Convergence of mixing times for sequences of random walks on finite graphs

TL;DR

This paper provides conditions under which the mixing times of random walks on sequences of finite graphs converge, using Gromov-Hausdorff convergence, with applications to various graph models including Erdős-Rényi, fractal graphs, and trees.

Contribution

It establishes a general framework for the convergence of mixing times based on geometric and measure convergence, extending previous results to new graph classes.

Findings

Convergence of mixing times on the largest component of Erdős-Rényi graphs in the critical window.

Convergence results for fractal graphs, Galton-Watson trees, and high-dimensional random walk ranges.

Sharpened understanding of mixing time behavior in complex graph sequences.

Abstract

We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdos-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk.