Random Matrix Research Articles

Laser ultrasonic reconstruction model for additive manufacturing based on compressed sensing theory

Laser ultrasonic has unique advantages in online non-destructive testing for additive manufacturing. However, continuous online scanning can lead to issues such as large data volumes, difficult storage and transmission. This paper proposes a deep learning reconstruction algorithm based on compressed sensing that aims to reconstruct compressed data with high performance, called LUNet. Firstly, ultrasonic signals are obtained from specimens containing different defects, Gaussian random matrix is used to compress the signals, and then the compressed signals are quickly reconstructed by the proposed method. The model utilizes dilated convolutions and skip connections to learn the latent relationship between the compressed signals and original signals. Different dilation rates of the dilated convolutions enhance the model’s receptive field, while skip connections preserve low-level information and ensure that the model’s performance will not degrade. Compared to four reconstruction algorithms, the proposed method demonstrates advantages in reconstruction quality and speed across in comparative experiments.

Vulnerability analysis on random matrix theory for power grid with flexible impact loads

The stochastic volatility of the rail transit load brings greater uncertainty to the vulnerability of the power grid. To solve the problem of the inaccurate results caused by the incomplete time-domain simulation model of the power system with rail transit load integration, this paper proposes a vulnerability analysis method for the power system with rail transit load integration based on the random matrix theory. In this paper, we first constructed a rail transit load model based on Deep Convolutional Generative Adversarial Networks (DCGAN) to simulate the situation that massive rail transit load merged into the Grid Scenario. Then, we generate a high-dimensional random matrix based on the power flow of the grid-connected system under different rail transit loads. Then, we construct a vulnerability analysis model combining the random matrix theory and the real-time separation window. Finally, we take the IEEE-39 bus system and a regional power grid in China as examples to evaluate the vulnerability of the grid-connected system. The results show that our method quantifies not only the impact of the rail transit load volatility on the system vulnerability, but the system endurance under different capacities of the rail transit load connected to grid. Moreover, it also provides a new way for system planning and safety monitoring in the power system with rail transit load integration.

Infrared detectors based on graphene lattice: An environmental dependent approach

Abstract The article discusses the development of an infrared detector using a graphene lattice, focusing on how environmental temperature and external voltage affect its electrical conductivity. The study models environmental temperature as a phonon bath and uses random matrix theory and quantum chaos to analyze the impact of impurities and structural defects. Impurities and defects in the graphene lattice significantly impact both the multifractal dimension and conductivity. The applied external voltage increases the spread of the multifractal dimension, indicating a tendency toward localization at higher voltages. Additionally, the multifractal dimension narrows for the metal state and converges to a single point for localized states. The research highlights the importance of studying these factors to understand the conductivity phase and the ability to detect infrared photons. The findings suggest that changes in electrical current at specific bias voltages can be used to detect infrared photons, with a graphene lattice containing boron impurities operating effectively at room temperature.

Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures

We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition between stable and unstable states. In the stable regime, the spectral density has finite and positive support, whereas negative eigenvalues emerge in the unstable regime. We determine the critical line separating these regimes and show that the spectral density exhibits a power-law tail at marginal stability, with an exponent independent of the temperature distribution. Additionally, we compute the spectral density of the lagged covariance matrix characterizing the stationary states of linear transformations of the original dynamical variables. Our random-matrix model is potentially interesting to understand the spectral properties of empirical correlation matrices appearing in the study of complex systems. Published by the American Physical Society 2025

Time almost periodicity for solutions of Toda lattice equation with almost periodic initial datum

Abstract This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schrödinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schrödinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift’s conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419–430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl.13 (2017), Paper no. 016].

Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking

Open quantum systems have complex energy eigenvalues which are expected to follow non-Hermitian random matrix statistics, when chaotic, or two-dimensional (2d) Poisson statistics, when integrable. We investigate the spectral properties of a many-body quantum spin chain, i.e., the Hermitian Heisenberg model with imaginary disorder. Its rich complex eigenvalue statistics is found to separately break both Hermiticity and integrability at different scales of the disorder strength. With no disorder, the system is integrable and Hermitian, with spectral statistics corresponding to the 1d Poisson point process. At very small disorder, we find a transition from 1d Poisson statistics to an effective D-dimensional Poisson point process, showing Hermiticity breaking. At intermediate disorder, we find integrability breaking, as inferred from the statistics matching that of non-Hermitian complex symmetric random matrices in class AI†. For large disorder, as the spins align, we recover the expected integrability (now in the non-Hermitian setup), indicated by 2d Poisson statistics. These conclusions are based on fitting the spin-chain data of numerically generated nearest- and next-to-nearest-neighbor spacing distributions to an effective 2d Coulomb gas description at inverse temperature β. We confirm that such an effective description of random matrices also applies in classes AI† and AII† up to next-to-nearest-neighbor spacings. Published by the American Physical Society 2025

Correlation Structure of the Spanish Stock Market Around COVID-19 Using Random Matrix Theory

Correlation Structure of the Spanish Stock Market Around COVID-19 Using Random Matrix Theory

Optimal network sizes for most robust Turing patterns

Many cellular patterns exhibit a reaction-diffusion component, suggesting that Turing instability may contribute to pattern formation. However, biological gene-regulatory pathways are more complex than simple Turing activator-inhibitor models and generally do not require fine-tuning of parameters as dictated by the Turing conditions. To address these issues, we employ random matrix theory to analyze the Jacobian matrices of larger networks with robust statistical properties. Our analysis reveals that Turing patterns are more likely to occur by chance than previously thought and that the most robust Turing networks have an optimal size, consisting of only a handful of molecular species, thus significantly increasing their identifiability in biological systems. Broadly speaking, this optimal size emerges from a trade-off between the highest stability in small networks and the greatest instability with diffusion in large networks. Furthermore, we find that with multiple immobile nodes, differential diffusion ceases to be important for Turing patterns. Our findings may inform future synthetic biology approaches and provide insights into bridging the gap to complex developmental pathways.

Information limits and Thouless-Anderson-Palmer equations for spiked matrix models with structured noise

We consider a prototypical problem of Bayesian inference for a structured spiked model: a low-rank signal is corrupted by additive noise. While both information-theoretic and algorithmic limits are well understood when the noise is a Gaussian Wigner matrix, the more realistic case of structured noise still remains challenging. To capture the structure while maintaining mathematical tractability, a line of work has focused on rotationally invariant noise. However, existing studies either provide suboptimal algorithms or are limited to a special class of noise ensembles. In this paper, using tools from statistical physics (replica method) and random matrix theory (generalized spherical integrals) we establish the characterization of the information-theoretic limits for a noise matrix drawn from a general trace ensemble. Remarkably, our analysis unveils the asymptotic equivalence between the rotationally invariant model and a surrogate Gaussian one. Finally, we show how to saturate the predicted statistical limits using an efficient algorithm inspired by the theory of adaptive Thouless-Anderson-Palmer (TAP) equations. Published by the American Physical Society 2025

Dualities for characteristic polynomial averages of complex symmetric and self dual non-Hermitian random matrices

Abstract Ensembles of complex symmetric, and complex self dual random matrices are known to exhibit local statistical properties distinct from those of the non-Hermitian Ginibre ensembles. On the other hand, in distinction to the latter, the joint eigenvalue probability density function of these two ensembles are not known. Nonetheless, as carried out in the recent works of Liu and Zhang, Akemann et al. and Kulkarni et al., by considering averages of products of characteristic polynomials, analytic progress can be made. Here we show that an approach based on the theory of zonal polynomials provides an alternative to the diffusion operator or supersymmetric Grassmann integrations methods of these works. It has the advantage of not being restricted to a Gaussian unitary invariant measure on the matrix spaces. To illustrate this, as an extension, we consider averages of products and powers of characteristic polynomials for complex symmetric, and complex self dual random matrices subject to a spherical measure. In the case of powers, when comparing against the corresponding real Ginibre, respectively quaternion Ginibre averages with a spherical measure, one finds the qualitative feature of a decreasing (increasing) profile as the magnitude of the argument of the characteristic polynomial increases. This is analogous to the findings of the second two of the cited works in the Gaussian case.

Hammerstein equations for sparse random matrices

Abstract Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.

Meta-analyses of mouse and human prostate single-cell transcriptomes reveal widespread epithelial plasticity in tissue regression, regeneration, and cancer

BackgroundDespite extensive analysis, the dynamic changes in prostate epithelial cell states during tissue homeostasis as well as tumor initiation and progression have been poorly characterized. However, recent advances in single-cell RNA-sequencing (scRNA-seq) technology have greatly facilitated studies of cell states and plasticity in tissue maintenance and cancer, including in the prostate.MethodsWe have performed meta-analyses of new and previously published scRNA-seq datasets for mouse and human prostate tissues to identify and compare cell populations across datasets in a uniform manner. Using random matrix theory to denoise datasets, we have established reference cell type classifications for the normal mouse and human prostate and have used optimal transport to compare the cross-species transcriptomic similarities of epithelial cell populations. In addition, we have integrated analyses of single-cell transcriptomic states with copy number variants to elucidate transcriptional programs in epithelial cells during human prostate cancer progression.ResultsOur analyses demonstrate transcriptomic similarities between epithelial cell states in the normal prostate, in the regressed prostate after androgen-deprivation, and in primary prostate tumors. During regression in the mouse prostate, all epithelial cells shift their expression profiles toward a proximal periurethral (PrU) state, demonstrating an androgen-dependent plasticity that is restored to normal during androgen restoration and gland regeneration. In the human prostate, we find substantial rewiring of transcriptional programs across epithelial cell types in benign prostate hyperplasia and treatment-naïve prostate cancer. Notably, we detect copy number variants predominantly within luminal acinar cells in prostate tumors, suggesting a bias in their cell type of origin, as well as a larger field of transcriptomic alterations in non-tumor cells. Finally, we observe that luminal acinar tumor cells in treatment-naïve prostate cancer display heterogeneous androgen receptor (AR) signaling activity, including a split between AR-positive and AR-low profiles with similarity to PrU-like states.ConclusionsTaken together, our analyses of cellular heterogeneity and plasticity provide important translational insights into the origin and treatment response of prostate cancer. In particular, the identification of AR-low tumor populations suggests that castration-resistance and predisposition to neuroendocrine differentiation may be pre-existing properties in treatment-naïve primary tumors that are selected for by androgen-deprivation therapies.

Characteristic Polynomials of Sparse Non-Hermitian Random Matrices

Characteristic Polynomials of Sparse Non-Hermitian Random Matrices

Tridiagonal Hamiltonians modeling the density of states of the double-scaled SYK model

By analyzing the global density of states (DOS) in the Double-Scaled Sachdev-Ye-Kitaev (DSSYK) model, we construct a finite-dimensional Hamiltonian that replicates this DOS. We then tridiagonalize the Hamiltonian to determine the mean Lanczos coefficients within the parameter range. The bulk Lanczos coefficients, especially the Lanczos descent can be analytically expressed as a particular q-deformation of the logarithm. Our numerical results are further corroborated by semi-analytical findings, a random matrix potential construction in the bulk, and the analytic results at the edge of the Lanczos spectra using the method of moments.

Rate of convergence in multiple SLE using random matrix theory

In this paper, we provide a rate of convergence for a version of the Carathéodory convergence for the multiple SLE model with a Dyson Brownian motion driver towards its hydrodynamic limit, for [Formula: see text] and [Formula: see text]. The results are obtained by combining techniques from the field of Schramm–Loewner Evolutions with modern techniques from random matrices. Our approach shows how one can apply modern tools used in the proof of universality in random matrix theory to the field of Schramm–Loewner Evolutions.

A matrix model of a non-Hermitian β-ensemble

We introduce the first random matrix model of a complex [Formula: see text]-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite [Formula: see text]-ensembles discovered by [I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002) 5830–5847]. The main feature of the model is that the exponent [Formula: see text] of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in [Formula: see text] However, when [Formula: see text], the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.

