Random Matrix Theory

Physics-Informed Singular-Value Learning for Cross-Covariances Forecasting in Financial Markets ArXiv ID: 2601.07687 “View on arXiv” Authors: Efstratios Manolakis, Christian Bongiorno, Rosario Nunzio Mantegna Abstract A new wave of work on covariance cleaning and nonlinear shrinkage has delivered asymptotically optimal analytical solutions for large covariance matrices. Building on this progress, these ideas have been generalized to empirical cross-covariance matrices, whose singular-value shrinkage characterizes comovements between one set of assets and another. Existing analytical cross-covariance cleaners are derived under strong stationarity and large-sample assumptions, and they typically rely on mesoscopic regularity conditions such as bounded spectra; macroscopic common modes (e.g., a global market factor) violate these conditions. When applied to real equity returns, where dependence structures drift over time and global modes are prominent, we find that these theoretically optimal formulas do not translate into robust out-of-sample performance. We address this gap by designing a random-matrix-inspired neural architecture that operates in the empirical singular-vector basis and learns a nonlinear mapping from empirical singular values to their corresponding cleaned values. By construction, the network can recover the analytical solution as a special case, yet it remains flexible enough to adapt to non-stationary dynamics and mode-driven distortions. Trained on a long history of equity returns, the proposed method achieves a more favorable bias-variance trade-off than purely analytical cleaners and delivers systematically lower out-of-sample cross-covariance prediction errors. Our results demonstrate that combining random-matrix theory with machine learning makes asymptotic theories practically effective in realistic time-varying markets. ...

From sectorial coarse graining to extreme coarse graining of S&P 500 correlation matrices ArXiv ID: 2511.05463 “View on arXiv” Authors: Manan Vyas, M. Mijaíl Martínez-Ramos, Parisa Majari, Thomas H. Seligman Abstract Starting from the Pearson Correlation Matrix of stock returns and from the desire to obtain a reduced number of parameters relevant for the dynamics of a financial market, we propose to take the idea of a sectorial matrix, which would have a large number of parameters, to the reduced picture of a real symmetric $2 \times 2$ matrix, extreme case, that still conserves the desirable feature that the average correlation can be one of the parameters. This is achieved by averaging the correlation matrix over blocks created by choosing two subsets of stocks for rows and columns and averaging over each of the resulting blocks. Averaging over these blocks, we retain the average of the correlation matrix. We shall use a random selection for two equal block sizes as well as two specific, hopefully relevant, ones that do not produce equal block sizes. The results show that one of the non-random choices has somewhat different properties, whose meaning will have to be analyzed from an economy point of view. ...

Denoising Complex Covariance Matrices with Hybrid ResNet and Random Matrix Theory: Cryptocurrency Portfolio Applications ArXiv ID: 2510.19130 “View on arXiv” Authors: Andres Garcia-Medina Abstract Covariance matrices estimated from short, noisy, and non-Gaussian financial time series are notoriously unstable. Empirical evidence suggests that such covariance structures often exhibit power-law scaling, reflecting complex, hierarchical interactions among assets. Motivated by this observation, we introduce a power-law covariance model to characterize collective market dynamics and propose a hybrid estimator that integrates Random Matrix Theory (RMT) with deep Residual Neural Networks (ResNets). The RMT component regularizes the eigenvalue spectrum in high-dimensional noisy settings, while the ResNet learns data-driven corrections that recover latent structural dependencies encoded in the eigenvectors. Monte Carlo simulations show that the proposed ResNet-based estimators consistently minimize both Frobenius and minimum-variance losses across a range of population covariance models. Empirical experiments on 89 cryptocurrencies over the period 2020-2025, using a training window ending at the local Bitcoin peak in November 2021 and testing through the subsequent bear market, demonstrate that a two-step estimator combining hierarchical filtering with ResNet corrections produces the most profitable and well-balanced portfolios, remaining robust across market regime shifts. Beyond finance, the proposed hybrid framework applies broadly to high-dimensional systems described by low-rank deformations of Wishart ensembles, where incorporating eigenvector information enables the detection of multiscale and hierarchical structure that is inaccessible to purely eigenvalue-based methods. ...

