A note on the Poincar\'e and Cheeger inequalities for simple random walk on a connected graph

TL;DR

This paper examines the Poincaré and Cheeger inequalities for simple random walks on connected graphs, providing examples, conditions, and path choices that influence which bound is tighter.

Contribution

It offers insights into when the Poincaré bound outperforms the Cheeger bound and demonstrates that suitable path choices can ensure this superiority.

Findings

An example where Cheeger bound is better than Poincaré

Conditions under which Poincaré bound is superior

Existence of path choices that favor Poincaré bound

Abstract

In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincar\'e and Cheeger bounds, and they raised the question as to whether the Poincar\'e bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincar\'e, establish some sufficient conditions on the canonical paths for the Poincar\'e bound to triumph, and show that there is always a choice of paths for which this happens.