Applications of the Dotsenko-Fateev Integral in Random-Matrix Models

TL;DR

This paper explores how Dotsenko-Fateev integrals from conformal field theory can be applied to compute complex quantities in random-matrix models, providing explicit calculations and asymptotic behaviors.

Contribution

It demonstrates the use of Dotsenko-Fateev integrals to evaluate physical quantities in random-matrix theory, linking conformal field theory techniques to random-matrix calculations.

Findings

Calculated mean squared S-matrix elements for Gaussian orthogonal ensemble.

Derived asymptotic behavior of density-density correlator in Calogero-Sutherland model.

Established a connection between conformal field theory integrals and random-matrix models.

Abstract

The characteristic multi-dimensional integrals that represent physical quantities in random-matrix models, when calculated within the supersymmetry method, can be related to a class of integrals introduced in the context of two-dimensional conformal field theories by Dotsenko & Fateev. Known results on these Dotsenko-Fateev integrals provide a means by which to perform explicit calculations (otherwise difficult) in random-matrix theory. We illustrate this by (i) an evaluation of the mean squared S-matrix elements for the Gaussian orthogonal ensemble coupled with M external channels, and (ii) a direct derivation of the asymptotic behaviour of the dynamical density-density correlator in the limit of large spatial and temporal separation for the Calogero-Sutherland model which, at certain couplings, is known to map onto the parameter-dependent random matrix ensembles.