Unitary Ensemble Of Random Matrices Research Articles

Random-matrix model for thermalization

We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions Tr(Aρ(t)) in the ensemble thermalize: For N→∞ every such function tends to the value Tr(Aρeq(∞))+Tr(Aρ(0))g2(t) . Here ρ(t) is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and ρeq(∞) is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as 1/|t|3/2 for large t. With g(t)=g(−t) , thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen. 32 1163) shows overall agreement but raises significant questions.

Sine‐kernel determinant on two large intervals

AbstractWe consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.

The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature β = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions Rd and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions Cd, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For Cd with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Investigation of the two-cut phase region in the complex cubic ensemble of random matrices

We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential V(M)=−13M3+tM, where t is a complex parameter. As proven in our previous paper [Bleher et al., J. Stat. Phys. 166, 784–827 (2017)], the whole phase space of the model, t∈C, is partitioned into two phase regions, Oone−cut and Otwo−cut, such that in Oone−cut the equilibrium measure is supported by one Jordan arc (cut) and in Otwo−cut by two cuts. The regions Oone−cut and Otwo−cut are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In Bleher et al. [J. Stat. Phys. 166, 784–827 (2017)], the one-cut phase region was investigated in detail. In the present paper, we investigate the two-cut region. We prove that in the two-cut region, the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter t, but not of the parameter t itself (so that the Cauchy–Riemann equations are violated for the endpoints). We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann–Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of S-curves and quadratic differentials.

How much can the eigenvalues of a random Hermitian matrix fluctuate?

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations—or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices, the Gaussian unitary ensemble (GUE) being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher–Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

Entanglement Dynamics of Random GUE Hamiltonians

In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential n-point correlation functions in the gas of GUE eigenvalues.

A note on pair-dependent linear statistics with a slowly increasing variance

We prove Gaussian fluctuations for pair-counting statistics of the form $$\Sigma_{1\le i\ne j\le N}f(\theta_i-\theta_j)$$ for the circular unitary ensemble of random matrices in the large- $$N$$ limit under the condition that the variance increases slowly as $$N$$ increases.

Splitting of a gap in the bulk of the spectrum of random matrices

We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle.

Gap Probability Distribution of the Jacobi Unitary Ensemble: An Elementary Treatment, from Finite n to Double Scaling

AbstractIn this paper, we study the gap probability problem of the (symmetric) Jacobi unitary ensemble of Hermitian random matrices, namely, the probability that the interval is free of eigenvalues. Using the ladder operator technique for orthogonal polynomials and the associated supplementary conditions, we derive three quantities instrumental in the gap probability, denoted by , , and . We find that each one satisfies a second‐order differential equation. We show that after a double scaling, the large second‐order differential equation in the variable a with n as parameter satisfied by can be reduced to the Jimbo–Miwa–Okamoto σ form of the Painlevé V equation.

Generic appearance of objective results in quantum measurements

Measurement is of central interest in quantum mechanics as it provides the link between the quantum world and the world of everyday experience. One of the features of the latter is its robust, objective character, contrasting the delicate nature of quantum systems. Here we analyze in a completely model-independent way the celebrated von Neumann measurement process, using recent techniques of information flow, studied in open quantum systems. We show the generic appearance of objective results in quantum measurements, provided we macroscopically coarse-grain the measuring apparatus and wait long enough. To study genericity, we employ the widely-used Gaussian Unitary Ensemble of random matrices and the Hoeffding inequality. We derive generic objectivization timescales, given solely by the interaction strength and the systems' dimensions. Our results are manifestly universal and are a generic property of von Neumann measurements.

Statistics of eigenvectors in the deformed Gaussian unitary ensemble of random matrices

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices , where is a random matrix from the 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.

