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Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

Jérémie Brieussel, Romain Tessera, Tianyi Zheng

TL;DR

The paper establishes Gaussian concentration bounds for the maximal displacement of compactly supported random walks on groups of polynomial growth, without requiring symmetry or centering. It develops a filtration- and norm-based framework on nilpotent Lie groups and their semidirect extensions, proving subgaussian concentration for an adapted norm and then transferring those bounds to word norms via structural embeddings. A key innovation is the introduction of the essential average and a quantitative splitting theorem, which yield uniform concentration in finite-by-nilpotent extensions and enable applications to AU-extensions, almost connected amenable Lie groups, polycyclic and finite-Prüfer-rank solvable groups. Consequently, centred walks on broad classes of groups are maximally diffusive, and laws of iterated logarithm follow in several natural settings. The results significantly extend prior partial findings and provide a robust toolkit for understanding diffusion-like behavior of random walks on complex groups.

Abstract

We prove Gaussian concentration inequalities for maximal displacement of compactly supported random walks on a compactly generated locally compact group with polynomial growth. Concentration inequalities with different exponents hold for non-centred random walks as well, after correction by the drift. When the support of the measure generates a virtually nilpotent group, we provide an effective version of this result. These more refined estimates rely on the existence of a ``quantitative splitting'' of a virtually simply connected nilpotent group, a result which may be of independent interest. As applications, we deduce that the same concentration inequalities hold for centred random walks on the following classes of groups: amenable connected Lie groups (including non-unimodular ones), polycyclic and more generally finitely generated solvable groups with finite Prüfer rank. This shows in particular that centred random walk are diffusive on such groups. For polycyclic groups, this strengthens and completes partial results previously obtained by Russ Thompson in 2011.

Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

TL;DR

The paper establishes Gaussian concentration bounds for the maximal displacement of compactly supported random walks on groups of polynomial growth, without requiring symmetry or centering. It develops a filtration- and norm-based framework on nilpotent Lie groups and their semidirect extensions, proving subgaussian concentration for an adapted norm and then transferring those bounds to word norms via structural embeddings. A key innovation is the introduction of the essential average and a quantitative splitting theorem, which yield uniform concentration in finite-by-nilpotent extensions and enable applications to AU-extensions, almost connected amenable Lie groups, polycyclic and finite-Prüfer-rank solvable groups. Consequently, centred walks on broad classes of groups are maximally diffusive, and laws of iterated logarithm follow in several natural settings. The results significantly extend prior partial findings and provide a robust toolkit for understanding diffusion-like behavior of random walks on complex groups.

Abstract

We prove Gaussian concentration inequalities for maximal displacement of compactly supported random walks on a compactly generated locally compact group with polynomial growth. Concentration inequalities with different exponents hold for non-centred random walks as well, after correction by the drift. When the support of the measure generates a virtually nilpotent group, we provide an effective version of this result. These more refined estimates rely on the existence of a ``quantitative splitting'' of a virtually simply connected nilpotent group, a result which may be of independent interest. As applications, we deduce that the same concentration inequalities hold for centred random walks on the following classes of groups: amenable connected Lie groups (including non-unimodular ones), polycyclic and more generally finitely generated solvable groups with finite Prüfer rank. This shows in particular that centred random walk are diffusive on such groups. For polycyclic groups, this strengthens and completes partial results previously obtained by Russ Thompson in 2011.
Paper Structure (26 sections, 43 theorems, 153 equations)

This paper contains 26 sections, 43 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Concentration inequalities for random walks on groups of polynomial growth
  3. Uniform constants in finite-by-nilpotent groups
  4. Application to extensions by approximately ultrametric subgroups
  5. Law of iterated logarithm
  6. Review of concentration inequalities
  7. Characterisations of concentration inequalities
  8. Stability under operations
  9. Filtrations and norms on nilpotent Lie groups
  10. Filtrations on nilpotent Lie groups
  11. Norms adapted to the filtration
  12. The Baker-Campbell-Hausdorff formula
  13. Gaussian concentration for the adapted norm
  14. Random walks on semi-direct products
  15. Centering the nilpotent part
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $G$ be a locally compact, compactly generated group of polynomial growth. Let $S$ be a compact symmetric generating set, and $\mu$ be a compactly supported measure on $G$. Denote $(w_n)_{n\ge 0}$ the associated random walk starting at the neutral element.

Theorems & Definitions (92)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4: Maximally diffusive random process
  • Definition 1.5: AU subgroups
  • Proposition 1.6
  • Definition 1.7: Polynomial-by-AU
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 82 more