Stationary hitting times on vertex-transitive graphs

Nathanaël Berestycki; Jonathan Hermon; Lucas Teyssier

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arXiv · 2601.03864

Stationary hitting times on vertex-transitive graphs

Nathanaël Berestycki·Jonathan Hermon·Lucas Teyssier

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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.

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Nathanaël Berestycki, Jonathan Hermon, Lucas Teyssier. 2026-01-07. Stationary hitting times on vertex-transitive graphs. https://arxiv.org/abs/2601.03864

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