The moving particle lemma for the exclusion process on a weighted graph

TL;DR

This paper establishes a version of the moving particle lemma for the exclusion process on finite weighted graphs, extending its applicability to non-translationally invariant graphs like fractals.

Contribution

It introduces a new version of the moving particle lemma for exclusion processes on arbitrary finite weighted graphs, based on the octopus inequality.

Findings

Proves the moving particle lemma for general weighted graphs.

Shows the lemma's potential optimality based on related spectral gap results.

Applies the lemma to demonstrate local ergodicity in complex graph structures.

Abstract

We prove a version of the moving particle lemma for the exclusion process on any finite weighted graph, based on the octopus inequality of Caputo, Liggett, and Richthammer. In light of their proof of Aldous' spectral gap conjecture, we conjecture that our moving particle lemma is optimal in general. Our result can be applied to graphs which lack translational invariance, including, but not limited to, fractal graphs. An application of our result is the proof of local ergodicity for the exclusion process on a class of weighted graphs, the details of which are reported in a follow-up paper [arXiv:1705.10290].