Eigenvalue Density Research Articles

Efficient computational method using random matrices describing critical thermodynamics

Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.

From spectral to scattering form factor

We propose a novel indicator for chaotic quantum scattering processes, the scattering form factor (ScFF). It is based on mapping the locations of peaks in the scattering amplitude to random matrix eigenvalues, and computing the analog of the spectral form factor (SFF). We compute the spectral and scattering form factors of several non-chaotic systems. We determine the ScFF associated with the phase shifts of the leaky torus, closely related to the distribution of the zeros of Riemann zeta function. We compute the ScFF for the decay amplitude of a highly excited string states into two tachyons. We show that it displays the universal features expected from random matrix theory - a decline, a ramp and a plateau - and is in general agreement with the Gaussian unitary ensemble. It also shows some new features, owning to the special structure of the string amplitude, including a “bump” before the ramp associated with gaps in the average eigenvalue density. The “bump” is removed for highly excited string states with an appropriate state dependent unfolding. We also discuss the SFF for the Gaussian β-ensemble, writing an interpolation between the known results of the Gaussian orthogonal, unitary, and symplectic ensembles.

Spectral moments of the real Ginibre ensemble

The moments of the real eigenvalues of real Ginibre matrices are investigated from the viewpoint of explicit formulas, differential and difference equations, and large N expansions. These topics are inter-related. For example, a third-order differential equation can be derived for the density of the real eigenvalues, and this can be used to deduce a second-order difference equation for the general complex moments M2pr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{2p}^\ extrm{r}$$\\end{document}. The latter are expressed in terms of the 3F2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_3 F_2$$\\end{document} hypergeometric functions, with a simplification to the 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2 F_1$$\\end{document} hypergeometric function possible for p=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=0$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, allowing for the large N expansion of these moments to be obtained. The large N expansion involves both integer and half-integer powers of 1/N. The three-term recurrence then provides the large N expansion of the full sequence {M2pr}p=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{ M_{2p}^\ extrm{r} \\}_{p=0}^\\infty $$\\end{document}. Fourth- and third-order linear differential equations are obtained for the moment generating function and for the Stieltjes transform of the real density, respectively, and the properties of the large N expansion of these quantities are determined.

Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices

AbstractWe consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal up to the factor . The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the ‐dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near‐optimal lower tail estimate for the small singular values of , is of independent interest.

Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations.

Absorption-invariant focusing efficiency for wavefront-shaping controlled reflection from absorbing disordered media.

We numerically model the influence of absorption on wavefront-shaping controlled reflection from absorbing disordered media and provide experimental verification of our model. We find that absorption modifies the reflection eigenvalue density, the average reflectance, and the reflection matrix element density. However, we also find that despite these effects, the efficiency of wavefront-shaping controlled reflection is invariant with absorption.

Finite size corrections for real eigenvalues of the elliptic Ginibre matrices

In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.

Improved spectral cluster bounds for orthonormal systems

Abstract We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ {p=\infty} , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) {[\lambda^{2},(\lambda+1)^{2})} to [ λ 2 , ( λ + ϵ ⁢ ( λ ) ) 2 ) {[\lambda^{2},(\lambda+\epsilon(\lambda))^{2})} , where ϵ ⁢ ( λ ) {\epsilon(\lambda)} is a function of λ that goes to 0 as λ goes to ∞ {\infty} . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].

On the Density of Complex Eigenvalues of Wigner Reaction Matrix in a Disordered or Chaotic System with Absorption

On the Density of Complex Eigenvalues of Wigner Reaction Matrix in a Disordered or Chaotic System with Absorption

Local Marchenko–Pastur law at the hard edge of the sample covariance ensemble

Consider an N by N matrix X of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of X*X converges to the Marchenko–Pastur law on the optimal scale with probability 1. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under the assumption that the entries have a finite fourth moment and are truncated at N1/4, or alternatively with exploding moments. In this work we simplify and adapt the methods from prior papers of Götze–Tikhomirov [Probab. Relat. Fields 165(1–2), 163–233 (2016)] and Cacciapuoti–Maltsev–Schlein [Probab. Theory Relat. Fields 163(1–2), 1–59 (2015)] to covariance matrices.

Exploring Transition from Stability to Chaos through Random Matrices

This study explores the application of random matrices to track chaotic dynamics within the Chirikov standard map. Our findings highlight the potential of matrices exhibiting Wishart-like characteristics, combined with statistical insights from their eigenvalue density, as a promising avenue for chaos monitoring. Inspired by a technique originally designed for detecting phase transitions in spin systems, we successfully adapted and applied it to identify analogous transformative patterns in the context of the Chirikov standard map. Leveraging the precision previously demonstrated in localizing critical points within magnetic systems in our prior research, our method accurately pinpoints the Chirikov resonance overlap criterion for the chaos boundary at K≈2.43, reinforcing its effectiveness. Additionally, we verified our findings by employing a combined approach that incorporates Lyapunov exponents and bifurcation diagrams. Lastly, we demonstrate the adaptability of our technique to other maps, establishing its capability to capture the transition to chaos, as evidenced in the logistic map.

