On the critical probability in percolation

TL;DR

This paper investigates the critical probability in percolation on finite transitive graphs, confirming a proposed characterization using the susceptibility's logarithmic derivative on Erdős-Rényi graphs and clarifying the location of its maximum.

Contribution

It validates Nachmias and Peres's characterization of critical probability for Erdős-Rényi graphs and refutes a previous speculation about the maximizer's location.

Findings

The logarithmic derivative's maximizer is inside the critical window p=1/n+Θ(n^{-4/3}).

The inverse of the maximum value matches the critical window's width.

The maximizer is not at p=1/n or p=1/(n-1), contradicting earlier speculation.

Abstract

For percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erd\H{o}s-R\'enyi random graph G_{n,p}, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window p=1/n+\Theta(n^{-4/3}), and (ii) the inverse of its maximum value coincides with the \Theta(n^{-4/3})-width of the critical window. We also prove that the maximizer is not located at p=1/n or p=1/(n-1), refuting a speculation of Peres.