Quadratic Forms in Random Matrices with Applications in Spectrum Sensing.

Quadratic forms with random kernel matrices are ubiquitous in applications of multivariate statistics, ranging from signal processing to time series analysis, biomedical systems design, wireless communications performance analysis, and other fields. Their statistical characterization is crucial to both design guideline formulation and efficient computation of performance indices. To this end, random matrix theory can be successfully exploited. In particular, recent advancements in spectral characterization of finite-dimensional random matrices from the so-called polynomial ensembles allow for the analysis of several scenarios of interest in wireless communications and signal processing. In this work, we focus on the characterization of quadratic forms in unit-norm vectors, with unitarily invariant random kernel matrices, and we also provide some approximate but numerically accurate results concerning a non-unitarily invariant kernel matrix. Simulations are run with reference to a peculiar application scenario, the so-called spectrum sensing for wireless communications. Closed-form expressions for the moment generating function of the quadratic forms of interest are provided; this will pave the way to an analytical performance analysis of some spectrum sensing schemes, and will potentially assist in the rate analysis of some multi-antenna systems.

Applications of Random Matrix Theory in Neuroscience and Neural Network Analysis: Unraveling High-Dimensional Connectivity

Random Matrix Theory (RMT) has emerged as a powerful mathematical framework to study large-scale, high-dimensional data across various scientific fields. In neuroscience, it provides unparalleled tools to elucidate the functional connectivity and dynamic properties underlying neural activity. By modeling connectivity matrices, synaptic weight distributions, and neural correlation structures through random matrices, researchers can dissect large-scale neural systems’ stability, detect phase transitions in neural firing patterns, and explain emergent phenomena such as criticality in brain networks. Similarly, in the realm of artificial neural networks, RMT guides the understanding of weight initialization, spectral properties of the network’s Jacobian matrices, generalization, and optimization landscapes. This article provides a comprehensive introduction to Random Matrix Theory’s fundamental principles, their application in contemporary neuroscience, and their intersection with modern neural network paradigms. We introduce the theoretical underpinnings and methodological steps for applying RMT to neuroscience data, including sequentially numbered didactic equations. We further present an illustrative code snippet to generate and visualize RMT-based graphs, discussing the results and connecting them to cutting-edge research. In a detailed discussion, we explore the benefits, limitations, and future directions of RMT methodologies in neuroscience, including ongoing challenges and emerging frontiers. This work aims to equip researchers with a robust theoretical and methodological framework, encouraging the integration of RMT into the neuroscience toolbox and catalyzing further breakthroughs in understanding complex neural systems.

Stochastic weight matrix dynamics during learning and Dyson Brownian motion

We demonstrate that the update of weight matrices in learning algorithms can be described in the framework of Dyson Brownian motion, thereby inheriting many features of random matrix theory. We relate the level of stochasticity to the ratio of the learning rate and the minibatch size, providing more robust evidence to a previously conjectured scaling relationship. We discuss universal and nonuniversal features in the resulting Coulomb gas distribution and identify the Wigner surmise and Wigner semicircle explicitly in a teacher-student model and in the (near-)solvable case of the Gaussian restricted Boltzmann machine. Published by the American Physical Society 2025

Performance analysis of wireless‐powered cell‐free massive multiple‐input multiple‐output system with spatial correlation in Internet of Things network

AbstractThe massive multiple‐input multiple‐output (mMIMO) approach is promising for the Internet of Things (IoT) owing to its massive connectivity and high data rate. We introduce a wireless‐powered cell‐free mMIMO system, in which ground IoT devices transmit pilot and uplink information by harvesting downlink power from multiantenna access points. Considering the spatial correlation, we derive closed‐form expressions for the average harvested power with a nonlinear energy‐harvesting model per IoT device and achievable data rate according to the random matrix theory. The analytical expressions show that spatial correlation has a negative effect on the data rate owing to the increasing interference power. In contrast, the average received power improves with increasing spatial correlation. Simulation results demonstrate that the derived analytical expressions are consistent with results from the Monte Carlo method.