Applied Mathematics Colloquium
Princeton University
Monday October 1, 2001
The Fractal Central Limit Theorem in Percolation
Martin Z. Bazant
Dept. of Mathematics, MIT
Abstract:
"Percolation" is the standard model for random connectivity in many
applications such as secondary oil recovery, polymer gelation, and
epidemic spreading. The key feature of the model is its "phase
transition" in the infinite-system limit: Above a critical bond
concentration, there is an infinite cluster of connected bonds, 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 poorly understood distribution is the
subject of this talk.
Away from the phase transition, this distribution obeys classical
limit theorems for independent random variables, namely the
Fisher-Tippett theorem for extremes (subcritical) and the Central
Limit Theorem for sums (supercritical). In the critical regime,
however, long-range correlations exist, and various unusual
distributions are observed. It is argued that these distributions can
be explained by a simple probabalistic model based on self-similar
random sums of random variables. The limiting behavior of these sums
is governed by a ``Fractal Central Limit Theorem'' which predicts a
non-universal central region (at the scale of the mean) and universal
stretched exponential tails. These predictions are in excellent
agreement with numerical simulations of critical percolation on the
square site lattice, as well as known properties of the Ising model
and random graphs.