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The subleading order of two dimensional cover times. (English) Zbl 1365.60071
The $$\varepsilon$$-cover time of the two-dimensional torus by Brownian motion is the time it takes for the process to come within distance $$\varepsilon > 0$$ from any point. A. Dembo et al. [Ann. Math. (2) 160, No. 2, 433–464 (2004; Zbl 1068.60018)] proved the law of large numbers: $$\frac{{{C_\varepsilon }}}{{\frac{1}{\pi }\log {\varepsilon ^{ - 1}}}} = (1 + o(1))\log{\varepsilon ^{ - 2}}$$, $$\varepsilon \to 0$$. The authors improve this result: $$\frac{{{C_\varepsilon }}}{{\frac{1}{\pi }\log {\varepsilon ^{ - 1}}}} = \log {\varepsilon ^{ - 2}} - (1 + o(1))\log\log{\varepsilon ^{ - 1}}$$ in probability, as $$\varepsilon \to 0$$.

##### MSC:
 60J65 Brownian motion 60G70 Extreme value theory; extremal stochastic processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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