A Central Limit Theorem for the Mean Starting Hitting Time for a Random Walk on a Random Graph

Abstract

We consider simple random walk on a realization of an Erdős–Rényi graph with n vertices and edge probability p_n. We assume that n p^2_n/(log mathrm{n})^{16 xi } rightarrow infty for some xi >1 defined below. This in particular implies that the graph is asymptotically almost surely (a.a.s.) connected. We show a central limit theorem for the average starting hitting time, i.e., the expected time it takes the random walker on average to first hit a vertex j when starting in a fixed vertex i. The average is taken with respect to pi _j, the invariant measure of the random walk.