Community-level Contagion among Diverse Financial Assets ArXiv ID: 2509.15232 “View on arXiv” Authors: An Pham Ngoc Nguyen, Marija Bezbradica, Martin Crane Abstract As global financial markets become increasingly interconnected, financial contagion has developed into a major influencer of asset price dynamics. Motivated by this context, our study explores financial contagion both within and between asset communities. We contribute to the literature by examining the contagion phenomenon at the community level rather than among individual assets. Our experiments rely on high-frequency data comprising cryptocurrencies, stocks and US ETFs over the 4-year period from April 2019 to May 2023. Using the Louvain community detection algorithm, Vector Autoregression contagion detection model and Tracy-Widom random matrix theory for noise removal from financial assets, we present three main findings. Firstly, while the magnitude of contagion remains relatively stable over time, contagion density (the percentage of asset pairs exhibiting contagion within a financial system) increases. This suggests that market uncertainty is better characterized by the transmission of shocks more broadly than by the strength of any single spillover. Secondly, there is no significant difference between intra- and inter-community contagion, indicating that contagion is a system-wide phenomenon rather than being confined to specific asset groups. Lastly, certain communities themselves, especially those dominated by Information Technology assets, tend to act as major contagion transmitters in the financial network over the examined period, spreading shocks with high densities to many other communities. Our findings suggest that traditional risk management strategies such as portfolio diversification through investing in low-correlated assets or different types of investment vehicle might be insufficient due to widespread contagion. ...

Wealth Thermalization Hypothesis and Social Networks ArXiv ID: 2506.17720 “View on arXiv” Authors: Klaus M. Frahm, Dima L. Shepelyansky Abstract In 1955 Fermi, Pasta, Ulam and Tsingou performed first numerical studies with the aim to obtain the thermalization in a chain of nonlinear oscillators from dynamical equations of motion. This model happend to have several specific features and the dynamical thermalization was established only later in other studies. In this work we study more generic models based on Random Matrix Theory and social networks with a nonlinear perturbation leading to dynamical thermalization above a certain chaos border. These systems have two integrals of motion being total energy and norm so that the theoretical Rayleigh-Jeans thermal distribution depends on temperature and chemical potential. We introduce the wealth thermalization hypothesis according to which the society wealth is associated with energy in the Rayleigh-Jeans distribution. At relatively small values of total wealth or energy there is a formation of the Rayleigh-Jeans condensate, well studied in physical systems such as multimode optical fibers. This condensation leads to a huge fraction of poor households at low wealth and a small oligarchic fraction which monopolizes a dominant fraction of total wealth thus generating a strong inequality in human society. We show that this thermalization gives a good description of real data of Lorenz curves of US, UK, the whole world and capitalization of companies at Stock Exchange of New York SE (NYSE), London and Hong Kong. It is also shown that above a chaos border the dynamical Rayleigh-Jeans thermalization takes place also in social networks with the Lorenz curves being similar to those of wealth distribution in world countries. Possible actions for inequality reduction are briefly discussed. ...

Multivariate Distributions in Non-Stationary Complex Systems II: Empirical Results for Correlated Stock Markets ArXiv ID: 2412.11602 “View on arXiv” Authors: Unknown Abstract Multivariate Distributions are needed to capture the correlation structure of complex systems. In previous works, we developed a Random Matrix Model for such correlated multivariate joint probability density functions that accounts for the non-stationarity typically found in complex systems. Here, we apply these results to the returns measured in correlated stock markets. Only the knowledge of the multivariate return distributions allows for a full-fledged risk assessment. We analyze intraday data of 479 US stocks included in the S&P500 index during the trading year of 2014. We focus particularly on the tails which are algebraic and heavy. The non-stationary fluctuations of the correlations make the tails heavier. With the few-parameter formulae of our Random Matrix Model we can describe and quantify how the empirical distributions change for varying time resolution and in the presence of non-stationarity. ...

