## 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

### Keywords:

soft local time; hitting time; simple random walk
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