Infinite Products of Random Matrices and Repeated Interaction Dynamics

Consider two types of products of independent random matrices, including products of Ginibre matrices and inverse Ginibre matrices and products of truncated Haar unitary matrices and inverse truncated Haar matrices. Each product matrix has $m$ multiplicands of $n$ by $n$ square matrices, and the empirical distribution based on the $n$ eigenvalues of the product matrix is called empirical spectral distribution of the matrix. In this paper, we investigate the limiting empirical...

For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent o...

Products of $M$ i.i.d. non-Hermitian random matrices of size $N \times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix theory (finite $M$ and large $N$). The remaining task is to study local eigenvalue statistics as $M$ and $N$ tend to infinity simultaneously, which lies at the heart of understanding two kinds of universal patterns. For products of i.i.d. compl...

We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, $U(t)=M_t\cdots M_1$, with the factors $M_i$ drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of $U(t)$, its largest eigenvalue and its spectrum are obtained exactly for $N=2$ and numerically ...

Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical distributions of the eigenvalues of two types of $n$ by $n$ random matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of $m$ independent Ha...

We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N \times N$ random matrix ensembles. As $N \to \infty$, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for $M$ fixed and has a white noise component for $M \to \infty$ jointly with $N$. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for whi...

We consider a quantum system S interacting sequentially with independent systems E_m, m=1,2,... Before interacting, each E_m is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asym...

We present a brief introduction to the theory of operator limits of random matrices to non-experts. Several open problems and conjectures are given. Connections to statistics, integrable systems, orthogonal polynomials, and more, are discussed.

We study the ergodic properties of a class of measures on $\Sigma^{\mathbb{Z}}$ for which $\mu_{\mathcal{A},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\left \|A_{x_{0}}\cdots A_{x_{n-1}}\right \| ^{t}$, where $\mathcal{A}=(A_{0}, \ldots , A_{M-1})$ is a collection of matrices. The measure $\mu_{\mathcal{A},t}$ is called a matrix Gibbs state. In particular we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques t...

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ behaves like a sum of i.i.d.\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logari...