Bulk Eigenvalues Research Articles

Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices

AbstractWe consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal up to the factor . The latter result complements the very recent matching lower bound obtained by Cipolloni et al. and improves the ‐dependence of the upper bounds by Banks et al. and Jain et al. Our main ingredient, a near‐optimal lower tail estimate for the small singular values of , is of independent interest.

Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations

Consider two random vectors x˜=Az+C11∕2x∈Rp and y˜=Bz+C21∕2y∈Rq, where x∈Rp, y∈Rq and z∈Rr are independent random vectors with i.i.d. entries of zero mean and unit variance, C1 and C2 are p×p and q×q deterministic population covariance matrices, and A and B are p×r and q×r deterministic factor loading matrices. With n independent observations of x˜ and y˜, we study the sample canonical correlations between them. Under the sharp fourth moment condition on the entries of x, y and z, we prove the BBP transition for the sample canonical correlation coefficients (CCCs). More precisely, if a population CCC is below a threshold, then the corresponding sample CCC converges to the right edge of the bulk eigenvalue spectrum of the sample canonical correlation matrix and satisfies the famous Tracy-Widom law; if a population CCC is above the threshold, then the corresponding sample CCC converges to an outlier that is detached from the bulk eigenvalue spectrum. We prove our results in full generality, in the sense that they also hold for near-degenerate population CCCs and population CCCs that are close to the threshold.

CORRELATION MATRIX OF EQUI-CORRELATED NORMAL POPULATION: FLUCTUATION OF THE LARGEST EIGENVALUE, SCALING OF THE BULK EIGENVALUES, AND STOCK MARKET

Given an N-dimensional sample of size [Formula: see text] form a sample correlation matrix [Formula: see text]. Suppose that N and T tend to infinity with [Formula: see text] converging to a fixed finite constant [Formula: see text]. If the population is a factor model, then the eigenvalue distribution of [Formula: see text] almost surely converges weakly to Marčenko–Pastur distribution such that the index is Q and the scale parameter is the limiting ratio of the specific variance to the ith variable [Formula: see text]. For an N-dimensional normal population with equi-correlation coefficient [Formula: see text], which is a one-factor model, for the largest eigenvalue [Formula: see text] of [Formula: see text], we prove that [Formula: see text] converges to the equi-correlation coefficient [Formula: see text] almost surely. These results suggest an important role of an equi-correlated normal population and a factor model in Laloux et al. [(2000) Random matrix theory and financial correlations, International Journal of Theoretical and Applied Finance3 (3), 391–397]: the histogram of the eigenvalue of sample correlation matrix of the returns of stock prices fits the density of Marčenko–Pastur distribution of index [Formula: see text] and scale parameter [Formula: see text]. Moreover, we provide the limiting distribution of the largest eigenvalue of a sample covariance matrix of an equi-correlated normal population. We discuss the phase transition as to the decay rate of the equi-correlation coefficient in N.

The conjugate gradient algorithm on a general class of spiked covariance matrices

We consider the conjugate gradient algorithm applied to a general class of spiked sample covariance matrices. The main result of the paper is that the norms of the error and residual vectors at any finite step concentrate on deterministic values determined by orthogonal polynomials with respect to a deformed Marchenko–Pastur law. The first-order limits and fluctuations are shown to be universal. Additionally, for the case where the bulk eigenvalues lie in a single interval we show a stronger universality result in that the asymptotic rate of convergence of the conjugate gradient algorithm only depends on the support of the bulk, provided the spikes are well-separated from the bulk. In particular, this shows that the classical condition number bound for the conjugate gradient algorithm is pessimistic for spiked matrices.

