Locality of critical percolation on expanding graph sequences

TL;DR

This paper proves that under certain expansion conditions, the critical percolation threshold on finite graphs converges to that of an infinite limit graph with finite expected degree, extending previous results to unbounded degree scenarios.

Contribution

It extends the locality of critical percolation results to graphs with unbounded degrees, requiring only finite expected root degree of the limit graph, relaxing previous uniform degree bounds.

Findings

Critical percolation thresholds coincide under expansion conditions.

Finite expected root degree suffices for convergence of critical probabilities.

Method uses pruning and second moment techniques to handle unbounded degrees.

Abstract

We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probability for the emergence of an infinite component on $G$ coincide with the critical probability for the emergence of a linear-sized component on $G_n$? In this short article we give a positive answer provided the graphs $G_n$ satisfy an expansion condition, and the limiting graph $G$ has finite expected root degree. The main result of Benjamini, Nachmias, and Peres (2011), where this question was first formulated, showed the result assuming the $G_n$ satisfy a uniform degree bound and uniform expansion condition, and converge to a deterministic limit $G$. Later work of Sarkar (2021) extended the result to allow for a random limit $G$, but still required a uniform degree bound and uniform expansion for $G_n$. Our result replaces the degree bound on $G_n$ with the (milder) requirement that $G$ must have finite expected root degree. Our proof is a modification of the previous results, using a pruning procedure and the second moment method to control unbounded degrees.