Related papers: The norm of products of free random variables

Suppose that $X_1,\...,X_n,\...$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let $\Pi_n=X_n\... X_1$. It is known that $n^{-1}\log |\Pi_n|$ converges to a nonrandom limit. We prove that under certain additional assumptions on…

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…

In this paper we review recently established results on the asymptotic behaviour of the trigonometric product $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ as $n\to \infty$. We focus on irrationals $\alpha$ whose continued fraction…

This article is devoted to the study of products of random operators of the form $M_{0,n}=M_0\cdots M_{n-1}$, where $(M_{n})_{n\in\mathbb{N}}$ is an ergodic sequence of positive operators on the space of signed measures on a space…

Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of…

Suppose $\{ X_k \}_{k \in \mathbb{Z}}$ is a sequence of bounded independent random matrices with common dimension $d\times d$ and common expectation $\mathbb{E}[ X_k ]= X$. Under these general assumptions, the normalized random matrix…

In this article, we are interested in the high dimensional normal approximation of $T_n =\Big(\sum_{i=1}^{n}X_{i1}/\Big(\sqrt{\sum_{i=1}^{n}X_{i1}^2}\Big),\dots,$ $\sum_{i=1}^{n}X_{ip}/\Big(\sqrt{\sum_{i=1}^{n}X_{ip}^2}\Big)\Big)$ in…

We study the $\ell^\infty \to \ell^\infty$ operator norm of products of independent random matrices with independent and identically distributed entries. For $n$-by-$n$ matrices whose entries are centered, have unit variance, and have a…

Let $\{X_{i,j}:(i,j)\in\mathbb N^2\}$ be a two-dimensional array of independent copies of a random variable $X$, and let $\{N_n\}_{n\in\mathbb N}$ be a sequence of natural numbers such that $\lim_{n\to\infty}e^{-cn}N_n=1$ for some $c>0$.…

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(\pi x_k)$ for $N \to \infty$, where the numbers $\omega=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$.…

Suppose that $\xi^{(n)}_1,\xi^{(n)}_2,...,\xi^{(n)}_n$ are i.i.d with $P(\xi^{(n)}_i=1)=p_n=1-P(\xi^{(n)}_i=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to…

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on…

We analyze the joint extremal behavior of $n$ random products of the form $\prod_{j=1}^m X_j^{a_{ij}}, 1 \leq i \leq n,$ for non-negative, independent regularly varying random variables $X_1, \ldots, X_m$ and general coefficients $a_{ij}…

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…

Let $P_n$ be the $n$-step right product $A_1\cdots A_n$, where $A_1,A_2,\dots$ is a given infinite sequence of $d\times d$ matrices with nonnegative entries. In a wide range of situations, the normalized matrix product $P_n/{\Vert…

Let $(X_i)_{1 \le i \le n}$ be independent and identically distributed (i.i.d.) standard Gaussian random variables, and denote by $X_{(n)} = \max_{1 \le i \le n} X_i$ the maximum order statistic. It is well-known in extreme value theory…

We consider matrix products of the form $A_1(A_2A_2)^\top\ldots(A_{m}A_{m}^\top)A_{m+1}$, where $A_i$ are normalized random Fourier-Walsh matrices. We identify an interesting polynomial scaling regime when the operator norm of the expected…

We present a non-asymptotic concentration inequality for the random matrix product \begin{equation}\label{eq:Zn} Z_n = \left(I_d-\alpha X_n\right)\left(I_d-\alpha X_{n-1}\right)\cdots \left(I_d-\alpha X_1\right), \end{equation} where…

In this paper, a connection between bi-free probability and the asymptotics of random quantum channels and tensor products of random matrices is established. Using bi-free matrix models, it is demonstrated that the spectral distribution of…

Let $\Phi(N)$ denote the number of products of matrices $[ \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}]$ and $[ \begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} ]$ of trace equal to $N$, and $\Psi(N)=\sum_{n=3}^N \Phi(n)$ be…