School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

University of California, Davis

In probability theory and statistical physics, bond percolation on the square lattice is

Abstract:

a natural model for a porous random medium and provides one of the simplest examples of a

phase transition. At criticality, it has remarkable long-range correlation properties (many of them

still conjectural). This talk will focus on "connectivity patterns" associated with a percolation

configuration, which are random noncrossing matchings of the integers that encode connectivity

information. Mysteriously, the probabilities for interesting events on connectivity patterns are

dyadic rational numbers such as 3/8, 59/1024 and 69693/2^21. This "rationality phenomenon"

can be proved in a few cases but remains mostly conjectural. I will describe the background

and history of the model as well as new results that shed light on the problem by reducing the

proof of the rationality phenomenon to a family of purely algebraic identities about coefficients

of certain multivariate polynomials.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006