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) in [4] 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.
Highlights
We show that the local weak limit of a sequence of finite expander graphs with uniformly bounded degree is an ergodic unimodular random graph
We obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) in [4] 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
Local weak convergence of a sequence of finite expander graphs has been studied in relation to locality of critical probability of percolation in [4]
Summary
Local weak convergence of a sequence of finite expander graphs has been studied in relation to locality of critical probability of percolation in [4]. 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. We obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) in [4] 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.
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