Horizons in Complex Systems
Conference in Honor of Gene Stanley's 60th Brithday
Messina, Italy
Dec 5-8, 2001

Title: The Fractal Central Limit Theorem in Percolation

Speaker: Martin Z. Bazant (Dept. of Mathematics, MIT)

Abstract: 

Within the context of a renormalization-group (RG) theory for
percolation (building upon classical work of P. Reynolds, W. Klein,
H. E. Stanley from the 1970s), it is shown that the critical limiting
distribution of incipient infinite cluster masses obeys a "Fractal
Central Limit Theorem" (FCLT).  The FCLT describes mass fluctuations
in random unifractal sets. Unlike the normal CLT, however, which holds
in the supercritical phase and predicts a universal Gaussian "central
region" (at the scale of the standard deviation) with non-universal
tails, the FCLT predicts a non-universal central region (at the
scale of the mean) with universal stretched exponential tails.  The RG
theory can also describe the crossover between the critical (FCLT) and
supercritical (CLT) regimes. These predictions compare well with
numerical results for the distribution of the largest cluster mass in
2d percolation, as well as previous simulations of critical spanning
clusters in different geometries.  The general relevance of the FCLT
for critical phenomena is further evidenced by comparisons with exact
results for the Ising model and random graphs.