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Product matrix processes are multi-level point processes formed by thesingular values of random matrix products. In this paper we study suchprocesses where the products of up to $m$ complex random matrices are no longerindependent, by introducing a coupling term and potentials for each product. Weshow that such a process still forms a multi-level determinantal pointprocesses, and give formulae for the relevant correlation functions in terms ofthe 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 acontour integral representation for the correlation kernels suitable for anasymptotic analysis of large matrix size $n$. Here, the correlations betweenthe 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 edgescaling limit at the origin of the spectra of all products we find threedifferent asymptotic regimes. The first regime corresponding to weak couplingagrees with the multi-level process for the product of $m$ independent complexGaussian matrices for all levels, including the $m$-th. This process wasintroduced by one of the authors and can be understood as a multi-levelextension of the Meijer $G$-kernel introduced by Kuijlaars and Zhang. In thesecond asymptotic regime at strong coupling the point process on level $m$collapses onto level $m-1$, thus leading to the process of $m-1$ independentmatrices. Finally, in an intermediate regime where the coupling is proportionalto $n^{\frac12}$, we obtain a family of parameter dependent kernels,interpolating between the limiting processes in the weak and strong couplingregime. These findings generalise previous results of the authors and theircoworkers for $m=2$.

Product matrix processes for coupled multi-matrix models and their hard edge scaling limits