Joint Distribution Of Eigenvalues Research Articles

Hermitian and non-Hermitian perturbations of chiral Gaussian β-ensembles

We compute the joint eigenvalue distribution for the rank one Hermitian and non-Hermitian perturbations of chiral Gaussian β-ensembles (β > 0) of random matrices.

General truncated linear statistics for the top eigenvalues of random matrices

Invariant ensembles, which are characterised by the joint distribution of eigenvalues P(λ 1, …, λ N ), play a central role in random matrix theory. We consider the truncated linear statistics with 1 ⩽ K ⩽ N, λ 1 > λ 2 > ⋯ > λ N and f a given function. This quantity has been studied recently in the case where the function f is monotonous. Here, we consider the general case, where this function can be non-monotonous. Motivated by the physics of cold atoms, we study the example f(λ) = λ 2 in the Gaussian ensembles of random matrix theory. Using the Coulomb gas method, we obtain the distribution of the truncated linear statistics, in the limit N → ∞ and K → ∞, with κ = K/N fixed. We show that the distribution presents two essential singularities, which arise from two infinite order phase transitions for the underlying Coulomb gas. We further argue that this mechanism is universal, as it depends neither on the choice of the ensemble, nor on the function f.

Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in addition to the usual level repulsion. The universal fluctuations of Gaussian Unitary and Ginibre Unitary universality classes in the large-matrix-size limit are shown to be well described by Wigner-like surmises for small-size matrices with eigenvalues on the circle and on the two-torus, respectively. To study the latter case, we introduce the Toric Unitary Ensemble, characterized by a flat joint eigenvalue distribution on the two-torus. Second, we study different physical situations where nonhermitian matrices arise: dissipative quantum systems described by a Lindbladian, non-unitary quantum dynamics described by nonhermitian Hamiltonians, and classical stochastic processes. We show that known integrable models have a flat distribution of complex spacing ratios whereas generic cases, expected to be chaotic, conform to Random Matrix Theory predictions. Specifically, we were able to clearly distinguish chaotic from integrable dynamics in boundary-driven dissipative spin-chain Liouvillians and in the classical asymmetric simple exclusion process and to differentiate localized from delocalized phases in a nonhermitian disordered many-body system.

Universal Behavior of the Corners of Orbital Beta Processes

Abstract There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1> \dots > a_N$. The joint eigenvalue distribution of the $(N-1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of the base field, of the explicit density formulas, we define the orbital beta processes. We prove the universal behavior of the virtual eigenvalues of the smallest $m$ principal submatrices, when $m$ is independent of $N$ and the eigenvalues $a_1> \dots > a_N$ grow linearly in $N$ and in such a way that the rescaled empirical measures converge weakly. The limiting object is the Gaussian beta corners process. As a byproduct of our approach, we prove a theorem on the asymptotics of multivariate Bessel functions.

Bit Error Probability for MMSE Receiver in GFDM Systems

In this letter, we consider the minimum mean-square error receiver for the generalized frequency division multiplexing system (GFDM) over frequency selective fading channels. We derive an approximate probability density function for the signal-to-interference-plus-noise ratio. This expression allows us to obtain a new approximate, but rather accurate formulation for the bit error probability for a $\mathcal {M}$ -quadrature amplitude modulation scheme. Our results resort on the pivotal properties exhibited by eigenvalues of a circulant matrix. Since the entries of the channel matrix $\text {H}_{\text {ch}}$ are complex Gaussian distributed, and the eigenvalues are given as a weighted sum of its entries, the joint eigenvalue distribution is also Gaussian. Comparisons of the simulated and analytical results validate our formulation and allow a quick and efficient tool to compute the bit error rate for the GFDM system.

Replica-symmetric approach to the typical eigenvalue fluctuations of Gaussian random matrices

We discuss an approach to compute the first and second moments of the number of eigenvalues IN that lie in an arbitrary interval of the real line for Gaussian random matrices. The method combines the standard replica-symmetric theory with a perturbative expansion of the saddle-point action up to (), leading to the correct logarithmic scaling of the variance as well as to an analytical expression for the correction to the average . Standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval. The limitations of the replica-symmetric method are unveiled by comparing our results with those derived through exact methods. The present work represents an important step to study the fluctuations of IN in non-invariant random matrix ensembles, where the joint distribution of eigenvalues is not known.

The Differential Entropy of the Joint Distribution of Eigenvalues of Random Density Matrices

We derive exactly the differential entropy of the joint distribution of eigenvalues of Wishart matrices. Based on this result, we calculate the differential entropy of the joint distribution of eigenvalues of random mixed quantum states, which is induced by taking the partial trace over the environment of Haar-distributed bipartite pure states. Then, we investigate the differential entropy of the joint distribution of diagonal entries of random mixed quantum states. Finally, we investigate the relative entropy between these two kinds of distributions.

A new joint eigenvalue distribution of finite random matrix for cognitive radio networks

A new joint eigenvalue distribution (JED) based on dual extreme eigenvalues of finite random matrix is proposed in this study. Different from conventional JED based on K(K ≥ 2) variables, only two variables are included in the proposed formulation. The upper and lower bounds of the new JED are determined. The new JED provides a simple and efficient way to deduce the distributions of key characteristics of finite random matrix, such as the extreme (largest and smallest) eigenvalues, standard condition number, and scaled largest eigenvalue. Moreover, a novel cooperative spectrum sensing (CSS) scheme based on the new JED is proposed for cognitive radio networks. The simulation results verify the proposed JED and the proposed CSS scheme can improve sensing performance.

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random matrices. Typically ensembles with correlations among the matrix elements are much more difficult to solve. Using a combination of supersymmetry, superbosonisation and bi-orthogonal functions we are able to determine all spectral k-point density correlation functions of H for arbitrary matrix size N. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the recent results by Kumar for the joint eigenvalue distribution of H serve as our starting point. In this case the ensemble has a bi-orthogonal structure and we explicitly determine its kernel, providing its exact solution for finite N. The kernel follows from computing the expectation value of a single characteristic polynomial. In the general non-degenerate case the generating function for the k-point resolvent is determined from a supersymmetric evaluation of the expectation value of k ratios of characteristic polynomials. Numerical simulations illustrate our findings for the spectral density at finite N and we also give indications how to do the asymptotic large-N analysis.

POLARIMETRIC TARGET DETECTION USING STATISTIC OF THE DEGREE OF POLARIZATION

The degree of polarization (DoP) can be utilized as a detection statistic in the polarimetric radar to achieve target detection performance improvement. In this paper, a polarimetric radar model is established, which includes reflections from both target and clutter at first. Then, probability density functions (PDFs) of the estimated DoP are expressed in closed form, which is derived from joint eigenvalue distributions of complex noncentral Wishart matrices. The detector is developed and evaluated theoretically on the basis of the statistical properties of the DoP. Finally, a comparison between the new DoP detector and single-polarization detector is presented against real data. The performance improvement is demonstrated by the comparison results.

Multi-matrix models at general coupling

The eigenvalue distribution of Hoppe’s two-matrix model is investigated in detail as a function of the model’s coupling. For small couplings it is a perturbed Wigner semicircle, while for large couplings it is a parabolic distribution which crosses over to a Wigner semicircle for eigenvalues within an approximately inverse coupling from the boundary of the distribution. The model is approximately commuting at large couplings and we find the joint eigenvalue distribution of the two matrices. We also study a related three-matrix model finding the corresponding three-dimensional eigenvalue distribution there also. The techniques developed here are more widely applicable to other multi-matrix models.

Near commuting multi-matrix models

We investigate the radial extent of the eigenvalue distribution for Yang-Mills type matrix models. We show that, a three matrix Gaussian model with complex Myers coupling, to leading order in strong coupling is described by commuting matrices whose joint eigenvalue distribution is uniform and confined to a ball of radius R=(3Pi/2g)^(1/3). We then study, perturbatively, a 3-component model with a pure commutator action and find a radial extent broadly consistent with numerical simulations.

The Diversity-Multiplexing Tradeoff of the MIMO Half-Duplex Relay Channel

The fundamental diversity-multiplexing tradeoff of the three-node, multi-input, multi-output (MIMO), quasi-static, Rayleigh faded, half-duplex relay channel is characterized for an arbitrary number of antennas at each node and in which opportunistic scheduling (or dynamic operation) of the relay is allowed, i.e., the relay can switch between receive and transmit modes at a channel dependent time. In this most general case, the diversity-multiplexing tradeoff is characterized as a solution to a simple, two-variable optimization problem. This problem is then solved in closed form for special classes of channels defined by certain restrictions on the numbers of antennas at the three nodes. The key mathematical tool developed here that enables the explicit characterization of the diversity-multiplexing tradeoff is the joint eigenvalue distribution of three mutually correlated random Wishart matrices. Besides being relevant here, this distribution result is interesting in its own right. Previously, without actually characterizing the diversity-multiplexing tradeoff, the optimality in this tradeoff metric of the dynamic compress-and-forward protocol based on the classical compress-and-forward scheme of Cover and El Gamal was shown by Yuksel and Erkip. However, this scheme requires global channel state information at the relay. In this paper, the so-called quantize-map and forward (QMF) coding scheme is adopted as the achievability scheme with the added benefit that it achieves optimal tradeoff with only the knowledge of the (channel dependent) switching time at the relay node. Moreover, in special classes of the MIMO half-duplex relay channel, the optimal tradeoff is shown to be attainable even without this knowledge. Such a result was previously known only for the half-duplex relay channel with a single antenna at each node, also via the QMF scheme. More generally, the explicit characterization of the tradeoff curve in this study enables the in-depth comparisons herein of full-duplex versus half-duplex relaying as well as static versus dynamic relaying, both as a function of the numbers of antennas at the three nodes.

