Tightness for the Cover Time of the two dimensional sphere

Abstract

Let $C_{\epsilon ,S^2}^{\ast }$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $\epsilon$. We prove that $$\sqrt{C_{\epsilon ,S^2}^{\ast }}-\sqrt{\frac{2A_{S^2}}{\pi }}(\log {\epsilon }^{-1}-\frac{1}{4}\log \log {\epsilon }^{-1})$$ is tight, where $A_{S^2}=4\pi$ denotes the Riemannian area of $S^2$.

arXiv commentThird version deals only with the sphere, because the reduction from general manifold to the sphere in the second version contains a mistake. V5 corrects an error in the statement of Lemma 9.1, and its use in the first and second moment estimates (replacing the former erroneous estimates (4.56) and (4.87)). V6 corrects minor typos, and added details to the statement and proof of Lemma 9.2