Narrow Results

This paper discusses the multivariate Behrens–Fisher problem, which tests the equality of two population mean vectors under heteroscedasticity. The classical Wald-type test is not applicable in high dimensions owing to the singularity of sample covariance matrices. A ridgelized Wald-type (RIWT) test is then proposed. Using the exact four-moment theorem, its asymptotic null distribution is derived under moment conditions. Thus, our test can accommodate non-Gaussian distributions with general covariance matrices. Simulation results demonstrate the good performance of the proposed test.

Consider the random matrix $W_n=B_n+n^{-1}{X}_n^{\ast }{A}_n{X}_n$, where $A_n$ and $B_n$ are Hermitian matrices of dimensions $p\times p$ and $n\times n$, respectively, and $X_n$ is a $p\times n$ random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that $p$ and $n$ grow to infinity proportionally, and that the spectral measures of $A_n$ and $B_n$ converge as $p,n\to \infty$ towards two probability measures $\mathcal{𝒜}$ and $\mathcal{ℬ}$. Building on the groundbreaking work of [V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. 114(4) (1967) 507–536], which demonstrated that the empirical spectral distribution of $W_n$ converges towards a probability measure $F$ characterized by its Stieltjes transform, this paper investigates the properties of $F$ when $\mathcal{ℬ}$ is a general measure. We show that $F$ has an analytic density at the region near where the Stieltjes transform of $\mathcal{ℬ}$ is bounded. The density closely resembles $C\sqrt{|x-x_0|}$ near certain edge points $x_0$ of its support for a wide class of $\mathcal{𝒜}$ and $\mathcal{ℬ}$. We provide a complete characterization of the support of $F$. Moreover, we show that $F$ can exhibit discontinuities at points where $\mathcal{ℬ}$ is discontinuous.

The supersymmetric approach has proved to be a powerful tool for the study of random systems where classical techniques do not seem to apply. It also seems promising for rigorous analysis. In this context, we consider the GUE density of states, and show that, using the supersymmetric approach, we can rigorously re-derive the results obtained by classical techniques (orthogonal polynomials), in all energy regions (inside the spectrum, at the edge and outside the spectrum).

We apply the mathematics of cognitive radio to a single receiver to obtain a new coherent energy metric. This allows us to derive the time correlation law separating Gaussian colored noise from coherent signal energy.

Rotationally Invariant Estimators (RIE) are a new family of covariance matrix estimators based on random matrix theory and free probability. The family RIE has been proposed to improve the performance of an investment portfolio in the Markowitz model’s framework. Here, we apply state-of-the-art RIE techniques to improve the estimation of financial states via the correlation matrix. The Synthesized Clustering (SYNCLUS) and a dynamic programming algorithm for optimal one-dimensional clustering were employed to that aim. We found that the RIE estimations of the minimum portfolio risk increase the Active Information Storage (AIS) in the American and European markets. AIS’s local dynamic also mimics financial states’ behavior when estimating under the one-dimensional clustering algorithm. Our results suggest that in times of financial turbulence, RIE estimates can be of great advantage in minimizing risk exposure.

This work aims to address the optimal allocation instability problem of Markowitz’s modern portfolio theory in high dimensionality. We propose a combined strategy that considers covariance matrix estimators from Random Matrix Theory (RMT) and the machine learning allocation methodology known as Nested Clustered Optimization (NCO). The latter methodology is modified and reformulated in terms of the spectral clustering algorithm and Minimum Spanning Tree (MST) to solve internal problems inherent in the original proposal. Markowitz’s classical mean-variance allocation and the modified NCO machine learning approach are tested on the financial instruments that compose the S&P 500 index listed on the New York Stock Exchange (NYSE) from 2012 to 2021. The modified NCO algorithm achieves stable allocations by incorporating RMT covariance estimators. A particular combination of covariance estimators, known as two-step estimators involving a hierarchical clustering technique, achieves the best portfolio performance. Surprisingly, a common pattern suggests that the improvement in performance is agnostic to the allocation strategy and, in fact, relies heavily on the covariance estimator selection. Our results suggest that the amount of leverage can be drastically reduced in investment strategies through RMT inference and statistical learning techniques.

We derive a formula for the entropy of two-dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Q_k = ∫ ω(x)^kd²x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. By taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; we also predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value 〈n〉, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension δ is a dynamical observable in our model, and plays the role of an order parameter. The computation of 〈δ〉 is discussed and an upper bound is found, 〈δ〉 < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

The random matrix model approach to quantum chromodynamics (QCD) with nonvanishing chemical potential is reviewed. The general concept using global symmetries is introduced, as well as its relation to field theory, the so-called epsilon regime of chiral perturbation theory (∊χPT). Two types of matrix model results are distinguished: phenomenological applications leading to phase diagrams, and an exact limit of the QCD Dirac operator spectrum matching with ∊χPT. All known analytic results for the spectrum of complex and symplectic matrix models with chemical potential are summarised for the symmetry classes of ordinary and adjoint QCD, respectively. These include correlation functions of Dirac operator eigenvalues in the complex plane for real chemical potential, and in the real plane for imaginary isospin chemical potential. Comparisons of these predictions to recent lattice simulations are also discussed.

