Electronic Journal of Probability

On the critical probability in percolation

Svante Janson and Lutz Warnke

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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 $p=1/n+\Theta (n^{-4/3})$, 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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 1, 25 pp.

Received: 25 November 2016
Accepted: 27 March 2017
First available in Project Euclid: 12 January 2018

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60C05: Combinatorial probability 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

random graph percolation phase transition critical probability critical window

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Janson, Svante; Warnke, Lutz. On the critical probability in percolation. Electron. J. Probab. 23 (2018), paper no. 1, 25 pp. doi:10.1214/17-EJP52. https://projecteuclid.org/euclid.ejp/1515726029

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