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

Francis Comets (Université Paris-7)
Christophe Gallesco (UNICAMP)
Serguei Popov (UNICAMP)
Marina Vachkovskaia (UNICAMP)


Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}{\pi}\gamma n^2\ln^2 n]=\exp(-n^{2(1-\sqrt{\gamma})+o(1)})$ 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.

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Pages: 1-18

Publication Date: November 6, 2013

DOI: 10.1214/EJP.v18-2856


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