Two-species k-body embedded Gaussian unitary ensembles: q-normal form of the eigenvalue density

The eigenvalue density generated by an embedded Gaussian unitary ensemble with k-body interactions for two-species (say π and ν ) fermion systems is investigated by deriving formulas for the lowest six moments. Assumed in constructing this ensemble, called EGUE(), is that the π fermions (m 1 in number) occupy N 1 number of degenerate single particle (sp) states, and similarly the ν fermions (m 2 in number) occupy N 2 number of degenerate sp states. The Hamiltonian is assumed to be k-body preserving . Formulas with finite corrections and asymptotic limit formulas both show that the eigenvalue density takes q-normal form with the q parameter defined by the fourth moment. The EGUE() formalism and results are extended to two-species boson systems. The results in this work show that the q-normal form of the eigenvalue density established only recently for identical fermion and boson systems extends to two-species fermion and boson systems.

The Hydraulic and Boundary Characteristics of a Dike Breach Based on Cluster Analysis

It is important to determine the hydraulic boundary eigenvalues of typical embankment breaches before carrying out research on their occurrence mechanisms and assessing their repair technology. However, it is difficult to obtain the hydraulic boundary conditions of the typical levee breaches accurately with minor or incomplete measured data due to the complexity and instability of the levee breach. Based on more than 100 groups of domestic and foreign test data of embankment/earth dam failures, the correlation between the hydraulic boundary eigenvalues of a breach was established based on the cluster analysis approach. Additionally, the missing values were imputed after correlating and fitting. Meanwhile, the hydraulic boundary parameters and the related equations of a generalized typical breach were obtained through the statistical analysis of the probability density of the dimensionless eigenvalues of the breach. The analysis showed that the width of the breach mainly ranges in 20~100 m, while the water head of the breach is 4~12 m, and the velocity of the breach is 2~8 m/s. The distribution probabilities of all them are about 64~71%. The probability density of the width-to-depth ratio and the Froude number of the breach are both subject to normal distribution characteristics. The distribution frequency of the width-to-depth ratio of 3~8 is approximately 55%, and the Froude number of 0.4~0.8 is approximately 60%. These methods and findings might provide valuable support for the statistical research of the boundary and hydraulic characteristics of the breach, and the closure technology of breach.

Ensemble averaging in JT gravity from entanglement in Matrix Quantum Mechanics

We consider the generalization of a matrix integral with arbitrary spectral curve ρ0(E) to a 0+1D theory of matrix quantum mechanics (MQM). Using recent techniques for 1D quantum systems at large-N, we formulate a hydrodynamical effective theory for the eigenvalues. The result is a simple 2D free boson BCFT on a curved background, describing the quantum fluctuations of the eigenvalues around ρ0(E), which is now the large-N limit of the quantum expectation value of the eigenvalue density operator hat{rho}(E) .The average over the ensemble of random matrices becomes a quantum expectation value. Equal-time density correlations reproduce the results (including non-perturbative corrections) of random matrix theory. This suggests an interpretation of JT gravity as dual to a one-time-point reduction of MQM.As an application, we compute the Rényi entropy associated to a bipartition of the eigenvalues. We match a previous result by Hartnoll and Mazenc for the c = 1 matrix model dual to two-dimensional string theory and extend it to arbitrary ρ0(E). The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The entropy is naturally finite and displays a large amount of short range entanglement, proportional to the microcanonical entropy. We also compute the reduced density matrix for a subset of n < N eigenvalues.

Late time behavior of n-point spectral form factors in Airy and JT gravities

We study the late time behavior of n-point spectral form factors (SFFs) in two-dimensional Witten-Kontsevich topological gravity, which includes both Airy and JT gravities as special cases. This is conducted in the small ħ expansion, where hbar sim {e}^{-1/{G}_{textrm{N}}} is the genus counting parameter and nonperturbative in Newton’s constant GN. For one-point SFF, we study its absolute square at two different late times. We show that it decays by power law at t ~ ħ−2/3 while it decays exponentially at t ~ ħ−1 due to the higher order corrections in ħ. We also study general n(≥2)-point SFFs at t ~ ħ−1 in the leading order of the ħ expansion. We find that they are characterized by a single function, which is essentially the connected two-point SFF and is determined by the classical eigenvalue density ρ0(E) of the dual matrix integral. These studies suggest that qualitative behaviors of n-point SFFs are similar in both Airy and JT gravities, where our analysis in the former case is based on exact results.

