2022/12/12 by Ren, Michael, Sun, Nike
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2212.05943
We study the locality of critical percolation on finite graphs: let Gn 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 Gn? In this short article we give a positive answer provided the graphs Gn 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 Gn 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 Gn. Our result replaces the degree bound on Gn 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.