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

References in corpus (5)

• Critical random graphs: Diameter and mixing time
• Random subgraphs of finite graphs: I. The scaling window under the triangle condition
• Susceptibility in subcritical random graphs
• The phase transition in bounded-size Achlioptas processes
• A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics

Cited by in corpus (6)

• The phase transition in bounded-size Achlioptas processes
• Preferential attachment without vertex growth: emergence of the giant component
• Expansion of percolation critical points for Hamming graphs
• Critical Percolation on Random Networks with Prescribed Degrees
• A branching process with deletions and mergers that matches the threshold for hypercube percolation
• The giant in random graphs is almost local