On large deviations for the cover time of two-dimensional torus

Abstract

Let ${\mathcal{T}}_n$ be the cover time of two-dimensional discrete torus ${\mathbb{Z}}_n^2={\mathbb{Z}}^2/n{\mathbb{Z}}^2$. We prove that $\mathbb{P}[{\mathcal{T}}_n\le \frac{4}{\pi }\gamma n^2{\ln }^{2}n]=\exp (-n^{2(1-\sqrt{\gamma })+o(1)})$ for $\gamma \in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.

25 pages, 5 figures

References in corpus (4)

• Soft local times and decoupling of random interlacements
• Tightness for a family of recursion equations
• Late points for random walks in two dimensions
• Extremal geometry of a Brownian porous medium

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• An improved decoupling inequality for random interlacements