Dynamical correlation functions for products of random matrices

Abstract

We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard-Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.Let us mention that recent works on products of random matrices (in particular, the works mentioned above) deal with the situation when the number of matrices in products under consideration is fixed. In this paper, we propose to consider sequences of products of random matrices instead of fixed products. The main idea behind this work is rather similar to that of Johansson and Nordenstam [21]. Johansson and Nordenstam [21] start with an infinite random matrix picked from the Gaussian Unitary Ensemble (GUE), and consider the eigenvalues of its minors. They show that all such eigenvalues form a determinantal process on N × R. Such determinantal processes are called the minor processes. Here we can start from a matrix product G m · · · G 1 , and consider squared singular values of subproducts G k · · · G 1 , k = 1, . . . , m. As a result we arrive to a family of determinantal processes on {1, . . . , m} × R >0 , which can be understood as time-dependent extensions of the processes studied previously.The minor processes and their generalizations are of great interest in Random Matrix Theory, and are closely related to random tiling models, and to certain percolation models on the Z + × Z + lattice. In particular, it was observed by Johansson and Nordenstam [21] (see also [28]) that the spectra of the principal minors of a GUE matrix behave as domino tilings of large size Aztec diamonds. We refer the reader to [3, 1, 2] and references therein for a description of these connections, and for recent results in this area of research. Many other time-dependent determinantal point processes appeared over the past decade in the context of growth models, and were applied to different problems in numerative combinatorics and statistical physics, see, for example, [11,10].The novelty of this paper is in the very observation that products of random matrices also lead to time-dependent determinantal point processes. These processes are rather different from those formed by eigenv...

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

“…In this paper we introduce and study a different multi-matrix model of statistically dependent random matrices, also satisfying these requirements. We show (see Theorem 2.7) that these product matrix processes generalise the Ginibre product process studied in Strahov [41]. For a particular choice of potentials the first m − 1 levels remain that of products of m − 1 independent random matrices, to which the m-th level is coupled.…”

“…, m}×R >0 . This process was introduced and studied in Strahov [41], calling it the Ginibre product process. It was shown in [41] that it is a multi-level determinantal point process.…”

“…This process was introduced and studied in Strahov [41], calling it the Ginibre product process. It was shown in [41] that it is a multi-level determinantal point process. Furthermore, a contour integral representation for the correlation kernel of this process was derived, and its hard edge scaling limit computed.…”

“…In the hard edge scaling limit at large matrix sizes we distinguish three different asymptotic regimes. In the first regime corresponding to weak coupling, the limiting product matrix process is the same as for the product of m independent complex Gaussian matrices, as described in Strahov [41]. This process can be called the m-level Meijer G-kernel process, a multi-level extension of the Meijer G-kernel process introduced by Kuijlaars and Zhang [31].…”

Product Matrix Processes as Limits of Random Plane Partitions

“…It was shown in Strahov [43] that this process is determinantal (and it can be viewed as a determinantal process with discrete time). 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].…”