A note on the local weak limit of a sequence of expander graphs

TL;DR

This paper proves that the local weak limit of finite expander graphs with bounded degree is an ergodic unimodular random graph, leading to new insights on percolation properties and improving previous theorems.

Contribution

It establishes the ergodic unimodular nature of the local weak limit for bounded degree expanders and removes a key assumption in existing percolation locality results.

Findings

Limit is an ergodic unimodular random graph

Critical percolation probability is constant almost surely

Improves previous percolation locality theorems

Abstract

We show that the local weak limit of a sequence of finite expander graphs with uniformly bounded degree is an ergodic (or extremal) unimodular random graph. In particular, the critical probability of percolation of the limiting random graph is constant almost surely. As a corollary, we obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) on locality of percolation probability for finite expander graphs with uniformly bounded degree where we can drop the assumption that the limit is a single rooted graph.