Related papers: Eigenvalues of truncated unitary matrices: disk co…

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of…

Let $U_m$ be an $m \times m$ Haar unitary matrix and $U_{[m,n]}$ be its $n \times n$ truncation. In this paper the large deviation is proven for the empirical eigenvalue density of $U_{[m,n]}$ as $m/n \to \lambda $ and $n \to \infty$. The…

We consider the class of non-Hermitian operators represented by infinite tridiagonal matrices, selfadjoint in an indefinite inner product space with one negative square. We approximate them with their finite truncations. Both infinite and…

We provide an elementary proof for a theorem due to Petz and R\'effy which states that for a random $n\times n$ unitary matrix with distribution given by the Haar measure on the unitary group U(n), the upper left (or any other) $k\times k$…

We study the asymptotic expansion in $n$ for the partition function of $\beta$ matrix models with real analytic potentials in the multi-cut regime up to the $O(n^{-1})$ terms. As a result, we find the limit of the generating functional of…

We obtain large $n$ asymptotics for the $m$-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble. We focus on the critical regime where all disk boundaries are merging at speed…

Expressions are given for the truncated fractional moments $E X_+^p$ of a general stable law. These involve families of special functions that arose out of the study of multivariate stable densities and probabilities. As a particular case,…

The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems by conjecturing statistical properties of matrix elements of typical operators in the (quasi-)energy…

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies "local" subordinate condition $\|B\varphi_k\|\leqslant b\mu_k^{\beta}$ with some $\beta <1$ and $b>0$. Here…

Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f_1) T_N^{-1}(f_2)) where f1 ({\theta}) = |1 - e^{i {\theta}} | ^{2{{\alpha}1}c1…

We continue the study of truncated matrix-valued moment problems begun in arXiv:2310.00957. Let $q\in\mathbb{N}$. Suppose that $(\mathcal{X},\mathfrak{X})$ is a measurable space and $\mathcal{E}$ is a finite-dimensional vector space of…

We consider compact Hankel operators realized in $ \ell^2(\mathbb Z_+)$ as infinite matrices $\Gamma$ with matrix elements $h(j+k)$. Roughly speaking, we show that if $h(j)\sim (b_{1}+ (-1)^j b_{-1}) j^{-1}(\log j)^{-\alpha}$ as $j\to…

The solution of QCD equations for generating functions of multiplicity distributions reveals new peculiar features of cumulant moments oscillating as functions of their rank. This prediction is supported by experimental data on $e^{+}e^{-},…

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$. In particular,…

Let $A_m^{(1)},\ldots, A_m^{(k)}$ be $m\times m$ left-uppermost blocks of $k$ independent $n\times n$ Haar unitary matrices where $\frac{n}{m}\to \alpha$ as $m\to \infty$, with $1<\alpha<\infty$. Using free probability and Brown measure…

In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random…

This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N…

In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity.…

Let $U^N$ be a family of $N\times N$ independent Haar unitary random matrices and their adjoints, $Z^N$ a family of deterministic matrices, and $P$ a self-adjoint noncommutative polynomial, i.e. for any $N$, $P(U^N,Z^N)$ is self-adjoint,…

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ We express the moments of the trace of any submatrix of $U$ as a sum over partitions whose terms count certain standard and semistandard Young…