lattices and high-dimensional finite graphs, the proofs of the existence of the corresponding limit trees rely on Wilson's algorithm. The situation is different in Z^d for d ≥ 5. In this case, the infinite-volume limit of uniform spanning trees of finite subgraphs is the uniform spanning forest (USF) with, almost surely, infinitely many trees. Sampling with Wilson's algorithm, one builds different trees simultaneously. Hence, studying the scaling limit of one tree in the USF requires a different approach. This talk will present recent advances in this direction for the USF of Z^d, d ≥ 5, based on ongoing joint work with Tom Hutchcroft.

Fluctuations of non-local branching Markov processes

We consider a general class of branching Markov processes whose offspring distribution is non-local. In the supercritical regime, the first order long-term behaviour is well understood via Perron Frobenius decompositions, laws of large numbers and the asymptotic behaviour of its moments. In this talk, we consider the second order fluctuations of the process in three different regimes, depending on the sign of the spectral gap. In each regime, we prove a functional limit theorem that characterises the long-term fluctuations of the process.

 Branching capacity of a random walk on Z^d

This talk is based on a joint work with Tianyi Bai and Jean-François Delmas. We study the branching capacity of the range of a simple random walk on Z^d. The branching capacity was defined by Zhu (2016) using a critical branching random walk conditioned on survival, similar to the classical capacity that uses a random walk on Z^d. Our main result provides the convergence in law for the branching capacity of the range in dimension 5, completing recent work of Schapira (2023+). The main step in the proof relies on a study of intersection probabilities between a critical branching random walk and an independent simple random walk. 

On decoupled random walks

We call a decoupled random walk a sequence $\hat S_1$, $\hat S_2,\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat S_n$ has the same distribution as the position at time $n$ of a standard random walk with nonnegative jumps. Similarly, we call a decoupled renewal process the counting process $(\hat N(t))_{t\geq 0}$ defined by $\hat N(t)=\sum_{n\geq 1}\1_{\{\hat S_n\leq t\}}$. I shall present a functional limit theorem for $(\hat N(t))_{t\geq 0}$, properly scaled, normalized and centered, as $t\to\infty$ under the assumption that the variance of $\hat S_1$ is positive and finite. Also, I shall discuss the asymptotic of $\log \mathbb{P}\{\min_{n\geq 1}\hat S_n>t\}$ as $t\to\infty$ under various assumptions imposed on the distribution of $\hat S_1$. Our interest to the so defined decoupled random walks was caused by their appearance in the particular case when $\hat S_1$ has an exponential distribution of unit mean in the context of infinite Ginibre point processes. The talk is based on a recent joint work with Gerold Alsmeyer and Zakhar Kabluchko (Muenster).

A variational approach to free probability 

Let A and B be large unitarily invariant independent Hermitian random matrices whose 

empirical spectra approximate probability measures $\mu$ and $\nu$ on the real line. The central results of free probability says that the empirical spectrum of the random sum $A+B$ approximates a certain measure $\mu \boxplus \nu$ on the real line, known as the additive free convolution of $\mu$ and $\nu$. 

We apply a large deviation principle on the symmetric group to a recent formula of Marcus, Spielman and Srivastava to prove a new variational description of the additive free convolution.

We prove similar formulas for other operations in free probability, as well as new inequalities relating free and classical operations.

This is joint work with Octavio Arizmendi (CIMAT, Mexico).

Lah distributions, Stirling numbers, and convex hulls of random walks

Let $X_1,X_2,\ldots$ be independent random vectors in $\mathbb R^d$ having an absolutely continuous distribution. Consider the random walk $S_k:=X_1+\ldots+X_k$, and let $P_n:=conv\{0,S_1,S_2,\ldots,S_n\}$ be the convex hull of its first $n$ points. We shall be interested in the number of the $k$-dimensional faces of the polytope $P_n$ and in particular, whether this number is equal to the maximal possible number $\binom {n+1}{k+1}$ with high probability, as $n$, $d$, and possibly also $k$ diverge to $\infty$. There is an explicit formula for the expected number of $k$-dimensional faces which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, whose definition involves Stirling numbers of both kinds. In this talk we shall discuss the properties of this and other distributions related to Stirling numbers. The talk is based on a joint work with Alexander Marynych: https://arxiv.org/abs/2105.11365 

Limit theorems for branching processes

Branching processes constitute an important class of stochastic processes with numerous practical and theoretical applications. In my talk, I will present recent limit theorems concerning these processes, which allow to determine asymptotic expansions up to Gaussian fluctuations. Based on recent joint collaborations with A.Iksanov, M. Meiners, and E.Sava-Huss.

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Random walks in sparse random environment

A random walk in sparse random environment is a model in which the particle performs a simple random walk on the set of integers. It moves symmetrically except for some randomly chosen sites, where random drift is imposed. During my talk I will present some properties of the model and the results on quenched limit laws obtained recently with D. Buraczewski and P. Dyszewski 

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The KPP traveling wave in the half-plane

H. Berestycki and C. Graham (2022) proved that the F-KPP reaction-diffusion equation $\partial_t u = \frac{1}{2} \Delta u + u(1-u)$ in the half-place with Dirichlet boundary conditions admit traveling waves for all speed $c \geq \sqrt{2}$. Using the duality between this PDE and the branching Brownian motion in the half-plane with absorption at the boundary, we prove that the minimal speed traveling wave is in fact unique (up to shift). Moreover, we give a probabilistic representation of this traveling wave $\Phi$ in terms of the Laplace transform of a certain "derivative martingale" of this branching Brownian motion. 

