Large-deviations/thermodynamic approach to percolation on the complete graph
Probability
2011-11-10 v3 Combinatorics
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.
Cite
@article{arxiv.math/0506255,
title = {Large-deviations/thermodynamic approach to percolation on the complete graph},
author = {Marek Biskup and Lincoln Chayes and S. Alex Smith},
journal= {arXiv preprint arXiv:math/0506255},
year = {2011}
}
Comments
16 pages, 1 figure; revised version accomodating literature remarks of the referees