Scaling Limit for the Random Walk on the Largest Connected Component of the Critical Random Graph

Abstract

A scaling limit for the simple random walk on the largest connected component of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ in the critical window, $p=n^{-1}+\lambda n^{-4/3}$, is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.