A limit law for the cover time of the two-dimensional discrete torus
Abstract
We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on (ℤ/Nℤ)2, let TN denote its cover time. We prove that TN/((2/π)N2log N)−2log N+loglog N Longrightarrow G+log(κZ), where G is a standard Gumbel random variable, Z is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, G and Z are independent, and κ>0 is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.
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BibTeX
@misc{zhou2026limit,
title = {{A limit law for the cover time of the two-dimensional discrete torus}},
author = {Yechi Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01852},
eprint = {2610.01852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01852},
}
RIS
TY - UNPB
TI - A limit law for the cover time of the two-dimensional discrete torus
AU - Yechi Zhou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01852
UR - https://arxiv.org/abs/2610.01852
AB - We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on (ℤ/Nℤ)^2, let T_N denote its cover time. We prove that T_N/((2/π)N^2log N)−2log N+loglog N Longrightarrow G+log(κZ), where G is a standard Gumbel random variable, Z is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, G and Z are independent, and κ>0 is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.
ER -