# 32 KAM Mathematical Colloquium

## Prof. JENNIFER CHAYES

### Microsoft

## BIRTH OF THE INFINITE CLUSTER: FINITE-SIZE SCALING IN PERCOLATION

April 7, 1998

Lecture Room S6, Charles University, Malostranske nam. 25,
Praha 1

10:30 AM

## Abstract

Percolation is the simplest and most widely studied model of a random
medium. Among the numerous applications of percolation are calculations
of the distribution of oil in a porous medium, determination of the
behavior of the diffusion coefficient in a turbulent plasma, and the
question of intractability in the satisfiability problem.
A fundamental feature of the percolation model is that it undergoes
a critical phase transition from a disordered phase to a phase with
long-range order and hence transport. Technically, the transition occurs
only in an infinite system. Beyond the transition point, the ordered phase
is characterized by the presence of an infinite cluster. The critical
regime has been studied extensively by both analytical and numerical
methods.
This work is a detailed study of the phase transition in percolation,
in particular of the question of finite-size scaling: Namely, how does
the critical transition behavior emerge from the behavior of large, finite
systems? Our results rigorously locate the proper window in which to
do critical computation and establish features of the phase transition.
No prior knowledge of percolation or phase transitions is assumed
in this talk.
(This is joint work with C. Borgs, H. Kesten and J. Spencer

This colloquium is organized by KAM jointly with DIMATIA and with the Center
for Theoretical Studies (Charles University)