Jennifer Chayes

University of California at Los Angeles, visiting AT&T

DIMACS Center - Rutgers University
CoRE Building Lecture Hall - Busch Campus
Piscataway, New Jersey
Friday, October 4, 1996, at 2pm-3pm

"Birth of the Infinite Cluster: Finite-Size Scaling in Percolation"


A fundamental question in the study of phase transitions is the question of finite-size scaling: How does thermodynamic behavior emerge from the behavior of large, finite systems? In these two lectures, we address this question in systems with both discontinuous and continuous phase transitions. We review the necessary prerequisites from statistical mechanics.

Lecture 2 (joint work with C. Borgs, H. Kesten and J. Spencer): We address finite-size scaling in the context of systems with continuous phase transitions by considering bond percolation in a finite box. As a function of the size of the box, we determine the scaling window about the transition point in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: non-unique incipient infinite clusters which give rise to the infinite cluster. We prove our results under a set of scaling axioms which we expect to hold in all dimensions $d < 6$ and which we explicitly establish in two dimensions. Our results are finite-dimensional analogs of recent results on the so-called dominant component of the Erd\"os-R\'enyi mean-field random graph model.

Document last modified on September 27, 1996