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

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. 

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.

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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.

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 

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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