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2018 On the critical probability in percolation
Svante Janson, Lutz Warnke
Electron. J. Probab. 23: 1-25 (2018). DOI: 10.1214/17-EJP52

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 $p=1/n+\Theta (n^{-4/3})$, 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.

Citation

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Svante Janson. Lutz Warnke. "On the critical probability in percolation." Electron. J. Probab. 23 1 - 25, 2018. https://doi.org/10.1214/17-EJP52

Information

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

zbMATH: 1387.05239
MathSciNet: MR3751076
Digital Object Identifier: 10.1214/17-EJP52

Subjects:
Primary: 05C80, 60C05, 60K35, 82B43

JOURNAL ARTICLE
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