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 \(\varepsilon \to 0\).
Reviewer: Oleg K. Zakusilo (Kyïv)
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 |
Keywords:
Brownian motion; torus; cover times; extremes; branching Brownian motion; second moment methodCitations:
Zbl 1068.60018References:
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