The partition function of log-gases with multiple odd charges

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 [Formula: see text] 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 [Formula: see text]-ensembles ([Formula: see text]) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.