We use this probabilistic representation to describe the asymptotic behavior of $\Phi$ away from the boundary of the domain, proving that 

\[ \lim_{y \to \infty} \Phi\left(x + \frac{1}{\sqrt{2}} \log y, y\right) = w(x) \] 

where $w$ is the usual one-dimensional traveling wave. This talk is based on joint work with Julien Berestycki, Cole Graham and Yujin H. Kim.

Branching products of random matrices 

In a branching random walk, we start with one particle positioned at the origin. It produces a random number of offspring that is moved away from the parent's position at random (the reproduction and displacement law being usually described by a point process N). Each particle reproduces in the same way according to the law of N; independently of all other particles. In this talk, we extend the model to a multidimensional setup, where moreover the displacements are not additive, but coming from the actions of random invertible matrices. We obtain a cloud of particles in the d-dimensional Euclidean space and want to describe its shape. In particular, we will focus on the maximal distance to the origin.

Upper large deviations for the number of edges in scale-free geometric graphs 

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. The strategy behind the large deviation event is based on a condensation effect for the vertex degrees. A finite number of vertices are selected at random and their power increased so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We also show that the empirical distribution of edge lengths under the conditioning splits into a bulk and travelling wave part of asymptotically positive proportions. The talk is based on a collaboration with Pim van der Hoorn, Remco van der Hofstad, Céline Kerriou and Neeladri Maitra.

The speed of biased random walk on weighted regular trees

We consider a random walk on a regular tree, where each edge is assigned a random weight or conductance. The random walk crosses an edge with probability proportional to the edge conductance and additionally, conductances of edges leading away from the root are increased by a bias factor. The distance to the root then satisfies a law of large numbers with limit the effective speed of the random walk, which depends in a complicated way on the conductance law and bias parameter. We are interested in properties of the speed as a function of the bias. For example, is the speed monotone increasing in the strength of the bias? This talk is based on joint work with Nina Gantert and Yuki Tokushige.

Fixation times for a multitype Lambda Wright-Fisher process

We derive stationary and fixation times for the multi-type Lambda Wright-Fisher process with and without mutations. Our method relies on a grand coupling of the process realized through the so-called lookdown-construction.

On the oscillating random walk on Z 

In this talk, we  present some recent results on the recurrence properties and the asymptotic behavior of the « oscillating random walk on Z ». We propose a simple criteria for recurrence and state an invariance principle for this process. The main tool is a renewal theorem for aperiodic sequences of operators, due to S. Gouezel.

Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials 

We establish exact formulae for the (positivity) persistence probabilities of an autoregressive sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results provide factorizations of general autoregressive models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to explicit asymptotic estimates for the persistence probabilities. This is a joint work with Gerold Alsmeyer, Alin Bostan and Thomas Simon (Adv. Appl. Math., 2023). 

Williams' path decomposition for $d$-dimensional self-similar Markov processes

The classical result of Williams states that a Brownian motion with positive drift "\mu" and issued from the origin is equal in law to a Brownian motion with unit negative drift, $-\mu$, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter $2\mu$, after which it behaves like a Brownian motion conditioned never to go below the aforesaid threshold (i.e. a Bessel-3 process, or equivalently a Brownian motion conditioned to stay positive, relative to the threshold). In this talk we will consider the analogue of Williams' path decomposition for a general self-similar Markov process (ssMp) on $\mathbb{R}^d$. In essence, Williams' path decomposition in the setting of a ssMp follows directly from an analogous decomposition for Markov additive processes (MAPs). The latter class are intimately related to the former via a space-time transform known as the Lamperti--Kiu transform. As a key feature of our proof of Williams' path decomposition, will obtain the analogue of Silverstein's duality identity for the excursion occupation measure for general Markov additive processes (MAPs). 

Markov equivalence classes of directed acyclic graphs 

Can we reconstruct a directed acyclic graph having only access to its v-structures, encoding conditional independence between the sites, but without knowing its edge directions? In this talk, we study the probability to have a unique way of such a reconstruction when the directed acyclic graph G is chosen uniformly at random on a fixed number of sites. More generally, we study the size of its Markov equivalence class, containing all graphs with the same edge set as G when forgetting the edge directions, and having the same v-structures. This talk is based on joint work with Allan Sly (Princeton University).

Persistence of autoregressive sequences with logarithmic tails 

We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbb P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential.

Limit theorems for the coefficients of products of random matrices and applications 

Let $(g_n)_{n \geq 1}$ be a sequence of independent and identically distributed $d \times d$ random matrices and consider the random walk $G_n = g_n \ldots g_1$. In this talk, we will present some recent progress on large and moderate deviations, Berry-Esseen bound and Edgeworth expansion for the coefficients of $G_n$. We will also give some applications to the study of limit theorems for the first passage time of multivariate perpetuity sequences. Mainly based on joint work with I. Grama, Q. Liu and S. Mentemeier.  

Stochastic Volterra Equations for the Local Times of Spectrally Positive Stable Processes 

In this talk, we main explore the macroevolution mechanism of local times of a spectrally positive stable process in the spatial direction. The main results state that conditioned on the finiteness of the first time at which the local time at zero exceeds a given value, the local times at positive half line are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson random measure whose intensity coincides with the L\’evy measure. This helps us to provide not only a simple proof for the H\”older regularity, but also a uniform upper bound for all moments of the H\”older coefficient as well as a maximal inequality for the local times. Moreover, based on this stochastic Volterra equation, we extend the method of duality to establish an exponential-affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra integral equation associated with the Laplace exponent of the stable process.