Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tom Hutchcroft, Isaac M. Lopez

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.

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

Abstract

We prove an inequality relating the isoperimetric profile of a graph to the decay of the random walk total variation distance . 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 and information , which may be of independent interest.

Paper Structure

This paper contains 4 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Consequences for graphs of non-negative Ollivier-Ricci curvature
  3. Upper tail estimates for the random walk displacement and information
  4. Construction of sets of small boundary

Key Result

Theorem 1.1

For each $M<\infty$, there exists a finite constant $C(M)$ such that the following holds. Let $G=(V,E)$ be an amenable connected graph with degrees bounded by $M$. For each $n\geq 1$, there exists a set $W_n \subseteq V$ with diameter and volume satisfying and whose boundary-to-volume ratio is bounded by $4\mathrm{TV}_n$: $|\partial W_n|/\sum_{w\in W}\deg(w)\leq 4 \mathrm{TV}_n$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Example 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof : Proof of \ref{['lem:Green_metric_upper_tail']}
  • Lemma 2.4
  • ...and 13 more