Journal article

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

Peter J Forrester, Shi-Hao Li

INTERNATIONAL MATHEMATICS RESEARCH NOTICES | OXFORD UNIV PRESS | Published : 2021

#### 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 Bu..

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#### Grants

Awarded by Australian Research Council [ARC grant] through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS)

#### Funding Acknowledgements

This work was supported by the Australian Research Council [ARC grant DP170102028] through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS).