Scaling limit for the random walk on the largest connected component of the critical random graph

TL;DR

This paper establishes a scaling limit for the simple random walk on the largest component of a critical Erdős-Rényi random graph, connecting it to a diffusion process on the continuum random tree.

Contribution

It introduces a new diffusion limit for the random walk on the critical random graph's largest component using resistance form techniques.

Findings

The limiting diffusion satisfies Brownian motion heat kernel asymptotics.

The approach uses resistance forms to construct the diffusion.

Connects discrete random walks to continuum random tree diffusions.

Abstract

A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window 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.