On the critical probability in percolation

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 Erdős-Rényi 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+Θ(n^{-4/3}), and (ii) the inverse of its maximum value coincides with the Θ(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.

22 pages, 1 figure