The mixing time of the giant component of a random graph

TL;DR

This paper proves that the mixing time of a simple random walk on the giant component of supercritical Erdős-Rényi graphs is on the order of log^2 n, using a novel structure theorem for these graphs.

Contribution

It introduces the concept of decorated expanders and establishes their role in analyzing the mixing time of the giant component.

Findings

Mixing time is log^2 n for the giant component.

Giant components are structured as decorated expanders.

The structure theorem provides new insights into graph expansion properties.

Abstract

We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" - an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations.