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:
- Mayer expansions; combinatoric aspects of convergence.
- A forest-root formula for directed branched polymers.
- 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.