A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier-Ricci curvature

TL;DR

This paper establishes a link between isoperimetric properties and the decay of total variation in random walks on graphs with non-negative Ollivier-Ricci curvature, with implications for graph expansion and new probabilistic estimates.

Contribution

It introduces a novel inequality connecting isoperimetry and total variation decay, and applies it to characterize non-expanding graphs with non-negative Ollivier-Ricci curvature.

Findings

Bounded-degree graphs with non-negative Ollivier-Ricci curvature are not expanders.

Universal tail estimates for random walk displacement and transition probabilities.

A quantitative version of Salez's theorem on graph expansion.

Abstract

We prove an inequality relating the isoperimetric profile of a graph to the decay of the random walk total variation distance $\sup_{x\sim y} ||P^n(x,\cdot)-P^n(y,\cdot)||_{\mathrm{TV}}$. This inequality implies a quantitative version of a theorem of Salez (GAFA 2022) stating that bounded-degree graphs of non-negative Ollivier-Ricci curvature cannot be expanders. Along the way, we prove universal upper-tail estimates for the random walk displacement $d(X_0,X_n)$ and information $-\log P^n(X_0,X_n)$, which may be of independent interest.