The partition function of multicomponent log-gases (Q2888579)

Motivated by the previous research of the author on ensembles of random Hermitian matrices and their links with the equilibrium properties of interacting charged particles in a multicomponent gas, an expression is given for the partition function of the one-dimensional log-gas on a line for a specific value \(\beta =1\) of the inverse (equilibrium) temperature. The major technical input (preferred here in view of its computational effectiveness) is the Berezin integration over Grassmann algebras. The central result of the paper is that the partition function of the grand canonical ensemble of the multicomponent ensemble can be expressed as the projection onto the determinantal line of the exponential of a non-homogeneous alternating tensor. This projection is conveniently expressed in terms of the Berezin integral. A discussion is provided of simplest examples of multicomponent ensembles.