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

Sourav Sarkar
2021 Electronic Communications in Probability  
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
more » ... e limit is a single rooted graph. of length o(1), that is, for any c ∈ (0, 1), the property that the random subgraph of an expander G = (V, E) after Bernoulli bond percolation contains a giant component of size c|V |, has a sharp threshold; removing the regularity and high-girth assumptions in [2]. We first recall local weak convergence of bounded degree finite graphs. A rooted graph is denoted as
doi:10.1214/21-ecp402 fatcat:7hhwmgakafglzh7cxtwzukx4ay