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On the universality of fluctuations for the cover time

Nathanaël Berestycki, Jonathan Hermon, Lucas Teyssier

Abstract

We consider random walks on finite vertex-transitive graphs $Γ$ of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel variable if and only if $\mathrm{Diam}(Γ)^2 = o(n/\log n)$, where $n = |Γ|$. We prove that this condition is furthermore equivalent to the decorrelation of the uncovered set. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which we leverage into strong heat kernel bounds, and refined quantitative estimates on Aldous and Brown's exponential approximation of hitting times, which are of independent interest.

On the universality of fluctuations for the cover time

Abstract

We consider random walks on finite vertex-transitive graphs of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel variable if and only if , where . We prove that this condition is furthermore equivalent to the decorrelation of the uncovered set. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which we leverage into strong heat kernel bounds, and refined quantitative estimates on Aldous and Brown's exponential approximation of hitting times, which are of independent interest.
Paper Structure (28 sections, 70 theorems, 345 equations)

This paper contains 28 sections, 70 theorems, 345 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Application to Cayley graphs
  5. Decorrelation in the uncovered set
  6. Diameter condition and local transience
  7. Discussion of proof ideas and organisation of paper
  8. Preliminaries
  9. A priori bounds on the heat kernel
  10. Hitting times and volume bounds
  11. Relaxation times and mixing times
  12. Off-diagonal heat kernel bounds
  13. Exponential approximation of stationary hitting times
  14. Convergence of the uncovered set
  15. Strategy
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(\Gamma)$ be a collection of finite (connected) vertex-transitive graphs of fixed degree, and let $\chi$ be a standard Gumbel variable, i.e. ${\mathbb P} ( \chi \le s) = e^{- e^{-s}}$ for $s \in {\mathbb R}$. Then if and only if

Theorems & Definitions (161)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3: Thin tori
  • Remark 1.4
  • Example 1.5: Heisenberg group
  • Example 1.6
  • Example 1.7: Product of a cycle and a Ramanujan Cayley graph
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 151 more