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

TL;DR

This paper connects Dyson gas partition functions to tau-functions of integrable hierarchies using fermionic operators, revealing deep links between statistical mechanics, integrable systems, and conformal maps.

Contribution

It provides a fermionic realization of Dyson gas partition functions as tau-functions, including for grand canonical ensembles, and explores their relation to integrable hierarchies and conformal maps.

Findings

Partition functions expressed as vacuum expectation values of fermionic operators.

Identification of partition functions with tau-functions of the 2D Toda lattice hierarchy.

Representation of grand canonical ensemble partition functions as Fredholm determinants.

Abstract

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