Random Matrix Research Articles

Universal Chern number statistics in random matrix fields

We investigate the probability distribution of Chern numbers (quantum Hall conductance integers) for a parametric version of the GUE random matrix ensemble, which is a model for a chaotic or disordered system. The numerically-calculated single-band Chern number statistics agree well with predictions based on an earlier study [O. Gat and M. Wilkinson, SciPost Phys., 10, 149, (2021)] of the statistics of the quantum adiabatic curvature, when the parametric correlation length is small. However, contrary to an earlier conjecture, we find that the gap Chern numbers are correlated, and that the correlation is weak but slowly-decaying. Also, the statistics of weighted sums of Chern numbers differs markedly from predictions based upon the hypothesis that gap Chern numbers are uncorrelated. All our results are consistent with the universality hypothesis described in the earlier paper, including in the previously unstudied regime of large correlation length, where the Chern statistics is highly non-Gaussian.

Limiting spectral distribution of stochastic block model

The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with [Formula: see text] communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.

The volume of random simplices from elliptical distributions in high dimension

Random simplices and more general random convex bodies of dimension p in Rn with p≤n are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if p→∞ and n→∞ in such a way that p/n→γ∈(0,1), a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of p×n random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.

Correction: Extreme eigenvalues of principal minors of random matrices with moment conditions

Correction: Extreme eigenvalues of principal minors of random matrices with moment conditions

A distributed consensus based algorithm for economic dispatch over time‐varying digraphs

AbstractIn this paper, a consensus based fully distributed optimization algorithm is proposed for solving economic dispatch problem (EDP) in smart grid. Since the incremental cost of all buses reach consensus when the optimal solution is achieved, it is selected as a consensus variable. An additional variable at each bus, called “surplus” is added to record the local power mismatch, which is used as a feedback variable to purse the balance between power supply and demand. Different from most of the existing distributed methods which require the communication network to be balanced, the algorithm uses a row random matrix and a column random matrix to precisely steer all the agents to asymptotically converge to a global optimal solution over a time‐varying directed communication network. Due to the use of a fixed step size, the proposed algorithm also outperforms other algorithms in terms of convergence speed. The graph and eigenvalue perturbation theories are employed for the algorithm convergence analysis, and the upper bound of the parameters required for convergence is given theoretically. Finally, the performance and scalability of the proposed distributed algorithm are verified by several case studies conducted on the IEEE 14‐bus power system and a 200‐node test system.

Counting and boundary limit theorems for representations of Gromov‐hyperbolic groups

AbstractGiven a Gromov‐hyperbolic group endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.

Asymmetric image encryption algorithm using a new chaotic map and an improved radial diffusion

In order to address the security risks associated with the theft of private images over open Internet of Things (IoT), this paper introduces a new Logistic-Sine-Coupling Map (LoSCM) in three dimensions with a better chaotic behavior, and proposes an image encryption algorithm using an improved Radial Diffusion (ImRD) to achieve a better diffusion effect in image encryption process. Firstly, a preprocessed image is obtained through an XOR operation between the original plain image and a random generated matrix, followed by Secure Hash Algorithm 512 (SHA-512) performed on it, producing a plaintext information with length of 512 bits. Then, a parameter transformation model (PaTM) is constructed, and the plaintext key associated with the plain image is computed by the plaintext property. Secondly, by employing public-key Rivest-Shamir-Adleman cryptogram, i.e., RSA, the ciphertext key is obtained through the plaintext key and opened to public. Then, a new initial value acquisition model (InVAM) is designed to calculate the initial values of LoSCM, thereby generating the corresponding keystream. Finally, XOR, row scrambling, ImRD diffusion, and column scrambling are performed on secret plain image, followed by modular addition with a random matrix to get the final cipher image. Furthermore, the simulation results and performance analysis also prove that the new suggested image encryption algorithm in this paper can effectively oppose chosen or known-plaintext attack. The highlights are: (1) Construct chaotic map LoSCM with better chaotic behavior. (2) Design diffusion ImRD with better diffusion effect. (3) Build models PaTM and InVAM with high security to produce keystream.

