Mathematical Physics Seminar, University of Arizona, April 19, 2000
Title: "A Limit Theorem for Subcritical Percolation"
Speaker: Martin Bazant (MIT)
Abstract:
Beginning with certain scaling assumptions related to the recent results
of Borgs, Chayes, Kesten and Spencer, it is shown that the probability
distribution of the largest-cluster size in subcritical percolation on a
finite lattice of size N converges to the Fisher-Tippett (or Gumbel)
distribution exp(-exp(-z)) in a certain weak sense as N -> oo. As a
corollary, it is shown that the variance of the largest-cluster size is
bounded. The proof uses ``renormalization group'' ideas adapted from the
statistical theory of extremes with correlations controlled by the FKG
inequality.
Note: this lecture addresses certain "gory details" behind the general
ideas to be presented in the Applied Mathematics Colloquium on April 21.