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On large deviations for the cover time of two-dimensional torus. (English) Zbl 1294.60066

Summary: 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 \(\operatorname{P}[\mathcal{T}_n\leq \frac{4}{\pi}\gamma n^2\ln^2 n]=\exp(-n^{2(1-\sqrt{\gamma})+o(1)})\) 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.

MSC:

60G50 Sums of independent random variables; random walks
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F10 Large deviations
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