Tightness for the Cover Time of the two dimensional sphere

Understand

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 $S^2$.