Product Matrix Processes for Coupled Multi-Matrix Models and Their Hard Edge Scaling Limits

Abstract

Product matrix processes are multi-level point processes formed by the singular values of random matrix products. In this paper we study such processes where the products of up to $m$ complex random matrices are no longer independent, by introducing a coupling term and potentials for each product. We show that such a process still forms a multi-level determinantal point processes, and give formulae for the relevant correlation functions in terms of the corresponding kernels. For a special choice of potential, leading to a Gaussian coupling between the $m$th matrix and the product of all previous $m-1$ matrices, we derive a contour integral representation for the correlation kernels suitable for an asymptotic analysis of large matrix size $n$. Here, the correlations between the first $m-1$ levels equal that of the product of $m-1$ independent matrices, whereas all correlations with the $m$th level are modified. In the hard edge scaling limit at the origin of the spectra of all products we find three different asymptotic regimes. The first regime corresponding to weak coupling agrees with the multi-level process for the product of $m$ independent complex Gaussian matrices for all levels, including the $m$-th. This process was introduced by one of the authors and can be understood as a multi-level extension of the Meijer $G$-kernel introduced by Kuijlaars and Zhang. In the second asymptotic regime at strong coupling the point process on level $m$ collapses onto level $m-1$, thus leading to the process of $m-1$ independent matrices. Finally, in an intermediate regime where the coupling is proportional to $n^{\frac12}$, we obtain a family of parameter dependent kernels, interpolating between the limiting processes in the weak and strong coupling regime. These findings generalise previous results of the authors and their coworkers for $m=2$.