$\beta$-WLZZ models from $\beta$-ensemble integrals directly

TL;DR

This paper introduces a direct evaluation method for $eta$-ensemble integrals in $eta$-WLZZ models, expanding the understanding of contour choices beyond previous Ward identity-based approaches.

Contribution

It provides a new direct evaluation technique for $eta$-ensemble integrals in $eta$-WLZZ models, demonstrating alternative contour choices and extending prior Ward identity methods.

Findings

Direct evaluation of $eta$-ensemble integrals using Macdonald's conjecture.

Confirmation of alternative contour choices for integrals.

Enhanced understanding of $eta$-WLZZ model realizations.

Abstract

Recently, we performed a two $\beta$-ensemble realization of the series of $\beta$-deformed WLZZ matrix models involving $\beta$-deformed Harish-Chandra-Itzykson-Zuber integrals. The realization was derived and studied by using Ward identities, which do not allow one to fix integration contours, these latter were chosen to be real axis for one $\beta$-ensemble and imaginary axis for the other one basing on some particular checks. Here, we evaluate the $\beta$-ensemble integrals directly using a conjecture by I.G. Macdonald, and explain that another choice of integration contours is also possible.