2022/02/04 by Nathanaël Berestycki, Berestycki, Nathanaël, Jonathan Hermon +3
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2202.02255
openalex publication_date 2022/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 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.