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Transient random walks on the space of lattices

Axel Péneau, Cagri Sert

TL;DR

The paper studies random walks on the space of lattices $X=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$ and demonstrates that Zariski-dense walks need not be recurrent in law; in particular it constructs a full-escape walk when $d\ge 2$. It further shows sharpness of the $L^p$ moment threshold by building a Zariski-dense walk with finite $L^p$-moment for $p\in(0,1)$ that still escapes, contrasting with recurrence results under stronger moment conditions. Beyond arithmetic examples, the work provides large-support and uncountably many divergent-start-point counterexamples, highlighting Diophantine-type mechanisms behind non-recurrence. Together with Eskin–Margulis and Bénard–de Saxcé results, the paper clarifies the boundary between recurrence and escape regimes in homogeneous dynamics and emphasizes the role of heavy-tailed and Diophantine structures in driving non-recurrence.

Abstract

Given $d\geq2$, we construct a Zariski-dense random walk on the space of lattices SL$_d(\mathbb{R})/$SL$_d(\mathbb{Z})$ that exhibits escape of mass. This negates the suggestion of recurrence made by Benoist [Ben14] (ICM 2014) and by Bénard-de Saxcé [BS22] (also asked in [BQ12]). For any $p \in (0,1)$, we also construct such a random walk with finite $L^p$-moment which shows that the moment assumption in [BS22] is sharp.

Transient random walks on the space of lattices

TL;DR

The paper studies random walks on the space of lattices and demonstrates that Zariski-dense walks need not be recurrent in law; in particular it constructs a full-escape walk when . It further shows sharpness of the moment threshold by building a Zariski-dense walk with finite -moment for that still escapes, contrasting with recurrence results under stronger moment conditions. Beyond arithmetic examples, the work provides large-support and uncountably many divergent-start-point counterexamples, highlighting Diophantine-type mechanisms behind non-recurrence. Together with Eskin–Margulis and Bénard–de Saxcé results, the paper clarifies the boundary between recurrence and escape regimes in homogeneous dynamics and emphasizes the role of heavy-tailed and Diophantine structures in driving non-recurrence.

Abstract

Given , we construct a Zariski-dense random walk on the space of lattices SLSL that exhibits escape of mass. This negates the suggestion of recurrence made by Benoist [Ben14] (ICM 2014) and by Bénard-de Saxcé [BS22] (also asked in [BQ12]). For any , we also construct such a random walk with finite -moment which shows that the moment assumption in [BS22] is sharp.

Paper Structure

This paper contains 9 sections, 15 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on rational lattices
  3. Construction of the rational counterexamples
  4. Laws with heavy records
  5. Laws with escape of mass and finite polynomial moment
  6. Diophantine counter-examples with no atoms
  7. Preliminaries
  8. Proof of Theorem \ref{['th:non-alg']}
  9. Proof of Theorem \ref{['th:non-alg-escape']}

Key Result

Theorem 1.1

Given $d \geq 2$, there exist a Zariski-dense probability measure $\mu$ on $\mathrm{SL}_d(\mathbb{R})$ and $x \in X=\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$ such that $\mu^{\otimes \mathbb{N}}$-a.s. $\gamma_0\cdots \gamma_{n-1}\cdot x \to \infty$ and $\gamma_{n-1}\cdots \gamma_{0}\cdot

Theorems & Definitions (37)

  • Theorem 1.1: Full escape
  • Theorem 1.2: Escape of mass under $L^p$-moment, $p \in (0,1)$
  • Theorem 1.3: Large support non-recurrent random walks
  • Theorem 1.4: Uncountably many divergent points
  • Definition 2.1: Systole of a lattice
  • Lemma 2.2
  • proof
  • Definition 2.3: Rational height of a matrix
  • Lemma 2.4
  • proof
  • ...and 27 more