We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_ω$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunski coefficients of constant modulus and correlated random phases.
This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements g…
This paper aims at presenting a few models of quantum dynamics whose description involves the analysis of random unitary matrices for which dynamical localization has been proven to hold. Some models come from physical approximations leading to effective descriptions of the dynamics of certain random systems that are popular in condensed matter theoretical physics, whereas others find their roots in more abstract considerations and generalizations. Although they may differ in details, the operators describing the models all have in common the following key features on which their analysis reli…
Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get rep…
This note describes recent results on the localization properties of Random Quantum Walks on the d-dimensional lattice in a regime analogous to the large disorder regime by means of the Fractional Moments Method adapted to the unitary framework.
Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary ensembles are recovered. At the same time, we study the perturbation invariability of the level densities of random matrix unitary ensembles. After the weight function associated with the 1-level correlation function is appended a polynomial multiplicative factor, the level density is invariant in the weak sense.
We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can also be considered as a unitary version of the Anderson model. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.
The explicit form of the unitary representation of the Poincaré group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators
We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as N(Q) is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary ensemble of random matrices may be destroyed) all spectral fluctuation measures have the same form as for the Gaussian unitary ensemble.
We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using the supersymmetry approach we calculate analytically the moments and the distribution function of the eigenvectors components for a generic matrix $W$. We show that specific choices of $W$ can modify significantly the nature of the eigenvectors changing them from extended to critical to localized. Our analytical results are supported by numerical simulations…
We investigate diverse relations of the colour-flavour transformations (CFT) introduced by Zirnbauer in \cite{Z1,Z2} to various topics in random matrix theory and multivariate analysis, such as measures on truncations of unitary random matrices, Jacobi ensembles of random matrices, Berezin reproducing kernels and a generalization of the Selberg integral due to Kaneko, Kadell and Yan involving the Schur functions. Apart from suggesting explicit formulas for bosonic CFT for the unitary group in the range of parameters beyond that in \cite{Z2}, we also suggest an alternative variant of the transf…
We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $ \Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.
A discussion of the method of multiscale analysis in the study of localization of random operators based on lectures given at \emph{Random Schrödinger operators: methods, results, and perspectives}, États de la recherche, Université Paris 13, June 2002
We prove exponential localization for the Schrödinger operator with a Poisson random potential at the bottom of the spectrum in any dimension. We also prove exponential localization in a prescribed interval for all large Poisson densities. In addition, we obtain dynamical localization and finite multiplicity of the eigenvalues.
We analyze composed quantum systems consisting of $k$ subsystems, each described by states in the $n$-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order $n^2$. The global evolution operator is represented by a unitary matrix of size $N=n^k$. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph…
We prove exponential and dynamical localization at low energies for the Schrödinger operator with an attractive Poisson random potential in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity.
Recently, the supersymmetry method was extended from Gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the Hubbard-Stratonovich transformation. Here, we complete this extension by including arbitrary orthogonally and unitary-symplectically invariant matrix ensembles. The results are equivalent to, but the approach is different from the superbosonization formula. We express our results in a unifying way. We also give explicit expressions for all one-point functions and discuss features of the higher order correlations.
We study the region of complete localization in a class of random operators which includes random Schrödinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prov…
These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random mat…
We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.
Random unitary matrices find a number of applications in quantum information science, and are central to the recently defined boson sampling algorithm for photons in linear optics. We describe an operationally simple method to directly implement Haar random unitary matrices in optical circuits, with no requirement for prior or explicit matrix calculations. Our physically-motivated and compact representation directly maps independent probability density functions for parameters in Haar random unitary matrices, to optical circuit components. We go on to extend the results to the case of random u…
We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization w…
In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^α{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~α,t>0. \end{equation*} By using the ladder operator approach, we obtain that an auxiliary quantity $R_n(t)$, which is closely related to the recurrence coefficients of monic polynomials orthogonal with $w(z)$, satisfies a particular Painlevé V equation.
We consider a two dimensional magnetic Schroedinger operator on a square lattice with a spatially stationary random magnetic field. We prove Anderson localization near the spectral edges. We use a new approach to establish a Wegner estimate that does not rely on the monotonicity of the energy on the random parameters.
We prove exponential and dynamical localization for the Schrödinger operator with a nonnegative Poisson random potential at the bottom of the spectrum in any dimension. We also conclude that the eigenvalues in that spectral region of localization have finite multiplicity. We prove similar localization results in a prescribed energy interval at the bottom of the spectrum provided the density of the Poisson process is large enough.
We consider the deformed Gaussian Ensemble $H_n=M_n+H^{(0)}_n$ in which $H_n^{(0)}$ is a diagonal Hermitian matrix and $M_n$ is the Gaussian Unitary Ensemble (GUE) random matrix. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ (both non-random and random) converges weakly to a measure $N^{(0)}$ with a bounded support we prove universality of the local eigenvalue statistics in the bulk of the limiting spectrum of $H_n$.
We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.
Among the discrete evolution equations describing a quantum system $\rH_S$ undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in $\RR^N$. The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group $U(\rH_0)$ of unitary operators on $\rH_0$.
Orthogonal - unitary and symplectic - unitary crossover ensembles of random matrices are relevant in many contexts, especially in the study of time reversal symmetry breaking in quantum chaotic systems. Using skew-orthogonal polynomials we show that the same generic form of n-level correlation functions are obtained for the Jacobi family of crossover ensembles, including the Laguerre and Gaussian cases. For large matrices we find universal forms of unfolded correlation functions when expressed in terms of a rescaled transition parameter with arbitrary initial level density.
