Recurrence of the twisted planar random walk

Random walk in a stratified medium

We study the recurrence properties of a random walk in a stratified medium.

Random walks on discrete Abelian groups

In the present paper we find necessary and sufficient conditions for recurrence of random walks on arbitrary subgroups of the group of rational numbers $\mathbb{Q}$.

On homogeneous and oscillating random walks on the integers

Considering homogeneous and oscillating random walks on the integers, we simplify classical works on recurrence of Spitzer and Kemperman, respectively. Some links with renewal theory are discussed.

Planar random walk in a stratified quasi-periodic environment

We study the recurrence of inhomogeneous Markov chains in the plane, when the environment is horizontally stratified and the heterogeneity of quasi-periodic type.

Twisted Recurrence via Polynomial Walks

In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathb…

Quantitative recurrence in two-dimensional extended processes

Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including $\ZZ^2$-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin.

Transience in law for symmetric random walks in infinite measure

We consider a random walk on a second countable locally compact topological space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has escape of mass for almost every starting point. We then apply this result in the context of homogeneous random walks on infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence theorem.

Random walk in a stratified independent random environment

We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results established in [2]. First of all the random walk is transient in dimension at least three. Focusing on dimension two, both recurrence and transience can happen, but transience remains by far the most general situation. We identify the critical scale of the local drift along the strata corresponding to the frontier between the two regimes.

遍历 $\mathbb{Z}^d$ 作用的原始平均、方向扩张性和定量扭曲回复性

Primitive Averages, Directional Expansivity, and Quantitative Twisted Recurrence for Ergodic $\mathbb{Z}^d$-Actions

本文为保概率 $\mathbb{Z}^d$ 作用建立了原始向量平均的 $L^2$ 极限公式，并给出了方向扩张集具有高下密度的谱条件，进而证明了扭曲多重回复定理的定量版本。