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 mth 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 mth 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 1 2 , 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. 16 3. Hard edge scaling limits of the multi-level determinantal processes 17 4. The interpolating multi-level determinantal process 20 5. An integration formula for coupled matrices 21 6. Proof of Theorem 2.1 25

From integrable to chaotic systems: Universal local statistics of Lyapunov exponents

“…The latter choice can be understood as a model for progressive scattering with a short memory, making it non-Markovian. For M = 2 it was considered in [39]. Indeed, we could also choose X j = 1 + γδt + A j √ δt with A j independent Ginibre matrices, the scalar γ being proportional to the variance of A j , and the time increment δt ∝ 1/M .…”

Product Matrix Processes as Limits of Random Plane Partitions

“…If, instead of cutting out corners of a single matrix, one starts adding independent GUE matrices, then the eigenvalues of the sums also form a determinantal process, and the number of matrices in the sum plays the role of discrete time, see Eynard, Mehta [19]. Another class of (dynamical) determinantal processes with discrete time can be constructed from products of random matrices, see Strahov [43], Akemann and Strahov [5]. (Determinantal processes in products of random matrices were first discovered in Akemann and Burda [2]).…”

“…Paper [43] gives a contour integral representation for the correlation kernel, together with its hard edge scaling limit, and generalizes results obtained in Akemann, Kieburg, and Wei [3], Akemann, Ipsen, and Kieburg [4], Kuijlaars and Zhang [31] to the multi-level situation. A more general class of product matrix processes related to certain multi-matrix models was introduced and studied in Akemann and Strahov [5]. In this class the matrices in the products are no longer independent, but in spite of that the product matrix processes are still determinantal.…”

“…. , m 2 − n, m 1 − n ν Szmigielski [8], equation(5). Starting from (9.12), and applying (9.13), we obtain (9.14)φ r,s (x, y) = 1 x s k=r+1 Γ (m k+l−1 − n − ν k+l−1 ) G s−r,0 s−r,s−r m r+l − n, .…”

Large n limit for the product of two coupled random matrices

“…In view of these interesting results obtained for products of independent complex Gaussian random matrices, a natural question to ask is how far such results remain valid, or yet if different ones arise, if some of the conditions on the models are relaxed. One attempt towards this direction is to drop the requirement of independence of the matrices in the product, as initiated by Akemann and Strahov [7] and further explored by them and Liu [5,6,42]. Following [7,42], let us consider a coupled two-matrix model defined by the probability distribution 1…”