High-dimensional covariance matrix estimators on simulated portfolios with complex structures ArXiv ID: 2412.08756 “View on arXiv” Authors: Unknown Abstract We study the allocation of synthetic portfolios under hierarchical nested, one-factor, and diagonal structures of the population covariance matrix in a high-dimensional scenario. The noise reduction approaches for the sample realizations are based on random matrices, free probability, deterministic equivalents, and their combination with a data science hierarchical method known as two-step covariance estimators. The financial performance metrics from the simulations are compared with empirical data from companies comprising the S&P 500 index using a moving window and walk-forward analysis. The portfolio allocation strategies analyzed include the minimum variance portfolio (both with and without short-selling constraints) and the hierarchical risk parity approach. Our proposed hierarchical nested covariance model shows signatures of complex system interactions. The empirical financial data reproduces stylized portfolio facts observed in the complex and one-factor covariance models. The two-step estimators proposed here improve several financial metrics under the analyzed investment strategies. The results pave the way for new risk management and diversification approaches when the number of assets is of the same order as the number of transaction days in the investment portfolio. ...

Kendall Correlation Coefficients for Portfolio Optimization ArXiv ID: 2410.17366 “View on arXiv” Authors: Unknown Abstract Markowitz’s optimal portfolio relies on the accurate estimation of correlations between asset returns, a difficult problem when the number of observations is not much larger than the number of assets. Using powerful results from random matrix theory, several schemes have been developed to “clean” the eigenvalues of empirical correlation matrices. By contrast, the (in practice equally important) problem of correctly estimating the eigenvectors of the correlation matrix has received comparatively little attention. Here we discuss a class of correlation estimators generalizing Kendall’s rank correlation coefficient which improve the estimation of both eigenvalues and eigenvectors in data-poor regimes. Using both synthetic and real financial data, we show that these generalized correlation coefficients yield Markowitz portfolios with lower out-of-sample risk than those obtained with rotationally invariant estimators. Central to these results is a property shared by all Kendall-like estimators but not with classical correlation coefficients: zero eigenvalues only appear when the number of assets becomes proportional to the square of the number of data points. ...

Signal inference in financial stock return correlations through phase-ordering kinetics in the quenched regime ArXiv ID: 2409.19711 “View on arXiv” Authors: Unknown Abstract Financial stock return correlations have been analyzed through the lens of random matrix theory to differentiate the underlying signal from spurious correlations. The continuous spectrum of the eigenvalue distribution derived from the stock return correlation matrix typically aligns with a rescaled Marchenko-Pastur distribution, indicating no detectable signal. In this study, we introduce a stochastic field theory model to establish a detection threshold for signals present in the limit where the eigenvalues are within the continuous spectrum, which itself closely resembles that of a random matrix where standard methods such as principal component analysis fail to infer a signal. We then apply our method to Standard & Poor’s 500 financial stocks’ return correlations, detecting the presence of a signal in the largest eigenvalues within the continuous spectrum. ...

Consistent Estimation of the High-Dimensional Efficient Frontier ArXiv ID: 2409.15103 “View on arXiv” Authors: Unknown Abstract In this paper, we analyze the asymptotic behavior of the main characteristics of the mean-variance efficient frontier employing random matrix theory. Our particular interest covers the case when the dimension $p$ and the sample size $n$ tend to infinity simultaneously and their ratio $p/n$ tends to a positive constant $c\in(0,1)$. We neither impose any distributional nor structural assumptions on the asset returns. For the developed theoretical framework, some regularity conditions, like the existence of the $4$th moments, are needed. It is shown that two out of three quantities of interest are biased and overestimated by their sample counterparts under the high-dimensional asymptotic regime. This becomes evident based on the asymptotic deterministic equivalents of the sample plug-in estimators. Using them we construct consistent estimators of the three characteristics of the efficient frontier. It it shown that the additive and/or the multiplicative biases of the sample estimates are solely functions of the concentration ratio $c$. Furthermore, the asymptotic normality of the considered estimators of the parameters of the efficient frontier is proved. Verifying the theoretical results based on an extensive simulation study we show that the proposed estimator for the efficient frontier is a valuable alternative to the sample estimator for high dimensional data. Finally, we present an empirical application, where we estimate the efficient frontier based on the stocks included in S&P 500 index. ...