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A Combinatorial Characterization of Constant Mixing Time

Lap Chi Lau, Raymond Liu

TL;DR

The paper asks when graphs can achieve constant mixing time through purely combinatorial properties rather than spectral radius alone. It introduces the alpha-bipartite density and delta-small-set variants as central conditions and proves sharp upper bounds on mixing time (both τ_{1/n} and τ_{1/3}) under these conditions, along with a matching lower bound arising from dense bipartite structures. A key methodological advance is deriving 2-norm contraction and translating it into variation-distance bounds without full spectral arguments. The authors also show that traditional expansion notions (edge conductance, small-set vertex expansion) are insufficient for constant mixing time by presenting counterexamples with embedded dense bipartite cores.

Abstract

Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set bipartite density condition, which is weaker than having near-optimal spectral radius and is stronger than having near-optimal small-set vertex expansion.

A Combinatorial Characterization of Constant Mixing Time

TL;DR

The paper asks when graphs can achieve constant mixing time through purely combinatorial properties rather than spectral radius alone. It introduces the alpha-bipartite density and delta-small-set variants as central conditions and proves sharp upper bounds on mixing time (both τ_{1/n} and τ_{1/3}) under these conditions, along with a matching lower bound arising from dense bipartite structures. A key methodological advance is deriving 2-norm contraction and translating it into variation-distance bounds without full spectral arguments. The authors also show that traditional expansion notions (edge conductance, small-set vertex expansion) are insufficient for constant mixing time by presenting counterexamples with embedded dense bipartite cores.

Abstract

Classical spectral graph theory characterizes graphs with logarithmic mixing time. In this work, we present a combinatorial characterization of graphs with constant mixing time. The combinatorial characterization is based on the small-set bipartite density condition, which is weaker than having near-optimal spectral radius and is stronger than having near-optimal small-set vertex expansion.

Paper Structure

This paper contains 8 sections, 7 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Upper Bounds on Mixing Time
  5. Bounding Mixing Time via 2-Norm
  6. Proof of \ref{['t:VD']}
  7. Lower Bounds on Mixing Time
  8. Expanders and Small-Set Vertex Expanders

Key Result

Theorem 1.2

Let $G=(V,E)$ be a $d$-regular graph with $d = n^\xi$ for some constant $\xi > 0$. If $G$ satisfies the $\alpha$-bipartite density condition for $\alpha\lesssim d/(\log d)^2$, then In particular, if $\alpha=d^{1-c}$ for some constant $c\in (0,\frac{1}{2}]$, this implies constant mixing time such that

Theorems & Definitions (21)

  • Definition 1.1: $\alpha$-Bipartite Density
  • Theorem 1.2: Upper Bounding Mixing Time by Bipartite Density
  • Theorem 1.3: Upper Bounding Mixing Time by Small-Set Bipartite Density
  • Theorem 1.4: Lower Bounding Mixing Time by Bipartite Density
  • Theorem 3.1: Upper Bounding Variation Distance by Small-Set Bipartite Density
  • proof : Proof of \ref{['t:weak-mixing-time']}
  • proof : Proof of \ref{['t:upper-bound']}
  • Proposition 3.2: $2$-Norm Decrease
  • proof
  • Lemma 3.3
  • ...and 11 more