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Stationary hitting times on vertex-transitive graphs

Nathanaël Berestycki, Jonathan Hermon, Lucas Teyssier

TL;DR

The paper advances the understanding of stationary hitting times for reversible Markov chains by refining the Aldous–Brown exponential approximation through a quasi-stationary framework, and then specializes these refinements to vertex-transitive graphs where bounds depend on volume growth and diameter. The authors establish a sharper bound involving the quasi-stationary parameter $R_M$ and derive a mixture-of-exponentials representation for hitting times, enabling precise tail and moment estimates. A general, robust bound for reversible chains is developed that does not require irreducibility of the complement set, using an auxiliary time and hitting-before-relaxation-time arguments. For vertex-transitive graphs, the results are sharpened further via spectral decompositions of killed chains and integral bounds, yielding bounds that scale with graph growth and diameter, with particularly strong improvements in low-dimensional torus-like settings; these bounds support the companion work on the fluctuations of the cover time on vertex-transitive graphs.

Abstract

We prove a refined version of the Aldous and Brown's exponential approximation of stationary hitting times. These are valid for all reversible Markov chains. We then specialise our estimates for vertex-transitive graphs, where we obtain improved bounds which depend on the growth of the graphs. The most delicate cases are when the diameter is comparable to that of low-dimensional tori. In particular, in "dimensions" less than four (up to logarithmic factors) our error terms are the square of those of Aldous and Brown. These improved bounds play a crucial role in the companion work arXiv:2202.02255 characterising the fluctuations of the cover time on vertex-transitive graphs.

Stationary hitting times on vertex-transitive graphs

TL;DR

The paper advances the understanding of stationary hitting times for reversible Markov chains by refining the Aldous–Brown exponential approximation through a quasi-stationary framework, and then specializes these refinements to vertex-transitive graphs where bounds depend on volume growth and diameter. The authors establish a sharper bound involving the quasi-stationary parameter and derive a mixture-of-exponentials representation for hitting times, enabling precise tail and moment estimates. A general, robust bound for reversible chains is developed that does not require irreducibility of the complement set, using an auxiliary time and hitting-before-relaxation-time arguments. For vertex-transitive graphs, the results are sharpened further via spectral decompositions of killed chains and integral bounds, yielding bounds that scale with graph growth and diameter, with particularly strong improvements in low-dimensional torus-like settings; these bounds support the companion work on the fluctuations of the cover time on vertex-transitive graphs.

Abstract

We prove a refined version of the Aldous and Brown's exponential approximation of stationary hitting times. These are valid for all reversible Markov chains. We then specialise our estimates for vertex-transitive graphs, where we obtain improved bounds which depend on the growth of the graphs. The most delicate cases are when the diameter is comparable to that of low-dimensional tori. In particular, in "dimensions" less than four (up to logarithmic factors) our error terms are the square of those of Aldous and Brown. These improved bounds play a crucial role in the companion work arXiv:2202.02255 characterising the fluctuations of the cover time on vertex-transitive graphs.
Paper Structure (21 sections, 21 theorems, 105 equations, 1 table)

This paper contains 21 sections, 21 theorems, 105 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Quasi-stationary distributions and hitting times: a simple approach
  4. Killed chains and quasi-stationary distributions
  5. A theorem of Aldous and Brown, remastered
  6. Hitting times and mixtures of exponentials
  7. Proof of Theorem \ref{['thm:ABrefinementintro']}
  8. A simple deduction of (a variant of) Theorem \ref{['T:ABintro']} from Aldous and Brown
  9. Improved lower bound in the Aldous--Brown approximation
  10. Interlacing relaxation times
  11. Sharpness of Theorem \ref{['thm:ABrefinementintro']}.
  12. A general bound for reversible Markov chains
  13. Hitting times when the walk starts at a given vertex
  14. General bound with an auxiliary time
  15. A general bound via hitting before the relaxation time
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(X_t)_{t\geq 0}$ be an irreducible reversible continuous-time Markov chain on a finite state space $S$, and denote its stationary distribution by $\pi$. Let $\eset \ne B\subsetneq S$ be such that the restriction of the chain to $B$ is irreducible, and set $A:= B^c$. We have that for all $t\geq 0$, and

Theorems & Definitions (49)

  • Theorem 1.1: AldousBrown1992
  • Theorem 1.2: A refinement of the AB Theorem
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5: A specialised quasi-stationary approximation for transitive graphs
  • Remark 1.6: Dependence on the degree
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 39 more