Concentration of hitting times in Erdős-Rényi graphs

Abstract

We consider Erdős-Rényi graphs $G\left(n,p\right)$ for $0<p<1$ fixed and $n\to \infty$ 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 Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and $H_{wv}=\left(1+o\left(1\right)\right)n$. Löwe-Terveer established a CLT for the Mean Starting Hitting Time suggesting $H_{wv}=n\pm O\left(\sqrt{n}\right)$. We prove the existence of a strong concentration phenomenon: $H_{wv}$ is given, up to a very small error of size $\lesssim {\left(\log n\right)}^{3∕2}∕\sqrt{n}$, by an explicit simple formula involving only the total number of edges $\mid E\mid$, the degree $\text{deg}\left(v\right)$ and the distance $d\left(v,w\right)$.