The Partition Function of Log-Gases with Multiple Odd Charges

TL;DR

This paper derives a general formula for the partition function of one-dimensional log-gases with multiple odd charges using shuffle algebra techniques, extending classical results to multicomponent ensembles.

Contribution

It introduces a unified framework that generalizes de Bruijn integral identities to multicomponent log-gases with multiple odd charges, removing previous restrictions.

Findings

Provides a formula for the partition function in terms of Berezin integrals.

Extends de Bruijn identities to multicomponent ensembles.

Unifies various integral identities within a single algebraic framework.

Abstract

We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $\beta$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $\beta$-ensembles ($\beta$ = 1, 2, 4) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.