Dyson's Circular Ensemble Research Articles

Matrix Models and Eigenvalue Statistics for Truncations of Classical Ensembles of Random Unitary Matrices

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 $\beta>0$, not merely $\beta=1,2,4$.

Loop equation analysis of the circular β ensembles

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.

Chern–Simons matrix models, two-dimensional Yang–Mills theory and the Sutherland model

We derive some new relationships between matrix models of Chern–Simons gauge theory and of two-dimensional Yang–Mills theory. We show that q-integration of the Stieltjes–Wigert matrix model is the discrete matrix model that describes q-deformed Yang–Mills theory on S2. We demonstrate that the semiclassical limit of the Chern–Simons matrix model is equivalent to the Gross–Witten model in the weak-coupling phase. We study the strong-coupling limit of the unitary Chern–Simons matrix model and show that it too induces the Gross–Witten model, but as a first-order deformation of Dyson's circular ensemble. We show that the Sutherland model is intimately related to Chern–Simons gauge theory on S3, and hence to q-deformed Yang–Mills theory on S2. In particular, the ground-state wavefunction of the Sutherland model in its classical equilibrium configuration describes the Chern–Simons free energy. The correspondence is extended to Wilson line observables and to arbitrary simply laced gauge groups.

Stein’s Method and Characters of Compact Lie Groups

Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.

Random matrix ensembles with an effective extensive external charge

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.

Joint densities of secular coefficients for unitary matrices

We determine the joint density of secular coefficients and of the traces of powers for random unitary matrices from Dyson's circular ensembles. A byproduct are workable `unitarity conditions', necessary and sufficient for all roots of a polynomial to be unimodular and different. These findings will be useful for combinations of random-matrix theory with semiclassical approximations in calculating spectral properties of quantum systems whose classical limit is globally chaotic. In a first such application we show that reasonable quasi-energy spectra result by throwing dice a la random-matrix theory to complement restricted knowledge (rigorous or from short periodic orbits) of the secular polynomial of a unitary Floquet matrix.

Impact of localization on Dyson's circular ensemble

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to provide certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.

Universal Correlations Near a Singularity of Random Matrix Spectrum

Asymptotic correlation functions of the eigenvalues in the limit of large dimension of random matrices are evaluated for two models related to the Jacobi polynomials. One of the models is a generalization of Dyson's circular ensemble. Both models show identical unfolded correlations near a singularity of the spectrum. A previously unnoticed universal behavior of random matrix ensembles near a singularity is discussed.

Generalized circular ensemble of scattering matrices for a chaotic cavity with nonideal leads.

We consider the problem of the statistics of the scattering matrix S of a chaotic cavity (quantum dot), which is coupled to the outside world by non-ideal leads containing N scattering channels. The Hamiltonian H of the quantum dot is assumed to be an M x N hermitian matrix with probability distribution P(H) ~ det[lambda^2 + (H - epsilon)^2]^[-(beta M + 2- beta)/2], where lambda and epsilon are arbitrary coefficients and beta = 1,2,4 depending on the presence or absence of time-reversal and spin-rotation symmetry. We show that this ``Lorentzian ensemble'' agrees with microscopic theory for an ensemble of disordered metal particles in the limit M -> infinity, and that for any M >= N it implies P(S) ~ |det(1 - \bar S^{\dagger} S)|^[-(beta M + 2 - beta)], where \bar S is the ensemble average of S. This ``Poisson kernel'' generalizes Dyson's circular ensemble to the case \bar S \neq 0 and was previously obtained from a maximum entropy approach. The present work gives a microscopic justification for the case that the chaotic motion in the quantum dot is due to impurity scattering.

Global fluctuation formulas and universal correlations for random matrices and log-gas systems at infinite density

It is shown how the universal correlation function of Brézin and Zee, and Beenakker, for random matrix ensembles of Wigner-Dyson type with density support on a finite interval can be derived using a linear response argument and macroscopic electrostatics. The analogous formula for ensembles of unitary random matrices is derived using the same method, and the corresponding global fluctuation formula for the variance of a linear statistic is presented. The result for the universal correlation is checked using an exact result for the finite-system two-point correlation in the Dyson circular ensemble at all even β.

An Integration Method on Generalized Circular Ensembles

Dyson's circular ensembles of random matrices have been successfully applied to account for the statistical properties of complex spectra. We present an integration method on generalized circular ensembles. In order to express the correlation functions of unitary ensembles, Szego's orthogonal polynomials on the unit circle are used. Further, we introduce skew orthogonal polynomials on the unit circle to derive the correlation functions of orthogonal and symplectic ensembles.

Stochastic versus semiclassical approach to quantum chaotic scattering

We explore the universal features of quantum scattering systems for which the classical scattering is chaotic. We do so by comparing the semiclassical approach and the stochastic approach, exhibiting their mutual overlap and their individual limits of applicability. We investigate in particular predictions of both approaches for the average cross section, for the autocorrelation function of a pair of S-matrix elements, its Fourier transform, the Wigner time delay, and the distribution of the poles of the S-matrix in the complex energy plane. The relationship of both approaches with Dyson's circular ensemble of S-matrices is also exhibited.

Random-matrix description of chaotic scattering: Semiclassical approach.

We present a semiclassical theory for the two-point correlation function of the eigenphases of the S matrix for chaotic scattering. It is expressed as a sum of contributions from unstable periodic orbits of the classical scatttering mapping. Backed by numerical results and for correlation ranges ${r}^{\mathrm{*}}$(\ensuremath{\sim}1/\ensuremath{\Elzxh}) we obtain a universal function which is consistent with the result for Dyson's circular ensemble. This result adds to the conjecture that universal fluctuations governed by random matrix theory are the quantum manifestation of classical chaos.