Embedded Random Matrix Ensembles with Lie Symmetries: Results from U(Ω) Wigner-Racah algebra

Random matrix ensembles for a system of m number of fermions or bosons in Ω number of single particle levels each r-fold degenerate and interacting with two-body forces are considered. The spectrum generating algebra for these systems is U(rΩ) and a subalgebra of interest is U(rΩ) ⊃ U(Ω) ⊗ SU(r) algebra. Now, for random two-body interactions preserving SU(r) symmetry, one can introduce embedded Gaussian unitary ensemble of random matrices with U(Ω)⊗SU(r) embedding and this class of ensembles are denoted by EGUE(2)-SU(r). Ensembles with r = 1,2 and 4 for fermions correspond to spinless fermions, fermions with spin and fermions with Wigner's spin-isospin SU(4) symmetry respectively. Similarly, for bosons r = 1, 2 and 3 correspond to spinless bosons, two species boson systems and bosons with spin one respectively. The distinction between fermions and bosons is in the U(Ω) irreducible representations. General formulation based on Wigner-Racah algebra for lower order moments of the one- and two-point functions in eigenvalues generated by EGUE(2)- SU(r) is briefly reviewed. The final formulas for the moments involve only SU(Ω) Racah coefficients. For the fourth moment of the one-point function for r > 1 and for the higher order (> 4) bivariate moments of the two-point function for r ≥ 1, formulas are not available for the SU(Ω) Racah coefficients that are needed. It is necessary to derive analytical formulas for these or develop methods that give asymptotic results (an example for this is given in the paper) or develop methods that allow for their numerical evaluation. This important open problem is discussed in some detail.

Eigenvalue Statistics as an Indicator of Integrability of Nonequilibrium Density Operators

We propose to quantify the complexity of nonequilibrium steady state density operators, as well as of long-lived Liouvillian decay modes, in terms of the level spacing distribution of their spectra. Based on extensive numerical studies in a variety of models, some solvable and some unsolved, we conjecture that the integrability of density operators (e.g., the existence of an algebraic procedure for their construction in finitely many steps) is signaled by a Poissonian level statistics, whereas in the generic nonintegrable cases one finds level statistics of a Gaussian unitary ensemble of random matrices. Eigenvalue statistics can therefore be used as an efficient tool to identify integrable quantum nonequilibrium systems.

Universal K-matrix distribution in β = 2 ensembles of random matrices

The K-matrix, also known as the ‘Wigner reaction matrix’ in nuclear scattering or the ‘impedance matrix’ in electromagnetic wave scattering, is given essentially by an M × M diagonal block of the resolvent (E − H)−1 of a Hamiltonian H. For chaotic quantum systems, the Hamiltonian H can be modelled by random Hermitian N × N matrices taken from invariant ensembles with the Dyson symmetry index β = 1, 2 or 4. For β = 2, we prove by explicit calculation a universality conjecture by Brouwer (1995 Phys. Rev. B 51 16878–84), which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution with density in the limit N → ∞, provided the parameter M is fixed and the spectral parameter E is taken within the support of the eigenvalue distribution of H. In particular, we show that for a broad class of unitary invariant ensembles of random matrices, finite diagonal blocks of the resolvent are Cauchy distributed. The cases β = 1 and β = 4 remain outstanding.

LargeNclassical dynamics of holographic matrix models

Using a numerical simulation of the classical dynamics of the plane-wave and flat space matrix models of M-theory, we study the thermalization, equilibrium thermodynamics and fluctuations of these models as we vary the temperature and the size of the matrices, N. We present our numerical implementation in detail and several checks of its precision and consistency. We show evidence for thermalization by matching the time-averaged distributions of the matrix eigenvalues to the distributions of the appropriate Traceless Gaussian Unitary Ensemble of random matrices. We study the autocorrelations and power spectra for various fluctuating observables and observe evidence of the expected chaotic dynamics as well as a hydrodynamic type limit at large N, including near-equilibrium dissipation processes. These configurations are holographically dual to black holes in the dual string theory or M-theory and we discuss how our results could be related to the corresponding supergravity black hole solutions.

