PreprintCC BY 4.0arXiv
Limit law for Brownian cover time of the two dimensional torus
Abstract
Let 𝒞ε denote the time it takes a Brownian motion to come within distance ε of every point in the two dimensional torus 𝕋2 of unit area. We prove that for an explicit non-random centering mε, the variables {√(𝒞ε) − mε } converge in distribution as ε → 0, to a randomly shifted Gumbel law.
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BibTeX
@misc{dembo2026limit,
title = {{Limit law for Brownian cover time of the two dimensional torus}},
author = {Amir Dembo and Jay Rosen and Ofer Zeitouni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03555},
eprint = {2610.03555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03555},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Limit law for Brownian cover time of the two dimensional torus
AU - Amir Dembo
AU - Jay Rosen
AU - Ofer Zeitouni
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.03555
UR - https://arxiv.org/abs/2610.03555
AB - Let 𝒞_ε denote the time it takes a Brownian motion to come within distance ε of every point in the two dimensional torus 𝕋^2 of unit area. We prove that for an explicit non-random centering m_ε, the variables {√(𝒞_ε) − m_ε } converge in distribution as ε → 0, to a randomly shifted Gumbel law.
ER -