We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes in 2001. Moreover, we give a probabilistic representation for the leading order coefficient in the asymptotic in terms of a real-valued random variable that plays an important role in the ergodic decomposition of the Hua-Pickrell measures. This enables us to establish connections between the characteristic function of this random variable and the $σ$-Painlevé III' equation.
We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary $t$-designs. Unitary $t$-designs are probability distributions that mimic Haar randomness up to $t$th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth $O(n t^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit th…
In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith, using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations…
We consider unitary analogs of $d-$dimensional Anderson models on $l^2(\Z^d)$ defined by the product $U_ω=D_ωS$ where $S$ is a deterministic unitary and $D_ω$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of $U_ω(U_ω-z)^{-1}$, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements…
In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $θ=π$ for a specific critical evolution time $t_c$.
We study a unitary version of the one-dimensional Anderson model, given by a five diagonal deterministic unitary operator multiplicatively perturbed by a random phase matrix. We fully characterize positivity and vanishing of the Lyapunov exponent for this model throughout the spectrum and for arbitrary distributions of the random phases. This includes Bernoulli distributions, where in certain cases a finite number of critical spectral values, with vanishing Lyapunov exponent, exists. We establish similar results for a unitary version of the random dimer model.
We consider a new class of non-Hermitian random matrices, namely the ones which have the form of sums of freely independent terms involving unitary matrices. To deal with them, we exploit the recently developed quaternion technique. After having derived some general identities describing additive properties of unitary matrices, we solve three particular models: CUE plus CUE, CUE plus... plus CUE (i. e. the sum of an arbitrary number of CUE matrices), and CUE plus GUE. By solution of a given model we mean here to calculate the borderline of the eigenvalues' two-dimensional domain, as well as th…
We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).
Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.
The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong-Ou-Mandel effect and the Boson Sampling problem. In that context, it would be useful to know the distribution of the permanent or other immanants for random matrices, but that seems a difficult problem.We advance this program by calculating the average of the squared modulus of a generic immanant for blocks from random matrices in the unitary group, in the orthogonal group and in the circular orthogonal ensemble. In the case of th…
In this paper, we study the behaviour of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings and Hayden-Winter to the additivity problems. In particular, we study in depth the difference of behaviour between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space,…
We consider the convergence of the empirical spectral measures of random $N \times N$ unitary matrices. We give upper and lower bounds showing that the Kolmogorov distance between the spectral measure and the uniform measure on the unit circle is of the order $\log N / N$, both in expectation and almost surely. This implies in particular that the convergence happens more slowly for Kolmogorov distance than for the $L_1$-Kantorovich distance. The proof relies on the determinantal structure of the eigenvalue process.
We study unitary random matrix ensembles in the critical regime where a new cut arises away from the original spectrum. We perform a double scaling limit where the size of the matrices tends to infinity, but in such a way that only a bounded number of eigenvalues is expected in the newborn cut. It turns out that limits of the eigenvalue correlation kernel are given by Hermite kernels corresponding to a finite size Gaussian Unitary Ensemble (GUE). When modifying the double scaling limit slightly, we observe a remarkable transition each time the new cut picks up an additional eigenvalue, leading…
The value of spectral form factor at the origin, called level compressibility, is an important characteristic of random spectra. The paper is devoted to analytical calculations of this quantity for different random unitary matrices describing models with intermediate spectral statistics. The computations are based on the approach developed by G. Tanner in [J. Phys. A: Math. Gen. 34, 8485 (2001)] for chaotic systems. The main ingredient of the method is the determination of eigenvalues of a transition matrix whose matrix elements equal squared moduli of matrix elements of the initial unitary ma…
This paper is concerned with complex eigenvalues of truncated unitary quaternion matrices equipped with the Haar measure. The joint eigenvalue probability density function is obtained for truncations of any size. We also obtain the spectral density and the eigenvalue correlation functions in various scaling limits. In the limit of strong non-unitarity the universal complex Ginibre form of the correlation functions is recovered in the spectral bulk off the real line after unfolding the spectrum. In the limit of weak non-unitarity we obtain the spectral density and eigenvalue correlation functio…
We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=α$, the empirical spectral measure is well-approximated by a deterministic measure $μ_α$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $μ_α$ is typically of order $\sqrt{\frac{\log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale,…
I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two "master equations," which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit …
Extremal spacings between eigenvalues of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N = 4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a composite quantum system consisting of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s_min) and P(s_max) reveal certai…
I consider random Schrödinger operators with exponentially decaying single site potential, which is allowed to change sign. For this model, I prove Anderson localization both in the sense of exponentially decaying eigenfunctions and dynamical localization. Furthermore, the results imply a Wegner-type estimate strong enough to use in classical forms of multi-scale analysis.
In this paper we consider the localization properties of coupled harmonic oscillators in random media. Each of these oscillators is restricted to the lattice $\mathbb{Z}^d$. We show that for most states and an arbitrary choice of the random media, the long time localization for the coupled system holds in a time scale larger than the polynomial one.
In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions $m$ and $n$ in the estimate with explicit constants. We are especially interested in the regime where $m$ grows with $n$ and our main result basically states that if $m\ll \sqrt{n}$, then the rate of convergence in the Gaussian approximatio…
We consider a d-dimensional random quantum walk with site-dependent random coin operators. The corresponding transition coefficients are characterized by deterministic amplitudes times independent identically distributed site-dependent random phases. When the deterministic transition amplitudes are close enough to those of a quantum walk which forbids propagation, we prove that dynamical localization holds for almost all random phases. This instance of Anderson localization implies that all quantum mechanical moments of the position operator are uniformly bounded in time and that spectral loca…
We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of "valleys" of height $\log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.
