Combinatorial Methods for Statistical Physics Models

Special Topics Course, Winter 1999

Many fundamental questions in statistical mechanics are inherently
combinatorial. We will introduce some basic models and examine
natural physical questions from a combinatorial perspective, including
the Ising model, the Potts model, monomer-dimer systems, self-avoiding
walks and percolation theory. Some highlights of the course will
include the Peierls argument (for the Ising model), Kasteleyn's theorem
(for dimer systems), the FKG inequality, and Reimer's new generalization
of the BK inequality (both used in percolation theory).

This course is intended to be introductory and no background in
statistical mechanics is required.

Time: Tuesday and Thursday 2-3:30 (tentatively).

Prerequisites: Some background in combinatorics and probability.

Text: None.

Some references:

- "Percolation," by Geoffrey Grimmett (Springer-Verlag)
- "Remarks on the FKG Inequalities," Richard Holley (in Comm. Math. Phys 36, 227-231, 1974)
- "The Van den Berg-Kesten-Reimer Inequality: A Review" (by C. Borgs, J. Chayes and D. Randall)
- "Gibbs Measures and Phase Transitions," by Hans-Otto Georgii (de Gruyter Studies in Mathematics)
- "Matching Theory," by L. Lovasz and M. D. Plummer

Lecture Notes: