Fox H-kernel and $\theta$-deformation of the Cauchy two-matrix model and Bures ensemble

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 as the 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 Laguerre-Bures ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding the Bures ensemble corresponds 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 be obtained.

24 pages

References in corpus (3)

• The Cauchy two-matrix model
• Singular value statistics of matrix products with truncated unitary matrices
• Integrable structure of products of finite complex Ginibre random matrices