Abstract:
In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of the partition function and proved a theorem which locates the zeros of the partition function of the "Ising model with free or periodic boundary condition" on the unit circle. There are several generalizations of the Lee-Yang theorem (by e.g. Ruelle, Newman, Heilmann-Lieb and Lieb-Sokal) which have found applications in combinatorics and complex analysis. Common for the proofs of these generalizations is the use of linear operators on multivariate polynomials preserving the property of being non-vanishing whenever all variables are in a prescribed set. We characterize such operators for open disks and half-planes, thus providing a framework for dealing with Lee-Yang problems and similar problems in combinatorics, statistical mechanics and other areas.

This is joint work with Julius Borcea.