John Imbrie

Combinatorial Aspects of Mayer Expansions, Forest Formulas, and Grassmann Integrals

Abstract. A number of traditional tools of statistical mechanics and quantum field theory have been redeveloped and combined in recent years to provide new insights into the behavior of generating functions in stochastic geometry. I will survey these tools and apply them to a number of interesting examples.  The following is an outline of the course:
  1. Mayer expansions; combinatoric aspects of convergence.
  2. A forest-root formula for directed branched polymers.
  3. Applications of Grassmann integrals to statistical mechanics.
    a. The matrix-tree theorem and a supersymmetric forest-root formula.  Branched polymers and dimensional reduction.
    b. Combinatoric aspects of the self-avoiding walk.

Suggested reading

Some of the lectures and slides from the conference on Combinatorial Identities and their Applications in Statistical Mechanics, April 7-11, 2008 may be helpful, especially the talks of Brydges, Kotecky, and myself.  See

A. Bovier and M. Zahradnk (2000): A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Statist. Phys. 100, 765?78.

J. Z. Imbrie: "Dimensional Reduction for Isotropic and Directed Branched Polymers," In: Proceedings of the International Conference in Mathematical Physics, Lisbon, 2003.

David Brydges: "Self-Avoiding Walk and Functional Integration." Lectures sponsored by PIMS.