학술논문
Tightness for the Cover Time of the two dimensional sphere
Document Type
Working Paper
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Abstract
Let $C^*_{\epsilon,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $\epsilon$. We prove that $$\sqrt{C^{*}_{\epsilon,S^2} } -\sqrt{\frac{2A_{S^2}}{\pi}}(\log \epsilon^{-1}-\frac14\log\log \epsilon^{-1})$$ is tight, where $A_{S^2}=4\pi$ denotes the Riemannian area of $S^2$.
Comment: Third 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
Comment: Third 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