Infinite Products of Random Matrices and Repeated Interaction Dynamics

Let $X_i$ denote free identically-distributed random variables. This paper investigates how the norm of products $\Pi_n=X_1 X_2 ... X_n$ behaves as $n$ approaches infinity. In addition, for positive $X_i$ it studies the asymptotic behavior of the norm of $Y_n=X_1 \circ X_2 \circ ...\circ X_n$, where $\circ$ denotes the symmetric product of two positive operators: $A \circ B=:A^{1/2}BA^{1/2}$. It is proved that if the expectation of $X_i$ is 1, then the norm of the symmetric...

We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent iid random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$. We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent iid random matrices. This is done in two steps. First, we ...

Quantum trajectories are Markov processes modeling the evolution of a quantum system subjected to repeated independent measurements. Inspired by the theory of random products of matrices, it has been shown that these Markov processes admit a unique invariant measure under a purification and an irreducibility assumptions. This paper is devoted to the spectral study of the underlying Markov operator. Using Quasi-compactness, it is shown that this operator admits a spectral gap ...

These are lectures notes for a 4h30 mini-course held in Ulaanbaatar, National University of Mongolia, August 5-7th 2015, at the summer school "Stochastic Processes and Applications". It aims at presenting an introduction to basic results of random matrix theory and some of its motivations, targeted to a large panel of students coming from statistics, finance, etc. Only a small background in probability is required.

The paper deals with the convergence properties of the products of random (row-)stochastic matrices. The limiting behavior of such products is studied from a dynamical system point of view. In particular, by appropriately defining a dynamic associated with a given sequence of random (row-)stochastic matrices, we prove that the dynamics admits a class of time-varying Lyapunov functions, including a quadratic one. Then, we discuss a special class of stochastic matrices, a class...

We consider the products of $m\ge 2$ independent large real random matrices with independent vectors $(X_{jk}^{(q)},X_{kj}^{(q)})$ of entries. The entries $X_{jk}^{(q)},X_{kj}^{(q)}$ are correlated with $\rho=\mathbb E X_{jk}^{(q)}X_{kj}^{(q)}$. The limit distribution of the empirical spectral distribution of the eigenvalues of such products doesn't depend on $\rho$ and equals to the distribution of $m$th power of the random variable uniformly distributed on the unit disc.

Inspired by a recent paper of I. Grama, E. Le Page and M. Peign\'e, we consider a sequence $(g_n)_{n \geq 1}$ of i.i.d. random $d\times d$-matrices with non-negative entries and study the fluctuations of the process $(\log \vert g_n\cdots g_1\cdot x\vert )_{n \geq 1}$ for any non-zero vector $x$ in $\mathbb R^d$ with non-negative coordinates. Our method involves approximating this process by a martingale and studying harmonic functions for its restriction to the upper half li...

We study stochastic properties of the norm cocycle associated with iid products of positive matrices. We obtain the almost sure invariance principle (ASIP) with rate o(n 1/p) under the optimal condition of a moment or order p > 2 and the Berry-Esseen theorem with rate O(1/ $\sqrt$ n) under the optimal condition of a moment of order 3. The results are also valid for the matrix norm. For the matrix coefficients, we also have the ASIP but we obtain only partial results for the B...

We consider products of independent random matrices with independent entries. The limit distribution of the expected empirical distribution of eigenvalues of such products is computed. Let $X^{(\nu)}_{jk},{}1\le j,r\le n$, $\nu=1,...,m$ be mutually independent complex random variables with $\E X^{(\nu)}_{jk}=0$ and $\E {|X^{(\nu)}_{jk}|}^2=1$. Let $\mathbf X^{(\nu)}$ denote an $n\times n$ matrix with entries $[\mathbf X^{(\nu)}]_{jk}=\frac1{\sqrt{n}}X^{(\nu)}_{jk}$, for $1\le...

If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n\to\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$ are nonnegative and that $V$ has positive entries. Denoting by $\Delta$ the diagonal matrix whose diagonal entries are the entries of $V$, the stochastic matrices $S_n=\...