Large Deviations of Cover Time of Tori in Dimensions d≥ 3

2024/11/25 by Xinyi Li, Li, Xinyi, Shi, Jialu +2
Mathematics · #05C81 #60F10 #60G70 #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2411.16398

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider large deviations of the cover time of the discrete torus (ℤ/Nℤ)^d, d ≥ 3 by simple random walk. We prove a lower bound on the probability that the cover time is smaller than γ∈ (0,1) times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for γ∈ ((d+2)/(2d),1). The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Prévost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs.

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