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Large-deviations/thermodynamic approach to percolation on the complete graph

Marek Biskup·Lincoln Chayes·S. Alex Smith
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Abstract

We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are ``small.'' One consequence is an immediate derivation of the ``cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only ``small'' components.

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Marek Biskup, Lincoln Chayes, S. Alex Smith. 2005-06-13. Large-deviations/thermodynamic approach to percolation on the complete graph. https://doi.org/10.1002/rsa.20169

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