The norm of products of free random variables

Let $\Psi_n$ be a product of $n$ independent, identically distributed random matrices $M$, with the properties that $\Psi_n$ is bounded in $n$, and that $M$ has a deterministic (constant) invariant vector. Assuming that the probability of $M$ having only the simple eigenvalue 1 on the unit circle does not vanish, we show that $\Psi_n$ is the sum of a fluctuating and a decaying process. The latter converges to zero almost surely, exponentially fast as $n\to\infty$. The fluctua...

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $...

We consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices and we show that its $L^p$-norm can be upper bounded, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M=\exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In pa...

Let $(g_{n})_{n\geq 1}$ be a sequence of independent identically distributed $d\times d$ real random matrices with Lyapunov exponent $\gamma$. For any starting point $x$ on the unit sphere in $\mathbb R^d$, we deal with the norm $ | G_n x | $, where $G_{n}:=g_{n} \ldots g_{1}$. The goal of this paper is to establish precise asymptotics for large deviation probabilities $\mathbb P(\log | G_n x | \geq n(q+l))$, where $q>\gamma $ is fixed and $l$ is vanishing as $n\to \infty$. W...

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices $\frac1N(D_n\circ X_n)(D_n\circ X_n)^*$, studied in Girko 2001. Here, $X_n=(x_{ij})$ is an $n\times N$ random matrix consisting of independent complex standardized random variables, $D_n=(d_{ij})$, $n\times N$, has nonnegative entries, and $\circ$ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of $X_n$ and $D_n$ which are different ...

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The maximum absolute values of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the...

Let $\xi_1,\xi_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi_{q_1(n)},\xi_{q_2(n)},...,\xi_{q_\ell(n)})$ where $0\leq q_1<q_2<...<q_\ell$ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to\infty$ of the singular values of the random matrix product $\Pi_N=X_N\cdots X_2X_1$ and show, in particular, that (under certai...

Let $\{x_{\alpha}\}_{\alpha \in \mathbb{Z}}$ and $\{y_{\alpha}\}_{\alpha \in \mathbb{Z}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix $X_n = ((x_{i - j}))_{1 \le i, j \le n}$ and a Hankel matrix $Y_n = ((y_{i + j}))_{1 \le i, j \le n}$, and let $M_n = X_n \odot Y_n$ be their elementwise/Schur-Hadamard product. In this article, we show that almost surely, $n^{-...

This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen-Ward, and they are similar in spirit to the results of Ahlswede-Winter and of Tropp for a sum of independent random matrices. The argument relies on the uniform smoothness properties of the Schatten trace classes.

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from ensembles of independent real, complex, or quaternionic Ginibre matrices, or truncated unitary matrices. Additional mixing within one ensemble between matrices and their inverses is also covered. Exact determinantal and Pfaffian expressions are...