Tel-Aviv University
School of Mathematical Sciences

Department Colloquium

Monday, March 18, 2013

Schreiber 006, 12:15

Dan Romik 

University of California, Davis

Random noncrossing matchings arising in percolation theory

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