The near-zero modes of the Dirac operator are connected to spontaneous breaking of chiral symmetry in QCD (SBCS) via the Banks–Casher relation. At the same time, the distribution of the near-zero modes is well described by the Random Matrix Theory (RMT) with the Gaussian Unitary Ensemble (GUE). Then, it has become a standard lore that a randomness, as observed through distributions of the near-zero modes of the Dirac operator, is a consequence of SBCS. The higher-lying modes of the Dirac operator are not affected by SBCS and are sensitive to confinement physics and related $SU{(2)}_{\mathrm{\text{CS}}}$ and $SU(2N_F)$ symmetries. We study the distribution of the near-zero and higher-lying eigenmodes of the overlap Dirac operator within $N_F=2$ dynamical simulations. We find that both the distributions of the near-zero and higher-lying modes are perfectly described by GUE of RMT. This means that randomness, while consistent with SBCS, is not a consequence of SBCS and is linked to the confining chromo-electric field.

We discuss the use of the replica ansatz in computing free energies in random matrix theory and confirm a conjectured condition on analytic continuation in the replica index at large-$N$.

We have investigated the transport properties of anisotropic disordered quantum wire by the method of energy level statistics and transfer matrix method. We have found a general average conductance relation in terms of the anisotropy parameter t, disorder strength W, and the cross-sectional area, A, of the wire. From the available numerical data, we have also shown that the anisotropic correlation length ξ_t can be related to the isotropic correlation length ξ as ξ_t ≃ t²ξ, for L ≃ A and ξ_t ≃ tξ, for L ≫ A. Here, L is the length of the wire. In addition, anisotropic dependence of conductance fluctuation and conductance distribution have been studied to a certain extent.

The latest global financial tsunami and its follow-up global economic recession has uncovered the crucial impact of housing markets on financial and economic systems. The Chinese stock market experienced a marked fall during the global financial tsunami and China’s economy has also slowed down by about 2%–3% when measured in GDP. Nevertheless, the housing markets in diverse Chinese cities seemed to continue the almost nonstop mania for more than 10 years. However, the structure and dynamics of the Chinese housing market are less studied. Here, we perform an extensive study of the Chinese housing market by analyzing 10 representative key cities based on both linear and nonlinear econophysical and econometric methods. We identify a common collective driving force which accounts for 96.5% of the house price growth, indicating very high systemic risk in the Chinese housing market. The 10 key cities can be categorized into clubs and the house prices of the cities in the same club exhibit an evident convergence. These findings from different methods are basically consistent with each other. The identified city clubs are also consistent with the conventional classification of city tiers. The house prices of the first-tier cities grow the fastest and those of the third- and fourth-tier cities rise the slowest, which illustrates the possible presence of a ripple effect in the diffusion of house prices among different cities.

We show numerically that the orthogonal transformation corresponding to the eigenvectors of (random) left circulant matrices, has similar performance in signal/image processing like the discrete cosine transform.

In this paper, covariance matrices of heights measured relative to the average height of growing self-affine surfaces in the steady state are investigated in the framework of random matrix theory. We show that the spectral density of the covariance matrix scales as ρ(λ) ~ λ^-ν deviating from the prediction of random matrix theory and has a scaling form, ρ(λ, L) = λ^-ν f(λ/L^ϕ) for the lateral system size L, where the scaling function f(x) approaches a constant for λ ≪ L^ϕ and zero for L^ϕ≪λ< λ_max. The values of exponents obtained by numerical simulations are ν ≈ 1.70 and ϕ ≈ 1.51 for the Edward–Wilkinson class and ν ≈ 1.61 and ϕ ≈ 1.76 for the Kardar–Parisi–Zhang class, respectively. The distribution of the largest eigenvalues follows a scaling form as ρ(λ_max, L) = 1/L^b f_max ((λ_max - L^a)/L^b), which is different from the Tracy–Widom distribution of random matrix theory while the exponents a and b are given by the same values for the two different classes.

We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT). Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix of price changes of the largest 1000 US stocks for the 2-year period 1994–1995. We find that the statistics of most of the eigenvalues in the spectrum of agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations. We demonstrate that shares universal properties with the Gaussian orthogonal ensemble of random matrices. In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the inverse participation ratio, and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum — a situation reminiscent of results in localization theory.

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

The correlation structure of a stock market contains important financial contents, which may change remarkably due to the occurrence of financial crisis. We perform a comparative analysis of the Chinese stock market around the occurrence of the 2008 crisis based on the random matrix analysis of high-frequency stock returns of 1228 Chinese stocks. Both raw correlation matrix and partial correlation matrix with respect to the market index in two time periods of one year are investigated. We find that the Chinese stocks have stronger average correlation and partial correlation in 2008 than in 2007 and the average partial correlation is significantly weaker than the average correlation in each period. Accordingly, the largest eigenvalue of the correlation matrix is remarkably greater than that of the partial correlation matrix in each period. Moreover, each largest eigenvalue and its eigenvector reflect an evident market effect, while other deviating eigenvalues do not. We find no evidence that deviating eigenvalues contain industrial sectorial information. Surprisingly, the eigenvectors of the second largest eigenvalues in 2007 and of the third largest eigenvalues in 2008 are able to distinguish the stocks from the two exchanges. We also find that the component magnitudes of the some largest eigenvectors are proportional to the stocks’ capitalizations.

Several approaches and concepts of physics, such as Random Matrix Theory, have been used to investigate the complexity of financial time series. This study aims to analyze the spectrum of stock correlation in the Brazilian stock market by applying Random Matrix Theory to the subprime and Asian financial crisis periods and their temporal neighborhoods. Results show evident synchronized market behavior during both crises. The results also show that the period preceding a crisis presents symptoms which may predict future crises. Thereby, the methodology presented here could be used by the market as a tool that helps to anticipate possible market fluctuations.