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The Analogue of Aldous' spectral gap conjecture for the generalized exclusion process

Kazuna Kanegae, Hidetada Wachi

TL;DR

The paper proves an Aldous-type spectral-gap analogue for the normal generalized exclusion process (GEP). By viewing GEP as a quotient of the interchange process on an extended graph under a group action, it leverages Aldous' equality between the interchange process and the random walk, together with results for expanded graphs, to show that the GEP spectral gap equals $k$ times the random-walk spectral gap on the base graph: $\lambda_1^{\mathrm{GEP}_{k,l}(X)}=k\lambda_1^{\mathrm{RW}}(X)$. This implies the normal GEP’s spectral gap is independent of the particle count $l$, providing a new structural route to compute relaxation times via simpler random-walk analyses. The approach unifies quotient-graph methods with classical Markov-process theory and extends Aldous-type correspondences to generalized exclusion dynamics on complete graphs.

Abstract

Caputo, Ligget, and Richthammer proved Aldous' spectral gap conjecture, which asserts that the spectral gaps of a random walk and an interchange process on the common weighted graph are equal. In this paper, we will prove an analogue of Aldous' spectral gap conjecture for generalized exclusion processes, which explicitly describes the spectral gap of a generalized exclusion process by the spectral gap of a random walk.

The Analogue of Aldous' spectral gap conjecture for the generalized exclusion process

TL;DR

The paper proves an Aldous-type spectral-gap analogue for the normal generalized exclusion process (GEP). By viewing GEP as a quotient of the interchange process on an extended graph under a group action, it leverages Aldous' equality between the interchange process and the random walk, together with results for expanded graphs, to show that the GEP spectral gap equals times the random-walk spectral gap on the base graph: . This implies the normal GEP’s spectral gap is independent of the particle count , providing a new structural route to compute relaxation times via simpler random-walk analyses. The approach unifies quotient-graph methods with classical Markov-process theory and extends Aldous-type correspondences to generalized exclusion dynamics on complete graphs.

Abstract

Caputo, Ligget, and Richthammer proved Aldous' spectral gap conjecture, which asserts that the spectral gaps of a random walk and an interchange process on the common weighted graph are equal. In this paper, we will prove an analogue of Aldous' spectral gap conjecture for generalized exclusion processes, which explicitly describes the spectral gap of a generalized exclusion process by the spectral gap of a random walk.
Paper Structure (10 sections, 16 theorems, 72 equations, 5 figures)

This paper contains 10 sections, 16 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Strategy
  4. Spectral gaps of complete weighted directed graphs
  5. Complete weighted directed graph
  6. Group action and quotient
  7. Interchange Process and the Generalized Exclusion Process
  8. Interchange Process
  9. Generalized exclusion process
  10. Main theorem

Key Result

Theorem 1.3

Let $\lambda_1^\mathrm{RW}$ be the spectral gap of a random walk on $X=(V,r)$. We have In particular, the spectral gap of the normal generalized exclusion process is independent of the number of particles $l$.

Figures (5)

  • Figure 1: GEP on $V=\{1,2,3,4,5\}$ with $l=2$, $k=2$. This picture shows the underlying graph and the transition of the jump on edge $(1,2)$.
  • Figure 2: The left column represents the morphism of \ref{['example:morphism1']}, and the right column represents the morphism of \ref{['example:morphism2']}.
  • Figure 3: IP on $V=\{1,2,3,4,5\}$. This picture shows the underlying graph and a transition of interchanging particles on vertices $1$ and $5$.
  • Figure 4: GEP on $V=\{1,2,3,4,5\}$ with $l=2$, $k=2$. This picture shows an underlying graph and a transition of the jump on edge $(1,2)$.
  • Figure 5: The lower part represents the particle jumps in GEP with $k=2$ and $l=2$, while the upper part illustrates an example of corresponding state transitions in SEP on the expanded graph.

Theorems & Definitions (45)

  • Definition 1.2
  • Theorem 1.3: \ref{['theorem-spectral-gap-gep']}
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6: CLR10*Theorem 1.1
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • ...and 35 more