Estimation of the Number of Spiked Eigenvalues in a Covariance Matrix by Bulk Eigenvalue Matching Analysis

ABSTRACT The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number of spiked eigenvalues, K. For estimation of K, most attention has focused on the use of top eigenvalues of sample covariance matrix, and there is little investigation into proper ways of using bulk eigenvalues to estimate K. We propose a principled approach to incorporating bulk eigenvalues in the estimation of K. Our method imposes a working model on the residual covariance matrix, which is assumed to be a diagonal matrix whose entries are drawn from a gamma distribution. Under this model, the bulk eigenvalues are asymptotically close to the quantiles of a fixed parametric distribution. This motivates us to propose a two-step method: the first step uses bulk eigenvalues to estimate parameters of this distribution, and the second step leverages these parameters to assist the estimation of K. The resulting estimator aggregates information in a large number of bulk eigenvalues. We show the consistency of under a standard spiked covariance model. We also propose a confidence interval estimate for K. Our extensive simulation studies show that the proposed method is robust and outperforms the existing methods in a range of scenarios. We apply the proposed method to analysis of a lung cancer microarray dataset and the 1000 Genomes dataset.

Fluctuations of extreme eigenvalues of sparse Erd\u0151s\u2013R\xe9nyi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph {{mathcal {G}}}(N,p). We show that if N^{varepsilon } leqslant Np leqslant N^{1/3-varepsilon } then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from Np geqslant N^{2/9 + varepsilon } down to the optimal scale Np geqslant N^{varepsilon }. The main technical achievement of our proof is a rigidity bound of accuracy N^{-1/2-varepsilon } (Np)^{-1/2} for the extreme eigenvalues, which avoids the (Np)^{-1}-expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np geqslant N^{varepsilon }.

Eigenvalue rigidity for truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an [Formula: see text] principal submatrix of an [Formula: see text] random unitary matrix distributed according to Haar measure. For [Formula: see text] and [Formula: see text] large with [Formula: see text], the empirical spectral measure is well approximated by a deterministic measure [Formula: see text] supported on the unit disc. In earlier work, we showed that for fixed [Formula: see text] and [Formula: see text], the bounded-Lipschitz distance [Formula: see text] between the empirical spectral measure and the corresponding [Formula: see text] is typically of order [Formula: see text] or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

Spectral statistics of sparse Erdős–Rényi graph Laplacians

Nous nous intéressons aux statistiques, dans l’intérieur du spectre, des valeurs propres de matrices laplaciennes de grands graphes d’Erdős–Rényi aléatoires dans le régime où $p\geq N^{\delta}/N$ pour un $\delta>0$ fixé arbitraire. Nous montrons une loi locale jusqu’à l’échelle optimale $\eta \gtrsim N^{-1}$ qui implique que les vecteurs propres sont délocalisés. Nous considérons les statistiques locales des valeurs propres et montrons que les statistiques des intervalles et les fonctions de corrélation moyennées coïncident avec le GOE dans l’intérieur du spectre.

The distribution of overlaps between eigenvectors of Ginibre matrices

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

Universal broadening of zero modes: A general framework and identification.

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian random matrix ensemble of size ν×ν, where ν is the number of zero modes. This observation unifies and extends a number of results within chiral random matrix theory and effective field theory and clarifies under which conditions they apply. The scaling of the former zero modes with the volume differs from the eigenvalues in the bulk, which we propose as an indicator to identify them in experiments. These results hold for all 10 symmetric spaces in the Altland-Zirnbauer classification and build on two facts. First, the broadened zero modes decouple from the bulk eigenvalues and, second, the mixing from eigenstates of the perturbation form a central limit theorem argument for matrices.

Overlaps between eigenvectors of correlated random matrices

We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum thermalisation to high dimensional statistics. We find that the overlaps only depend on measurable quantities, and do not require the knowledge of the underlying "true" (noiseless) matrices. We apply our results to the case of empirical correlation matrices, that allow us to estimate reliably the width of the spectrum of the true correlation matrix, even when the latter is very close to the identity. We illustrate our results on the example of stock returns correlations, that clearly reveal a non trivial structure for the bulk eigenvalues. We also apply our results to the problem of matrix denoising in high dimension.

