Random Matrix Research Articles

Density of Avoided Crossings and Diabatic Representation.

Electronic structure theory describes the properties of solids using Bloch states that correspond to highly symmetrical nuclear configurations. However, nuclear thermal motion destroys translation symmetry. Here, we describe two approaches relevant to the time evolution of electronic states in the presence of thermal fluctuations. On the one hand, the direct solution of the time-dependent Schrodinger equation for a tight-binding model reveals the diabatic nature of time evolution. On the other hand, because of random nuclear configurations, the electronic Hamiltonian falls into the class of random matrices, which have universal features in their energy spectra. In the end, we discuss combining two approaches to obtain new insights into the influence of thermal fluctuations on electronic states.

Joint distribution of the cokernels of random p-adic matrices

Abstract In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and let P 1 ⁢ ( t ) , … , P l ⁢ ( t ) ∈ ℤ p ⁢ [ t ] {P_{1}(t),\ldots,P_{l}(t)\in\mathbb{Z}_{p}[t]} be monic polynomials whose reductions modulo p in 𝔽 p ⁢ [ t ] {\mathbb{F}_{p}[t]} are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok ⁡ ( P 1 ⁢ ( A ) ) , … , cok ⁡ ( P l ⁢ ( A ) ) {{\operatorname{cok}}(P_{1}(A)),\ldots,{\operatorname{cok}}(P_{l}(A))} for a random n × n {n\times n} matrix A over ℤ p {\mathbb{Z}_{p}} with respect to the Haar measure as n → ∞ {n\rightarrow\infty} . By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels cok ⁡ ( A ) {{\operatorname{cok}}(A)} and cok ⁡ ( A + B n ) {{\operatorname{cok}}(A+B_{n})} become independent as n → ∞ {n\rightarrow\infty} , where B n {B_{n}} is a fixed n × n {n\times n} matrix over ℤ p {\mathbb{Z}_{p}} for each n and A is a random n × n {n\times n} matrix over ℤ p {\mathbb{Z}_{p}} .

Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

We analyse the stability of large, linear dynamical systems of variables that interact through a fully connected random matrix and have inhomogeneous growth rates. We show that in the absence of correlations between the coupling strengths, a system with interactions is always less stable than a system without interactions. Contrarily to the uncorrelated case, interactions that are antagonistic, i.e., characterised by negative correlations, can stabilise linear dynamical systems. In particular, when the strength of the interactions is not too strong, systems with antagonistic interactions are more stable than systems without interactions. These results are obtained with an exact theory for the spectral properties of fully connected random matrices with diagonal disorder.

Spacing ratio statistics of multiplex directed networks

Eigenvalues statistics of various many-body systems have been widely studied using the nearest neighbor spacing distribution under the random matrix theory framework. Here, we numerically analyze eigenvalue ratio statistics of multiplex networks consisting of directed Erdős-Rényi random networks layers represented as, first, weighted non-Hermitian random matrices and then weighted Hermitian random matrices. We report that the multiplexing strength rules the behavior of average spacing ratio statistics for multiplexing networks represented by the non-Hermitian and Hermitian matrices, respectively. Additionally, for both these representations of the directed multiplex networks, the multiplexing strength appears as a guiding parameter for the eigenvector delocalization of the entire system. These results could be important for driving dynamical processes in several real-world multilayer systems, particularly, understanding the significance of multiplexing in comprehending network properties.

The measurement in classical and quantum theory

The Bohigas-Giannoni-Schmit (BGS) conjecture states that the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble. Here, this conjecture is considered in the context of a recently discovered geometric relationship between classical and quantum mechanics. Motivated by BGS, we conjecture that the Hamiltonian of a system whose classical counterpart performs a random walk can be modeled by a family of independent random matrices from the Gaussian unitary ensemble. By accepting this conjecture, we find a relationship between the process of observation in classical and quantum physics, derive irreversibility of observation and describe the boundary between the micro and macro worlds.

Situation Awareness of Energy Internet of Things in Smart City Based on Digital Twin: From Digitization to Informatization

Rapid growth of diversity, uncertainty, and coupling effect of units in modern energy systems jointly challenge the traditional model-based situation awareness (SA) in energy internet of thing (EIoT). This work explores digital twin of EIoT (EIoT-DT), and then provides a novel data-driven SA paradigm, named DT-SA, as a promising alternative. Based on the combination of the latest data technologies and machine learning algorithms, DT-SA transfers those stubborn SA challenges to digital space, and then addresses them by building a domain-specific and data-friendly DT model upon massive data. The established model can be quantitatively tested via iterative virtual-real interaction, and thus be evaluated and updated through closed-loop feedback to improve its performance in the physical world. To this end, some engineering and scientific problems are raised: a) virtual-real interaction mechanism relevant to resource flow and data flow; b) unified modelling and analysis of heterogeneous spatial-temporal data; c) DT configuration and evolution; and d) domain-specific DT-SA characterization. To solve these problems, cloud-edge-terminal configuration, big data analytics (BDA), digital twin, and SA indicator systems are studied, respectively. Then, random matrix theory (RMT), overarching DT-SA framework are designed as a roadmap. Besides, some potential applications and under-going projects on the terminal, edge, or cloud are discussed, e.g., condition assessment of equipment, digital monitoring and diagnosis of power grid network, and EIoT construction in the smart city. Lastly, some perspectives and recommendations are proposed in conclusion for future researches. This research can be regarded as an efficient handbook for both energy engineering and data science, which may benefit enterprise digitization, smart city, etc.

An RMT-based generalized Bayesian information criterion for signal enumeration

This paper provides a method for enumerating signals impinging on an array of sensors based on the generalized Bayesian information criterion (GBIC). The proposed method motivates by a statistic for testing the sphericity of the covariance matrix when the sample size n is less than the dimension m. The statistic consists of the first four moments of sample eigenvalue distribution and relaxes the assumption of Gaussian distribution. We derive the asymptotical distribution of the statistic as m, n tends to infinity at the same ratio by random matrix theory and propose the expression of GBIC for determining the signal number. Numerical simulations demonstrate that the proposed method has a high probability of detection in both the Gaussian and the non-Gaussian noise, and performs better than other methods.

ReBLS Denoiser: An Effective Denoiser based on the Residual Broad Learning System for CT Images

Image denoising plays an important role in CT sampling. Although good results have been achieved by filter-based, model-based and deep learning-based denoising methods, there are still many demanding constraints in real applications. In order to improve accuracy and efficiency, a residual learning network based on broad learning system (ReBLS) is proposed. First, features are extracted from the noisy images to generate a feature node layer. Subsequently, through a random matrix after orthogonal normalization, the feature nodes are reinforced into an enhanced node layer. Finally, the noise will be removed by the residual network. Compared with commonly used deep learning scheme, the broad learning replaces depth with expansion in breadth, the training and process time of broad learning is much shorter. Meanwhile, the learning of residuals can simplify the learning process and enhance the feature extraction. Experiments on the open dataset fully prove the accuracy and efficiency by comparing with five state-of-the-art denoising methods.

Random normal matrices in the almost-circular regime

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow “band” around the unit circle of width proportional to 1∕n, where n is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.

Central limit theorem of linear spectral statistics of high-dimensional sample correlation matrices

A high-dimensional sample correlation matrix is an important random matrix in multivariate statistical analysis. Its central limit theory is one of the main theoretical bases for making statistical inferences on high-dimensional correlation matrices. Under the high-dimensional framework in which the data dimension tends to infinity proportionally with the sample size, we establish the central limit theorems (CLT) for the linear spectral statistics (LSS) of sample correlation matrices in two settings: (1) the population follows an independent component structure; (2) the population follows an elliptical structure, including some heavy-tailed distributions. The results show that the CLTs of the LSS of the sample correlation matrices are very different in the two settings. In particular, even if the population correlation matrix is an identity matrix, the CLTs are different in the two settings. An application of our two established CLTs is provided.

VICARIOUS EMBARRASSMENT SCALE: MORE OF CULTURE THAN EMPATHY

Traditionally, the vicarious embarrassment scale (VES) is developed by involving a protagonist with no relationship with the observer. This condition becomes problematic, especially in collectivistic contexts with interdependence in personal social relations. The protagonists of the original VES were modified from strangers to friends and family and were compared their psychometric properties and associations with empathy and self-construals. A total of 112 university students in Yogyakarta, Indonesia, filled out the online questionnaires voluntarily. Tests with Unrestricted Factor Analysis showed the retrieved eigenvalues of VES Other, VES Friend, and VES Family accounted for 61.54, 68.29, and 77.05, respectively. All VES were unidimensional, according to the parallel analysis with robust parallel using 500 random polychoric correlation matrices. Good internal consistency reliability was achieved, and fit criteria were met. This study supported previous findings of VES disassociation with empathy. The importance of cultural values reflected in self-construal showed interdependent self-construal relations with different protagonists with all VE scales. In contrast, independent self-construal was associated only with vicarious embarrassment with an unknown protagonist.

Deterministic Construction of Compressed Sensing Measurement Matrix with Arbitrary Sizes via QC-LDPC and Arithmetic Sequence Sets

It is of great significance to construct deterministic measurement matrices with good practical characteristics in Compressed Sensing (CS), including good reconstruction performance, low memory cost and low computing resources. Low-density-parity check (LDPC) codes and CS codes can be closely related. This paper presents a method of constructing compressed sensing measurement matrices based on quasi-cyclic (QC) LDPC codes and arithmetic sequence sets. The cyclic shift factor in each submatrix of QC-LDPC is determined by arithmetic sequence sets. Compared with random matrices, the proposed method has great advantages because it is generated based on a cyclic shift matrix, which requires less storage memory and lower computing resources. Because the restricted isometric property (RIP) is difficult to verify, mutual coherence and girth are used as computationally tractable indicators to evaluate the measurement matrix reconstruction performance. Compared with several typical matrices, the proposed measurement matrix has the minimum mutual coherence and superior reconstruction capability of CS signal according to one-dimensional (1D) signals and two-dimensional (2D) image simulation results. When the sampling rate is 0.2, the maximum (minimum) gain of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) is up to 2.89 dB (0.33 dB) and 0.031 (0.016) while using 10 test images. Meanwhile, the reconstruction time is reduced by nearly half.

The spherical ensemble and quasi-Monte-Carlo designs

The spherical ensemble is a well-known ensemble of N repulsive points on the two-dimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter s≤2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\le 2.$$\\end{document} The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.

TRP-Oriented Hyperspectral Remote Sensing Image Classification Using Entropy-Weighted Ensemble Algorithm

The problem that the randomly generated random projection matrix will lead to unstable classification results is addressed in this paper. To this end, a Tighter Random Projection-oriented entropy-weighted ensemble algorithm is proposed for classifying hyperspectral remote sensing images. In particular, this paper presents a random projection matrix selection strategy based on the separable information of a single class able to project the features of a certain class of objects. The projection result is measured by the degree of separability, thereby obtaining the low-dimensional image with optimal separability of the class. After projecting samples with the same random projection matrix, to calculate the distance matrix, the Minimum Distance classifier is devised, repeating for all classes. Finally, the weight of the distance matrix is considered in ensemble classification by using the information entropy. The proposed algorithm is tested on real hyperspectral remote sensing images. The experiments show an increase in both stability and performance.

A Note on Cumulant Technique in Random Matrix Theory.

