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

Abstract

AbstractWe consider simple random walk on a realization of an Erdős–Rényi graph with n vertices and edge probability $$p_n$$ p n . We assume that $$n p^2_n/(\log \mathrm{n})^{16 \xi } \rightarrow \infty $$ n p n 2 / ( log n ) 16 ξ → ∞ for some $$\xi >1$$ ξ > 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 $$\pi _j$$ π j , the invariant measure of the random walk.