Complex correlation approach for high frequency financial data

We propose a novel approach that allows the calculation of a Hilbert transform based complex correlation for unevenly spaced data. This method is especially suitable for high frequency trading data, which are of a particular interest in finance. Its most important feature is the ability to take into account lead-lag relations on different scales, without knowing them in advance. We also present results obtained with this approach while working on Tokyo Stock Exchange intraday quotations. We show that individual sectors and subsectors tend to form important market components which may follow each other with small but significant delays. These components may be recognized by analysing eigenvectors of complex correlation matrix for Nikkei 225 stocks. Interestingly, sectorial components are also found in eigenvectors corresponding to the bulk eigenvalues, traditionally treated as noise.

Bulk eigenvalue statistics for random regular graphs

AK was partly supported by Swiss National Science Foundation grant 144662. HTY was partly supported by NSF grant DMS-1307444 and a Simons Investigator fellowship.

Negativity spectrum of one-dimensional conformal field theories

The partial transpose $\rho_A^{T_2}$ of the reduced density matrix $\rho_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of $\rho_A^{T_2}$, that we dub negativity spectrum, in the ground sate of gapless one-dimensional systems described by a Conformal Field Theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of $\rho_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.

Protein States as Symmetry Transitions in the Correlation Matrices.

Over the last few years, there has been significant progress in the knowledge on protein folding. However, some aspects of protein folding still need further attention. One of these is the exact relationship between the folded and unfolded states and the differences between them. Whereas the folded state is well known, at least from a structural point of view (just think of the thousands of structures in online databases), the unfolded state is more elusive. Also, these are dynamic states of matter, and this aspect cannot be overlooked. Molecular dynamics-derived correlation matrices are an invaluable source of information on the protein dynamics. Here, bulk eigenvalue spectra of the correlation matrices obtained from the Trp-cage dynamics in the folded and unfolded states have been analyzed. The associated modes represent localized vibrations and are significantly affected by the fine details of the structure and interactions. Therefore, these bulk modes can be used as probes of the protein local dynamics in different states. The results of these analyses show that the correlation matrices describing the folded and unfolded dynamics belong to different symmetry classes. This finding provides new support to the phase-transition models of protein folding.

Bulk eigenvalue statistics for random regular graphs

We consider the uniform random $d$-regular graph on $N$ vertices, with $d \in [N^\alpha, N^{2/3-\alpha}]$ for arbitrary $\alpha > 0$. We prove that in the bulk of the spectrum the local eigenvalue correlation functions and the distribution of the gaps between consecutive eigenvalues coincide with those of the Gaussian Orthogonal Ensemble.

Large deviations of the shifted index number in the Gaussian ensemble

We show that, using the Coulomb fluid approach, we are able to derive a rate function of two variables that captures: (i) the large deviations of bulk eigenvalues; (ii) the large deviations of extreme eigenvalues (both left and right large deviations); (iii) the statistics of the fraction c of eigenvalues to the left of a position x. Thus, explains the full order statistics of the eigenvalues of large random Gaussian matrices as well as the statistics of the shifted index number. All our analytical findings are thoroughly compared with Monte Carlo simulations, obtaining excellent agreement. A summary of preliminary results has already been presented in Pérez Castillo (2014 Phys. Rev. E 90 040102) in the context of one-dimensional trapped spinless fermions in a harmonic potential.

Spectral Experiments+

ABSTRACTWe describe extensive computational experiments on spectral properties of random objects—random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere. We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.

RANDOM MATRICES: SHARP CONCENTRATION OF EIGENVALUES

Let [Formula: see text] be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval [-2, 2]. We prove a concentration bound for NI = NI(Wn), the number of eigenvalues of Wn in an interval I. Our result shows that NI decays exponentially with standard deviation at most O( log O(1) n). This is best possible up to the constant exponent in the logarithmic term. As a corollary, the bulk eigenvalues are localized to an interval of width O( log O(1) n/n); again, this is optimal up to the exponent. These results strengthen recent results of Erdős, Yau and Yin (under the extra assumption of vanishing third moment). Our proof is relatively simple and relies on the Lindeberg replacement argument.