Applied Mathematics Colloquium
University of Arizona, April 21, 2000
Title: Renormalization Groups and Limit Theorems in Percolation
Speaker: M. Z. Bazant, MIT
Abstract:
Percolation is used in almost every area of science as the simplest
model for spatial disorder. The model amounts to randomly coloring
each of N sites in a graph (e.g. the hypercubic lattice) either black
or white with probability p and then identifying "clusters" of
adjacent black sites. This very simple coin-flipping game has a rather
nontrivial "phase transition" in the limit N -> oo which is apparent
in the expected size of the largest cluster S: For p less than some
critical value p_c, the largest cluster is negligably small, E[S] =
O(log N), while for p > p_c it occupies a nonzero fraction of the
system E[S] = O(N). (At p = p_c it is a fractal.) Although this
scaling of the mean is well-known, however, the scaling of higher
moments and the limiting shapes of the probability distribution of S
are not as well understood.
Here, these quantities are derived analytically and checked
numerically on square lattices of up to 30 million sites. A
"renormalization-group" picture of percolation is proposed that draws
on classical ideas from probability theory as well as the modern
theory of critical phenomena.