Two $$\beta $$-ensemble realization of $$\beta $$-deformed WLZZ models

Abstract

AbstractWe consider a two $$\beta $$ β -ensemble realization of the series of $$\beta $$ β -deformed WLZZ matrix models. We demonstrate that such a realization involves $$\beta $$ β -deformed Harish-Chandra–Itzykson–Zuber integrals, one of them providing a coupling to the external field. We also construct Ward identities in the corresponding two $$\beta $$ β -ensemble model, which requires a set of identities for partition function of the one $$\beta $$ β -ensemble in the external field, and a set of identities for the $$\beta $$ β -deformed Itzykson-Zuber integral. These both sets of identities are formulated in terms of the Dunkl operators.