Approximate Solution of Painlevé Equation I by Natural Decomposition Method and Laplace Decomposition Method

Abstract The Painlevé equations and their solutions occur in some areas of theoretical physics, pure and applied mathematics. This paper applies natural decomposition method (NDM) and Laplace decomposition method (LDM) to solve the second-order Painlevé equation. These methods are based on the Adomain polynomial to find the non-linear term in the differential equation. The approximate solution of Painlevé equations is determined in the series form, and recursive relation is used to calculate the remaining components. The results are compared with the existing numerical solutions in the literature to demonstrate the efficiency and validity of the proposed methods. Using these methods, we can properly handle a class of non-linear partial differential equations (NLPDEs) simply. Novelty One of the key novelties of the Painlevé equations is their remarkable property of having only movable singularities, which means that their solutions do not have any singularities that are fixed in position. This property makes the Painlevé equations particularly useful in the study of non-linear systems, as it allows for the construction of exact solutions in certain cases. Another important feature of the Painlevé equations is their appearance in diverse fields such as statistical mechanics, random matrix theory and soliton theory. This has led to a wide range of applications, including the study of random processes, the dynamics of fluids and the behaviour of non-linear waves.

Epidemic thresholds and human mobility

A comprehensive view of disease epidemics demands a deep understanding of the complex interplay between human behaviour and infectious diseases. Here, we propose a flexible modelling framework that brings conclusions about the influence of human mobility and disease transmission on early epidemic growth, with applicability in outbreak preparedness. We use random matrix theory to compute an epidemic threshold, equivalent to the basic reproduction number R_{0}, for a SIR metapopulation model. The model includes both systematic and random features of human mobility. Variations in disease transmission rates, mobility modes (i.e. commuting and migration), and connectivity strengths determine the threshold value and whether or not a disease may potentially establish in the population, as well as the local incidence distribution.

The role of the core microorganisms in the microbial interactions in activated sludge

In order to gain a deeper understanding of the microbial interactions in wastewater treatment plants (WWTPs) in China and clarify the role of the core community in the microbial interactions in activated sludge (AS), this study used a molecular ecological network approach based on random matrix theory to construct co-occurrence networks of the core microorganisms (CoreN), the whole AS community (WholeN) and the microbial communities without the core microorganisms (OtherN), respectively. It was shown that the WholeN had more complex and tighter connections compared with the OtherN, because of its higher total number of nodes, higher average clustering coefficient, and shorter average geodesic distance. The proportions of positive links in the CoreN, WholeN and OtherN were gradually decreased, indicating that the core microorganisms promoted cooperation between AS microorganisms. Moreover, higher robustness after random removal of 50% of the nodes of the WholeN (0.2836 ± 0.0311) was observed than the robustness of the OtherN (0.1152 ± 0.0263). In addition, the vulnerability of OtherN (0.0514) is significantly higher than WholeN (0.0225). Meanwhile, the average ratio of negative/positive cohesion, was significantly decreased when the core microorganisms were removed. These results demonstrated that core community could strengthen the stability of the ecological network in AS. By discerning the key factors affecting ecological network, AS temperature was observed to have a strong correlation with all three networks. Moreover, pollutants in wastewater shown stronger correlations with the CoreN and WholeN, supporting the point that core community play a critical role in pollutant removal in WWTPs to a certain extent.

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.

Quality Analysis of Matrix Masking of Digital Audio Data

This article considered the guarantee of confidentiality task of audio information transmitted over a packet communication channel. Domain analysis showed, that in most cases cryptographical method used for guarantee of confidentiality of audio information, however, along with them, matrix methods of protective coding used, with implementation on FPGA or DSP processors. These methods, using random matrices, impose additional instrumental errors during decoding. The purpose of the study: is to assess the quality of protective coding ‒ masking of audio data by quasi-orthogonal structured matrices. The proposed method of masking digital audio information by quasi-orthogonal matrices differs from the known ones in predictable results and simplity of implementation. An alternative approach to the classical use of metrics is proposed for evaluating the result of masking. It is based on the analysis of the spectral component of the signal. It is shown that masking with a square Mersenne ‒ Walsh matrix brings digital audio information, presented in the form of a matrix, to a form close in spectrum to white noise. This reliably protects it in the communication channel from unauthorized access. The obtained results showed that the proposed method is promising and suggests the study of the influence of the choice of orthogonal matrix structures and their sizes on the results of masking.

