On the affine random walk on the torus

TL;DR

This paper studies the behavior of an affine random walk on the torus, proving uniqueness of the stationary measure and exponential convergence under certain conditions, and establishing limit theorems for functions along the walk.

Contribution

It extends previous work by showing the Lebesgue measure is the unique stationary measure for a broader class of affine random walks and proves several probabilistic limit results.

Findings

Lebesgue measure is the unique stationary measure under certain conditions.

Exponential convergence of expected values of Hölder functions.

Established law of large numbers and central limit theorem for the walk.

Abstract

Let $\mu$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, \[ \left\{\begin{array}{rcl} X_0 &=&x\\ X_{n+1} &=& a_{n+1} X_n + b_{n+1} \end{array}\right. \] where $((a_n,b_n))\in \mathbf{G}^\mathbb{N}$ is an iid sequence of law $\mu$. Then, we denote by $\mathbb{P}_x$ the measure on $(\mathbb{T}^d)^\mathbb{N}$ that is the image of $\mu^{\otimes \mathbb{N}}$ by the map $\left((g_n) \mapsto (x,g_1 x, g_2 g_1 x, \dots , g_n \dots g_1 x, \dots)\right)$ and for any $\varphi \in \mathrm{L}^1((\mathbb{T}^d)^\mathbb{N}, \mathbb{P}_x)$, we set $\mathbb{E}_x \varphi((X_n)) = \int \varphi((X_n)) \mathrm{d}\mathbb{P}_x((X_n))$. Bourgain, Furmann, Lindenstrauss and Mozes studied this random walk when $\mu$ is concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes\{0\}$ and this allowed us to study, for any h\"older-continuous function $f$ on the torus, the sequence $(f(X_n))$ when $x$ is not too well approximable by rational points. In this article, we are interested in the case where $\mu$ is not concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{Q}^d/\mathbb{Z}^d$ and we prove that, under assumptions on the group spanned by the support of $\mu$, the Lebesgue's measure $\nu$ on the torus is the only stationary probability measure and that for any h\"older-continuous function $f$ on the torus, $\mathbb{E}_x f(X_n)$ converges exponentially fast to $\int f\mathrm{d}\nu$. Then, we use this to prove the law of large numbers, a non-concentration inequality, the functional central limit theorem and it's almost-sure version for the sequence $(f(X_n))$. In the appendix, we state a non-concentration inequality for products of random matrices without any irreducibility assumption.