Random walkers versus random crowds: Diffusion of large matrices

Abstract

Abstract. We briefly review the random matrix theory for large N by N matrices viewed as free random variables in a context of stochastic diffusion. We establish a surprising link between the spectral properties of matrix-valued multiplicative diffusion processes for hermitian and unitary ensembles.

Free products of large random matrices – a short review of recent developments

“…2k k are equal to the Catalan numbers. One can use equation (5) to find that the corresponding potential is quadratic V (x) = x 2 /2, so the invariant measure (3) is Gaussian dµ(a) ∝ e −N Tra 2 /2 . Let us slightly modify the R transform and consider R(z) = α + σz.…”

“…In studies of dynamical systems one uses products of random matrices to derive limiting laws for Lyapunov exponents [2]; in information theory and wireless telecommunication -to calculate channel capacities for serial MIMO (multiple-input multiple-output) transmission [3]. With the help of products of random matrices one investigates unitary evolution [4], matrix diffusion [5] or phase transitions in quantum Yang-Mills theories [6]. One encounters products of random matrices in studies of quantum entanglement [7], financial engineering [8], and many other fields of research [9,10].…”

Eigenvector statistics of the product of Ginibre matrices

“…More precisely, we study the overlap between left and right eigenvectors for finite N and for N → ∞. In the first part of the paper, we adapt ideas developed in [78,79] to the product of random matrices by using the generalized Schur decomposition [45] for finite N , while in the second part we combine the generalized Green function method [81][82][83][84] with linearization (subordination) [10,28,37] to derive the limiting law for the overlap for N → ∞.…”

“…Products of random matrices have continuously attracted attention since the sixties [1][2][3][4][5]. They are of relevance in many fields of mathematics, physics and engineering including dynamical systems [2,6], disordered systems [7][8][9], statistical mechanics [10], quantum mechanics [11], quantum transport and mesoscopic systems [12,13], hidden Markov models [14], image processing [15], quantum chromodynamics [16], wireless telecommunication [17,18], quantitative finance [19][20][21] and many others [22]. Recently, an enormous progress has been made in the understanding of macroscopic and microscopic statistics of eigenvalues and singular values as well as of Lyapunov spectra for products of random matrices [67][68][69][70][71][72][73][74][75][76][77].…”