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

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

2005-06-13

arxiv.org/abs/math/0506255v3

Random Structures & Algorithms 31 (2007), no. 3, 354-370

Mathematics

Probability

16 pages, 1 figure; revised version accomodating literature remarks of the referees

Scientific paper

10.1002/rsa.20169

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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