A novel method to repair missing vibration data in rolling bearing vibration signals based on improved optimized measurement matrix

Missing data, caused by many factors such as equipment short circuits or data cleaning, affect the accuracy of condition monitoring for rotating machinery. To improve the precision of missing data recovery, a compressed sensing-based vibration data repair method is developed. First, based on the Gaussian random matrix, an improved optimized measurement matrix (OMM) is proposed to accurately sample data. Then, a sparse representation of the vibration signal, through a discrete cosine transform, is utilized to make the noisy vibration signal sparse. Finally, the orthogonal matching pursuit algorithm is employed to reconstruct the missing signal. The effectiveness of the proposed method is verified by analyzing constant and variable speed time series of rolling bearings. Compared with other data repair methods, it is shown that the OMM has a higher repair precision at different loss rates.

Productivity vs. Evenness in the U.S. Financial Market: A Business Ecosystem Perspective.

This paper starts by presenting an empirical finding in the U.S. stock market: Between 2001 and 2021, high productivity was achieved when the Shannon evenness-measuring the inverse of concentration-dropped. Conversely, when the Shannon evenness soared, productivity plunged. The same inverse relationship between evenness and productivity has been observed in several ecosystems. This suggests explaining this result by adopting the business ecosystem perspective, i.e., regarding the tangle of interactions between companies as an ecological network, in which companies play the role of species. A useful strategy to model such ecological communities is through ensembles of synthetic communities of pairwise interacting species, whose dynamics is described by the Lotka-Volterra generalized equations. Each community is specified by a random interaction matrix whose elements are drawn from a uniform distribution centered around 0. It is shown that the inverse relationship between productivity and evenness can be generated by varying the strength of the interaction between companies. When the strength increases, productivity increases and simultaneously the market evenness decreases. Conversely, when the strength decreases, productivity decreases and evenness increases. This strength can be interpreted as reflecting the looseness of monetary policy, thus providing a link between interest rates and market structure.

Correlation of local densities of states on mesoscopic energy scales in random band matrices

We are interested in the phase transition of the correlation function of local densities of states at mesoscopic scales of random band matrices of width W in dimension 2. As a result, we show that the local densities of states are alternately positively and negatively correlated in the diffusive regime O(log (L/W)) times, L being the size of the system.

Volcano transition in populations of phase oscillators with random nonreciprocal interactions.

Populations of heterogeneous phase oscillators with frustrated random interactions exhibit a quasiglassy state in which the distribution of local fields is volcanoshaped. In a recent work [Phys. Rev. Lett. 120, 264102 (2018)10.1103/PhysRevLett.120.264102], the volcano transition was replicated in a solvable model using a low-rank, random coupling matrix M. We extend here that model including tunable nonreciprocal interactions, i.e., M^{T}≠M. More specifically, we formulate two different solvable models. In both of them the volcano transition persists if matrix elements M_{jk} and M_{kj} are enough correlated. Our numerical simulations fully confirm the analytical results. To put our work in a wider context, we also investigate numerically the volcano transition in the analogous model with a full-rank random coupling matrix.

Random sparse generators of Markovian evolution and their spectral properties.

The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter φ, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with D vertices (number of system states) and 2φD edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as ∼φ, while its variance is ∼sqrt[φ]. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of D. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, φ=const, and find that starting already at φ⩾2, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.

A Riemann–Hilbert Approach to the Perturbation Theory for Orthogonal Polynomials: Applications to Numerical Linear Algebra and Random Matrix Theory

Abstract We establish a new perturbation theory for orthogonal polynomials using a Riemann–Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization, and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.

Reflective prolate‐spheroidal operators and the adelic Grassmannian

AbstractBeginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices

Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices