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"
Abstract:
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.
