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)