Microstructure modeling and experimental verification of isotropic magnetorheological elastomers based on edge-centered cubic structure

Aiming at the problem that the existing mechanism cannot accurately predict the shear properties of magnetorheological elastomers (MREs), in this research, a physical microscopic constitutive model of isotropic MREs based on an edge-centered cubic (ECC) lattice structure was constructed to accurately reflect the shear properties of isotropic MREs under quasi-static and dynamic shear modes and different magnetic fields. In order to reduce the error of the potential energy model in the modeling process, the accuracy of the MRE total potential energy theoretical model based on the ECC lattice structure was improved by transforming to under finite strain, and the maximum improvement rate of the model accuracy was 6%. Then, in order to construct a microstructure-based dynamic model, the Langevin equation was constructed based on the ECC lattice, and an exact solvable model of the topological regular network was constructed by the Rouse chain. The relationship between the relaxation time, eigenvalue and magnetic flux density was revealed, and then the updated shear modulus was solved. Then, the modulus values of the prepared isotropic MRE samples at different frequencies and magnetic flux densities were compared with the theoretical model values, and the model failure phenomenon under dynamic high magnetic field, in which the theoretical results are inconsistent with the experimental results, was modified by the method of piecewise function and optimization parameters. The final results show that the correlation coefficient between the experimental results and the theoretical results can reach 99%.

Almost-Hermitian random matrices and bandlimited point processes

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow “band” about the real line, of height proportional to tfrac{1}{N}, where N denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward’s equation and a property which we call “cross-section convergence”, which relates the large-N limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko–Pastur law for LUE. As an application of our approach, we prove the bulk universality of the almost-circular ensembles.

Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with k-body interactions.

Embedded random matrix ensembles with k-body interactions are well established to be appropriate for many quantum systems. For these ensembles the two point correlation function is not yet derived, though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues, say E and E^{'}. Fluctuation measures such as the number variance and Dyson-Mehta Δ_{3} statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with k-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called q-normal distribution. With this, the eigenvalue density can be expanded by starting with the q-normal form and using the associated q-Hermite polynomials He_{ζ}(x|q). Covariances S_{ζ}S_{ζ^{'}}[over ¯] (overline representing ensemble average) of the expansion coefficients S_{ζ} with ζ≥1 here determine the two-point function as they are a linear combination of the bivariate moments Σ_{PQ} of the two-point function. Besides describing all these, in this paper formulas are derived for the bivariate moments Σ_{PQ} with P+Q≤8, of the two-point correlation function, for the embedded Gaussian unitary ensembles with k-body interactions [EGUE(k)] as appropriate for systems with m fermions in N single particle states. Used for obtaining the formulas is the SU(N) Wigner-Racah algebra. These formulas with finite N corrections are used to derive formulas for the covariances S_{ζ}S_{ζ^{'}}[over ¯] in the asymptotic limit. These show that the present work extends to all k values, the results known in the past in the two extreme limits with k/m→0 (same as q→1) and k=m (same as q=0).

Spectral domain isolation of ballistic component in visible light OCT based on random matrix description

Visible light optical coherence tomography (vis-OCT) provides a unique tool for imaging both structure and oxygen metabolism in ophthalmology. Working in visible light bandwidth, it suffers from noises due to strong scattering, especially in the blood. This work established the random matrix (RM) description of vis-OCT’s k-space data as ballistic and multiple scattering components. The eigenvalue density of the hybrid RM follows a low-rank biased Marčenko–Pastur law. The ballistic component can thus be separated out using a generalized likelihood ratio test algorithm. The RM-based method was validated by both the Monte Carlo simulation and ex vivo pure blood phantom study. We further demonstrated that the RM-based method could significantly improve the imaging quality in the human fundus, showing more details of the layered structure than current vis-OCT with ∼23.6% increase of signal-to-noise ratio, measuring the blood oxygen value more accurately, and enabling better structure visualization than the traditional method, a 1.6-fold higher contrast-to-noise ratio in raster scan mode. The isolated ballistic component also fits the Beer–Lambert law better, giving more accurate oxygen saturation in arc scan mode. The RM-based method significantly improves the reconstruction quality in 3D and facilitates clinical diagnostics. As a general framework, random matrix description also provides a new separation strategy to estimate the ballistic component in other spectral domain OCT techniques.

On the Spectral Form Factor for Random Matrices

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models (Forrester in J Stat Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys 387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, supplementing the recently proven Wigner–Dyson universality (Cipolloni et al. in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7) to larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.