  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.
