Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

Abstract

Abstract A $\theta $-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib–Borodin ensemble. The $\theta $-deformed Cauchy–Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written in terms of particular Fox H-functions, generalising the Meijer G-function class known from the study of the case $\theta = 1$. In the $\theta =1$ case, it is shown that the Laguerre–Bures ensemble is related to the Laguerre–Cauchy two-matrix model, notwithstanding the Bures ensemble corresponding to a Pfaffian point process. This carries over to the $\theta $-deformed case, allowing explicit expressions involving Fox H-functions for the correlation kernel, and its hard edge scaling limit, to also be obtained for the $\theta $-deformed Laguerre–Bures ensemble. The hard edge scaling limit is in the Raney class $(\theta /2+1,1/2)$.