Harvard University
Applied Mechanics Colloquium
Wednesday May 23, 2001
Percolation and Renormalization
Martin Z. Bazant
Dept. of Mathematics, MIT
Abstract:
"Percolation" is the standard mathematical model for connectivity in
random systems. It finds applications in many areas of science and
engineering, such as secondary oil recovery, polymer gelation,
epidemic spreading, and the stability of computer networks. The key
feature of the model, which has been studied extensively, is its
"phase transition" in the infinite-system limit: Statistically
speaking, above a critical bond concentration, there is an infinite
cluster of connected sites, and below it there is not. Of course,
real systems are finite, so an important quantity for applications is
the size of the largest cluster, a random variable whose distribution
is not yet fully understood. In this work, a simple
renormalization-group technique is developed to approximate the
largest-cluster size distribution analytically, and the results are
compared with large-scale numerical simulations. The theory builds on
existing renormalization-group methods for percolation (which mainly
predict critical exponents) as well as classical limit theorems in
probability to quantify finite-size effects both at and away from
criticality.
4 PM in 209 Pierce Hall
(Coffee after the Colloquium, Brooks Room, Pierce Hall 213)
Host: Prof. E. Kaxiras
(kaxiras@kriti.harvard.edu)