A Central Limit Theorem for the average target hitting time for a random walk on a random graph

TL;DR

This paper proves a Central Limit Theorem for the average target hitting time of a simple random walk on Erdős-Rényi graphs, assuming an eigenvector delocalization conjecture, revealing statistical properties of hitting times.

Contribution

It establishes a CLT for the average target hitting time on Erdős-Rényi graphs under a conjecture, advancing understanding of random walk behavior on random graphs.

Findings

Proves a CLT for average target hitting time

Conditional on an eigenvector delocalization conjecture

Applicable to asymptotically almost surely connected Erdős-Rényi graphs

Abstract

Consider a simple random walk on a realization of an Erd\H{o}s-R\'enyi graph. Assume that it is asymptotically almost surely (a.a.s.) connected. Conditional on an eigenvector delocalization conjecture, we prove a Central Limit Theorem (CLT) for the average target hitting time. By the latter we mean the expected time it takes the random walk on average to first hit a vertex $j$ when starting in a fixed vertex $i$. The average is taken with respect to $\pi_i$, the invariant measure of the random walk.