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:

• January 5, 7: Basics of Percolation and the FKG Inequality
  
• January 12: The FKG Inequality (cont.)
  
• January 14, 19: The BK Inequality for increasing events
  
• January 21, 26: The general BK Inequality
  
• January 28, February 2: Percolation in Z^2
  
• February 4, 9, 11: Kasteleyn's algorithm
  
• February 18: The Peierl's argument for the Ising model
  
• February 23: The Swendsen-Wang Algorithm