Universality conjecture and results for a model of several coupled positive-definite matrices (Q2346846)

This clearly structured article splits into several parts. Section 1 is introductory, all main results are stated in Section 2. First, the authors introduce a Cauchy chain matrix model that generalises the Cauchy two-matrix model from the paper by \textit{M. Bertola} et al. [ibid. 287, No. 3, 983--1014 (2009; Zbl 1197.82037)] to the case of an arbitrary number of \(p\) positive definite \(n\times n\) Hermitian matrices \(M_1,M_2,\dots,M_p\). Their joint probability distribution function depends on the choice of \(p\) scalar functions, called the potentials. After discussing some general properties of such a model the relevant biorthogonal polynomials are expressed in terms of a Riemann-Hilbert problem. The solutions of this problem then give an expression of all kernels of the correlation functions. Next, for some choice of potentials and \(p=3\), the correlation function is studied in the scaling limit near the origin; the cases \(p=4,5,6\) are also addressed in some detail. The Meijer \(G\)-functions are used to express the limiting scaling fields. Several conjectures are stated for arbitrary \(p\). In Section 3, two main theorems are proved, whereas the final Section 4 contains a wealth of technical results, like the detailed asymptotical analysis of a Cauchy-Laguerre three matrix chain; these results are needed to accomplish the proofs of the remaining theorems.