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

### David A. Croydon

Warwick University, Coventry, UK

## 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.

## Cite this article

David A. Croydon, Scaling Limit for the Random Walk on the Largest Connected Component of the Critical Random Graph. Publ. Res. Inst. Math. Sci. 48 (2012), no. 2, pp. 279–338

DOI 10.2977/PRIMS/70