Special Seminar
Mathematical Sciences Theory Group 
Microsoft Research 
Microsoft Corporation, Redmond, WA

Title:   The Largest Cluster in Subcritical Percolation
                                                       
Speaker: Martin Z. Bazant  (Math, MIT)

Date:    March 10, 1999

Time:    10:30 - 12:00 

Abstract:

The asymptotic distribution of the largest cluster size in subcritical
percolation is derived using a "renormalization" argument borrowed from
the theory of extreme order statistics. Under very general conditions, the
cumulative distribution function approaches a universal shape given by the
well-known "extreme value distribution". The mean largest cluster size
grows logarithmically with system size, and the variance stays bounded on
the scale of the correlation size. These analytic predictions are
confirmed by Monte-Carlo simulations of 1D and 2D square lattices of up to
30 million sites (involving trillions of clusters), which also accurately
reveal finite-size scaling behavior and the cluster-size distribution.