# Local ergodicity in the exclusion process on an infinite weighted graph

## TL;DR

This paper proves a local ergodic theorem for the symmetric exclusion process on infinite weighted graphs, using resistance structures and harmonic analysis, applicable to fractal and random graphs, with implications for hydrodynamic limits.

## Contribution

It introduces an abstract local ergodic theorem for exclusion processes on infinite weighted graphs, extending to fractal and random graphs, and utilizes resistance structures and harmonic analysis techniques.

## Key findings

- Established local ergodic theorem for exclusion process on infinite graphs.
- Applied theorem to Sierpinski gasket for hydrodynamic limit analysis.
- Utilized resistance structure and harmonic analysis in proofs.

## Abstract

We establish an abstract local ergodic theorem, under suitable space-time scaling, for the (boundary-driven) symmetric exclusion process on an increasing sequence of balls covering an infinite weighted graph. The proofs are based on 1-block and 2-blocks estimates utilizing the resistance structure of the graph; the moving particle lemma established recently by the author; and discrete harmonic analysis. Our ergodic theorem applies to any infinite weighted graph upon which random walk is strongly recurrent in the sense of Barlow, Delmotte, and Telcs; these include many trees, fractal graphs, and random graphs arising from percolation. The main results of this paper are used to prove the joint density-current hydrodynamic limit of the boundary-driven exclusion process on the Sierpinski gasket, described in an upcoming paper with M. Hinz and A. Teplyaev.