Supremum of the Airy2 Process Minus a Parabola on a Half Line

Let $\mathcal {A}_{2}(t)$ be the Airy2 process. We show that the random variable $$\sup_{t\leq\alpha} \bigl\{\mathcal {A}_2(t)-t^2 \bigr\}+\min\{0,\alpha \}^2 $$ has the same distribution as the one-point marginal of the Airy2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution F GUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution F GOE(41/3 x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay $e^{-\frac{4}{3} x^{3/2} }$ .

KAPPA-DEFORMED RANDOM-MATRIX THEORY BASED ON KANIADAKIS STATISTICS

We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index κ (Boltzmann–Gibbs entropy is recovered in the limit κ → 0), we propose the non-Gaussian deformations (κ ≠ 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the κ-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invariant as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index κ as a measure for deviation from the state of chaos. We also introduce a κ-deformed Porter–Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution.

Janossy Densities for Unitary Ensembles at the Spectral Edge

For a broad class of unitary ensembles of random matrices, we demonstrate the universal nature of the Janossy densities of eigenvalues near the spectral edge, providing a different formulation of the probability distributions of the limiting second, third, etc. largest eigenvalues of the ensembles in question. The approach is based on a representation of the Janossy densities in terms of a system of orthogonal polynomials, plus the steepest descent method of Deift and Zhou for the asymptotic analysis of the associated Riemann.Hilbert problem.

Spectral properties of embedded Gaussian unitary ensemble of random matrices with Wigner’s SU(4) symmetry

For m fermions in Ω number of single particle orbitals, each fourfold degenerate, we introduce and analyze in detail embedded Gaussian unitary ensemble of random matrices generated by random two-body interactions that are SU(4) scalar [EGUE(2)- SU(4)]. Here the SU(4) algebra corresponds to the Wigner’s supermultiplet SU(4) symmetry in nuclei. Embedding algebra for the EGUE(2)- SU(4) ensemble is U(4 Ω) ⊃ U( Ω) ⊗ SU(4). Exploiting the Wigner–Racah algebra of the embedding algebra, analytical expression for the ensemble average of the product of any two m particle Hamiltonian matrix elements is derived. Using this, formulas for a special class of U( Ω) irreducible representations (irreps) {4 r , p}, p = 0, 1, 2, 3 are derived for the ensemble averaged spectral variances and also for the covariances in energy centroids and spectral variances. On the other hand, simplifying the tabulations of Hecht for SU( Ω) Racah coefficients, numerical calculations are carried out for general U( Ω) irreps. Spectral variances clearly show, by applying Jacquod and Stone prescription, that the EGUE(2)- SU(4) ensemble generates ground state structure just as the quadratic Casimir invariant ( C 2) of SU(4). This is further corroborated by the calculation of the expectation values of C 2[ SU(4)] and the four periodicity in the ground state energies. Secondly, it is found that the covariances in energy centroids and spectral variances increase in magnitude considerably as we go from EGUE(2) for spinless fermions to EGUE(2) for fermions with spin to EGUE(2)- SU(4) implying that the differences in ensemble and spectral averages grow with increasing symmetry. Also for EGUE(2)- SU(4) there are, unlike for GUE, non-zero cross-correlations in energy centroids and spectral variances defined over spaces with different particle numbers and/or U( Ω) [equivalently SU(4)] irreps. In the dilute limit defined by Ω → ∞, r ≫ 1 and r/ Ω → 0, for the {4 r , p} irreps, we have derived analytical results for these correlations. All correlations are non-zero for finite Ω and they tend to zero as Ω → ∞.

Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

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. By using a new method, we calculate directly the moments of the density (which has been obtained in the work of Nevai and Dehesa, Van Assche and others on asymptotic zero distribution), and prove that scaling eigenvalues converge weakly, in probability and almost surely to the Nevai–Ullmann measure. Furthermore, we can prove that the density is invariant when the weight function is perturbed by a polynomial.