On the critical probability in percolation

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 Gn,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.