Continuum Seminar
Department of Mechanical Engineering, MIT
Monday, March 19, 2001, 3-4pm in 5-324
Title: Renormalization Groups and Limit Theorems in Percolation
Speaker: Martin Z. Bazant (Dept. of Mathematics, MIT)
Abstract:
Percolation is used in almost every area of science as a simple model
for the connectivity of a disordered system. The model amounts to
randomly coloring each of N sites in a graph (for example, 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 fully 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.