Concentration of hitting times in Erdős-Rényi graphs

We consider Erd & odblac;s-R & eacute;nyi graphs G(n,p) $G(n,p)$ for 0<p<1 $0\lt p\lt 1$ fixed and n ->infinity $n\to \infty $ and study the expected number of steps, Hwv ${H}_{wv}$, that a random walk started in w $w$ needs to first arrive in v $v$. A natural guess is that an Erd & odblac;s-R & eacute;nyi random graph is so homogeneous that it does not really distinguish between vertices and Hwv=(1+o(1))n ${H}_{wv}=(1+o(1))n$. L & ouml;we-Terveer established a CLT for the Mean Starting Hitting Time suggesting Hwv=n +/- O(n) ${H}_{wv}=n\pm {\mathscr{O}}(\sqrt{n})$. We prove the existence of a strong concentration phenomenon: Hwv ${H}_{wv}$ is given, up to a very small error of size less than or similar to(logn)3/2/n $\lesssim {(\mathrm{log}n)}<^>{3\unicode{x02215}2}\unicode{x02215}\sqrt{n}$, by an explicit simple formula involving only the total number of edges divided by E divided by $| E| $, the degree deg(v) $\text{deg}(v)$ and the distance d(v,w) $d(v,w)$.