We study the effect of highly oscillatory potentials to the eigenvalues of a random matrix. Consider the circular unitary ensembles with an external potential which is periodic with the period comparable to the average spacing of the eigenvalues. We show that in this case the density of states is periodic and does not converge in the large matrix limit, but the local correlation functions converge to some simple combinations of the sine kernel and the potential. We evaluate the correlation functions exactly and also asymptotically.
This paper gives a rigorous proof of a conjectured statistical self-similarity property of the eigenvalues random matrices from the Circular Unitary Ensemble. We consider on the one hand the eigenvalues of an $n \times n$ CUE matrix, and on the other hand those eigenvalues $e^{iϕ}$ of an $mn \times mn$ CUE matrix with $|ϕ| \le π/ m$, rescaled to fill the unit circle. We show that for a large range of mesoscopic scales, these collections of points are statistically indistinguishable for large $n$. The proof is based on a comparison theorem for determinantal point processes which may be of indep…
In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the lim…
We apply the operation of random independent thinning on the eigenvalues of $n\times n$ Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as $n\to\infty$.
We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble ${\rm CUE}(N)$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, $N\rightarrow\infty$, we derive a ${\it\, concise\,}$ parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the ${\rm Sine}_2$ determinantal random point field. Further, we discuss a universality of the predicted power spectrum law an…
Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new c…
The partition function of the Chern-Simons theory on the three-sphere with the unitary group $U(N)$ provides a one-matrix model. The corresponding $N$-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as $q$-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the $N$-particle system to an infinite-dimensional determinantal point process in $N \…
Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$ submatrix of an $n$ by $n$ circular unitary matrix after deleting the last $n-p_n$ columns and rows. Jiang and Qi \cite{JiangQi2017} and Gui and Qi \cite{GQ2018} study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix. Some limiting distributions for the spectral radius for the truncated circular unitary matrix have been obtained under the following conditions: (1). $p_n/n$ is bounded away from $0$ and $1$; (2). $p_n\to\infty$ and $p_n/n\to 0$ as $…
Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$ submatrix of an $n$ by $n$ circular unitary matrix by deleting the last $n-p_n$ columns and rows. Jiang and Qi (2017) proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly normalized, converges in distribution to the Gumbel distribution if $p_n/n$ is bounded away from $0$ and $1$. In this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions: (1). $p_n\to\infty$ and $p_n/n\to 0$ as $n\to\infty$; …
In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular beta-ensemble, and we show how this joint density is related to known inter-relations between …
According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1-ξ$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that…
The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.
We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called "microscopic" level, that is we consider the characteristic polynomial at points which are of order $1/n$ distant. We prove this in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical…
Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values --- whereby only even, or odd, labels are observed --- for real symmetric random matrices with an orthogonal symmetry, and even weight. This requir…
Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
School of Mathematical Sciences \& Laboratory of Mathematics；Complex Systems of Ministry of Education, Beijing Normal University, 100875 Beijing, China.
Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.
Let $Z_N$ be a Ginibre ensemble and let $A_N$ be a Hermitian random matrix independent from $Z_N$ such that $A_N$ converges in distribution to a self-adjoint random variable $x_0$. For each $t>0$, the random matrix $A_N+\sqrt{t}Z_N$ converges in $\ast$-distribution to $x_0+c_t$, where $c_t$ is the circular variable of variance $t$, free from $x_0$. We use the Hamilton-Jacobi method to compute the Brown measure $ρ_t$ of $x_0+c_t$. The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of $x_0+c_t$ is related to the subo…
We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $β>0$, not merely $β=1,2,4$.
Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{δ,β}^{(n)} \prod_{1\leq k<l\leq n}| e^{\iiθ_k}-e^{\iiθ_l}|^β\prod_{j=1}^{n}(1-e^{-\iiθ_j})^δ (1-e^{\iiθ_j})^{\overlineδ} $$ with $\Re δ> -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(α_0, ..., α_{n-1})$. We introduce here a deformation $(γ_0, >..., γ_{n-1})$ of these coeff…
Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.
It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matr…
For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t in [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations.
Fill an n x n matrix with independent complex Gaussians of variance 1/n. As n approaches infinity, the eigenvalues {z_k} converge to a sum of an H^1-noise on the unit disk and an independent H^{1/2}-noise on the unit circle. More precisely, for C^1 functions of suitable growth, the distribution of sum_{k=1}^n (f(z_k)-E f(z_k)) converges to that of a mean-zero Gaussian with variance given by the sum of the squares of the disk H^1 and the circle H^{1/2} norms of f. Moreover, with p_n the characteristic polynomial, log|p_n|- E log|p_n| tends to the planar Gaussian free field conditioned to be har…
The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $β$-Ensemble (C$β$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$β$…
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.
The ensemble $\CUE^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q.$ $\CUE^{(q)}$ is an exactly solved model of random contraction matrices originally introduced in the context of scattering theory. In this article, we exhibit a connection between $\CUE^{(q)}$ and Fisher's random-turns vicious walker model from statistical mechanics. In particular, we show that the moment generating function of the trace of a random matrix from $\CUE^{(q)}$ is a generating series for the partition function of …
