Canonical and grand canonical partition functions of Dyson gases as tau-functions of integrable hierarchies and their fermionic realization

摘要

The partition function for a canonical ensemble of 2D Coulomb charges in abackground potential (the Dyson gas) is realized as a vacuum expectation valueof a group-like element constructed in terms of free fermionic operators. Thisrepresentation provides an explicit identification of the partition functionwith a tau-function of the 2D Toda lattice hierarchy. Its dispersionless(quasiclassical) limit yields the tau-function for analytic curves encoding theintegrable structure of the inverse potential problem and parametric conformalmaps. A similar fermionic realization of partition functions for grandcanonical ensembles of 2D Coulomb charges in the presence of an ideal conductoris also suggested. Their representation as Fredholm determinants is given andtheir relation to integrable hierarchies, growth problems and conformal maps isdiscussed.