We discuss the cumulant approach to spectral properties of large random matrices. In particular, we study in detail the joint cumulants of high traces of large unitary random matrices and prove Gaussian fluctuation for pair-counting statistics with non-smooth test functions.

Properties of eigenmodes and quantum-chaotic scattering in a superconducting microwave Dirac billiard with threefold rotational symmetry

We report on experimental studies that were performed with a microwave Dirac billiard (DB), that is, a flat resonator containing metallic cylinders arranged on a triangular grid, whose shape has a threefold rotational (${C}_{3}$) symmetry. Its band structure exhibits two Dirac points (DPs) that are separated by a nearly flat band. We present a procedure that we employed to identify eigenfrequencies and to separate the eigenstates according to their transformation properties under rotation by $\frac{2\ensuremath{\pi}}{3}$ into the three ${C}_{3}$ subspaces. This allows us to verify previous numerical results of Zhang and Dietz [Phys. Rev. B 104, 064310 (2021)], thus confirming that the properties of the eigenmodes coincide with those of artificial graphene around the lower DP, and they are well described by a tight-binding model for a honeycomb-kagome lattice of corresponding shape. Above all, we investigate the properties of the wave-function components in terms of the fluctuation properties of the measured scattering matrix, which are numerically not accessible. They are compared to random-matrix theory predictions for quantum-chaotic scattering systems exhibiting extended or localized states in the interaction region, that is, the DB. Even in regions where the wave functions are localized, the spectral properties coincide with those of typical quantum systems with chaotic classical counterparts.

Constraining Weil–Petersson volumes by universal random matrix correlations in low-dimensional quantum gravity

Based on the discovery of the duality between Jackiw–Teitelboim quantum gravity and a double-scaled matrix ensemble by Saad, Shenker and Stanford in 2019, we show how consistency between the two theories in the universal random matrix theory (RMT) limit imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil–Petersson (WP) volumes, solving a celebrated nonlinear recursion formula that is notoriously difficult to analyse. Since our results imply linear relations between the coefficients of the WP volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction. In this way, we propose a long-term program to improve the understanding of mathematically hard aspects concerning moduli spaces of hyperbolic manifolds by using universal RMT results as input.

On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size N goes to infinity. These quantities depend on two parameters k and q and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size N. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey [7]. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

Detection of tandem repeats in the Capsicum annuum genome

In this study, we modified the multiple alignment method based on the generation of random position weight matrices (RPWM) and used it to search for tandem repeats (TRs) in the Capsicum annuum genome. The application of the modified (m)RPWM method, which considers the correlation of adjusting nucleotides, resulted in the identification of 908,072 TR regions with repeat lengths from 2 to 200bp in the C. annuum genome, where they occupied ~29%. The most common TRs were 2 and 3bp long followed by those of 21, 4, and 15bp. We performed clustering analysis of TRs with repeat lengths of 2 and 21bp and created position-weight matrices (PWMs) for each group; these templates could be used to search for TRs of a given length in any nucleotide sequence. All detected TRs can be accessed through publicly available database (http : //victoria.biengi.ac.ru/capsicum_tr/). Comparison of mRPWM with other TR search methods such as Tandem Repeat Finder, T-REKS, and XSTREAM indicated that mRPWM could detect significantly more TRs at similar false discovery rates, indicating its superior performance. The developed mRPWM method can be successfully applied to the identification of highly divergent TRs, which is important for functional analysis of genomes and evolutionary studies.

Large deviations of extremal eigenvalues of sample covariance matrices

AbstractLarge deviations of the largest and smallest eigenvalues of $\mathbf{X}\mathbf{X}^\top/n$ are studied in this note, where $\mathbf{X}_{p\times n}$ is a $p\times n$ random matrix with independent and identically distributed (i.i.d.) sub-Gaussian entries. The assumption imposed on the dimension size p and the sample size n is $p=p(n)\rightarrow\infty$ with $p(n)={\mathrm{o}}(n)$ . This study generalizes one result obtained in [3].