On Eigenvalues of the Sum of Two Random Projections

We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.

On the Application of Character Expansions for MIMO Capacity Analysis

To evaluate the unitary integrals, such as the well-known Harish–Chandra–Itzykson–Zuber integral, character expansions were developed by Balantekin, where the matrix integrand is a group member; i.e., a square matrix with a nonzero determinant. Recently, this method has been exploited to derive the joint eigenvalue distributions of the Wishart matrices; i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\bf H}{\bf H}^{\ast}$</tex></formula> where <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\bf H}$</tex></formula> is the complex Gaussian random channel matrix of a multiple-input multiple-output (MIMO) system. The joint eigenvalue distributions are used to calculate the moment generating function of the mutual information (ergodic capacity) of a MIMO channel. In this paper, we show that the previous integration framework presented in the literature is not correct, and results in incorrect joint eigenvalue distributions for the Ricean and full-correlated Rayleigh MIMO channels. We develop a new framework to apply the character expansions for integrations over the unitary group, involving general rectangular complex matrices in the integrand. We derive the correct distribution functions and use them to obtain the capacity of the Ricean and correlated Rayleigh MIMO systems in a unified and straightforward approach. The integration technique proposed in this paper is general enough to be used for other unitary integrals in engineering, mathematics, and physics.

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.

Integrable random matrix ensembles

We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner–Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to the integrability of the underlying system. Formulae for spacing distributions and level compressibility are obtained for various instances of such ensembles.

Bulk Scaling Limit of the Laguerre Ensemble

We consider the $\beta$-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensembles exists for all $\beta&gt;0$ for a general family of parameters and it is the same as the bulk scaling limit of the corresponding $\beta$-Hermite ensemble.

Analytical Framework for Optimal Combining With Arbitrary-Power Interferers and Thermal Noise

The performance of multiple-input-multiple-output systems with optimum combining (OC) is studied in a Rayleigh fading environment with arbitrary-power cochannel interference and thermal noise. Based on the joint eigenvalue distributions of quadratic functions of complex Gaussian matrices, a closed-form expression for the exact distribution of the output signal-to-interference-plus-noise ratio (SINR) is derived. A closed-form expression for the exact moment-generating function (MGF) of the output SINR of single-input-multiple-output (SIMO) systems is also derived. From the exact MGF, the moments of the output SINR and the symbol error rate of various M-ary modulation schemes are obtained. We verify the accuracy of our analytical results with numerical examples. The new analytical framework provides a simple and accurate way to assess the effects of equal- and unequal-power cochannel interferers and thermal noise on the performance of OC.

Phases of thermal 𝒩 = 2 quiver gauge theories

We consider large N U(N)^M thermal N=2 quiver gauge theories on S^1 x S^3. We obtain a phase diagram of the theory with R-symmetry chemical potentials, separating a low-temperature/high-chemical potential region from a high-temperature/low-chemical potential region. In close analogy with the N=4 SYM case, the free energy is of order O(1) in the low-temperature region and of order O(N^2 M) in the high-temperature phase. We conclude that the N=2 theory undergoes a first order Hagedorn phase transition at the curve in the phase diagram separating these two regions. We observe that in the region of zero temperature and critical chemical potential the Hilbert space of gauge invariant operators truncates to smaller subsectors. We compute a l-loop effective potential with non-zero VEV's for the scalar fields in a sector where the VEV's are homogeneous and mutually commuting. At low temperatures the eigenvalues of these VEV's are distributed uniformly over an S^5/Z_M which we interpret as the emergence of the S^5/Z_M factor of the holographically dual geometry AdS_5 x S^5/Z_M. Above the Hagedorn transition the eigenvalue distribution of the Polyakov loop opens a gap, resulting in the collapse of the joint eigenvalue distribution from S^5/Z_M x S^1 into S^6/Z_M.