Concentration of Hitting Times in Erd\"os-R\'enyi graphs

Abstract

We consider Erd\H{o}s-R\'enyi graphs $G(n,p)$ for $0 < p < 1$ fixed and $n \rightarrow \infty$ and study the expected number of steps, $H_{wv}$, that a random walk started in $w$ needs to first arrive in $v$. A natural guess is that an Erd\H{o}s-R\'enyi random graph is so homogeneous that it does not really distinguish between vertices and $H_{wv} = (1+o(1)) n$. L\"owe-Terveer established a CLT for the Mean Starting Hitting Time suggesting $H_{w v} = n \pm \mathcal{O}(\sqrt{n})$. We prove the existence of a strong concentration phenomenon: $H_{w v}$ is given, up to a very small error of size $\lesssim \sqrt{\log{n}}/\sqrt{n}$, by an explicit simple formula involving only the total number of edges $|E|$, the degree of $v$ and the distance $d(v,w)$.