The subleading order of two dimensional cover times
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. \textit{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 \(\varepsilon \to 0\).
scientific article