Hard-Constraint Spin Systems: Some Results and Questions

Speaker

David Galvin
University of Notre Dame
http://math.nd.edu/people/faculty/david-galvin/

Description

In last week's Menger Lecture, Peter Winkler discussed the use of combinatorial methods in Statistical Physics. My talk will continue this theme, focusing on hard-constraint spin systems.

A hard-constraint spin system is one in which space, modeled by some lattice graph, is occupied by particles that each take one of a given set of ``spins''. There are some ``hard constraints'' in the system: certain pairs of spins are forbidden from appearing on neighboring vertices of the lattice. Each spin is given a weight, measuring how likely it is to appear in a configuration. This gives rise to a natural probability distribution on the space of configurations.

One may ask many interesting questions about a given model: How many configurations are there? What does a typical one look like? Does the model exhibit long-range correlation (that is, can revealing the appearance of a randomly chosen configuration at one side of the lattice give non-trivial information about the appearance at another distant site)?

In this talk, I'll discuss some of these questions for a few of the most-studied hard-constraint spin systems, such as the hard-core and zero-temperature Potts model, and mention some intriguing open questions.