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arXiv · 1611.08549

On the critical probability in percolation

Svante Janson·Lutz Warnke
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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.

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Svante Janson, Lutz Warnke. 2016-11-25. On the critical probability in percolation. https://doi.org/10.1214/17-ejp52

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