Hitting times in the binomial random graph

TL;DR

This paper provides a simple, precise formula for the expected hitting time between vertices in a binomial random graph, extending previous results to a broader range of edge probabilities.

Contribution

It introduces a new formula for hitting times in G(n,p) graphs that depends only on basic structural properties, covering a wider range of p values.

Findings

Derived a formula for expected hitting times in G(n,p)

Extended previous results to broader p ranges

Improved understanding of random walk behavior in sparse graphs

Abstract

Fix $k\geq 2$, choose $\frac{\log n}{n^{(k-1)/k}}\leq p\leq 1-\Omega(\frac{\log^4 n}{n})$, and consider $G\sim G(n,p)$. For any pair of vertices $v,w\in V(G)$, we give a simple and precise formula for the expected number of steps that a random walk on $G$ starting at $w$ needs to first arrive at $v$. The formula only depends on basic structural properties of $G$. This improves and extends recent results of Ottolini and Steinerberger, as well as Ottolini, who considered this problem for constant as well as for mildly vanishing $p$.