Tracy-Widom Law Research Articles

Independence of linear spectral statistics and the point process at the edge of Wigner matrices

Let [Formula: see text] be a Wigner matrix of dimension [Formula: see text] with eigenvalues [Formula: see text] and [Formula: see text] be an analytic function on [Formula: see text] with polynomial growth. It is known that Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] converges in distribution to a normal random variable with mean [Formula: see text] and a finite variance depending on [Formula: see text]. On the other hand, it is also known that [Formula: see text] converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that [Formula: see text] and Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300 ]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

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.

Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

On randomized sketching algorithms and the Tracy–Widom law

There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably chosen random projection can be used to embed the original dataset in a lower-dimensional space such that key properties of the original dataset are retained. These algorithms are often referred to as sketching algorithms, as the projected dataset can be used as a compressed representation of the full dataset. We show that random matrix theory, in particular the Tracy–Widom law, is useful for describing the operating characteristics of sketching algorithms in the tall-data regime when the sample size n is much greater than the number of variables d. Asymptotic large sample results are of particular interest as this is the regime where sketching is most useful for data compression. In particular, we develop asymptotic approximations for the success rate in generating random subspace embeddings and the convergence probability of iterative sketching algorithms. We test a number of sketching algorithms on real large high-dimensional datasets and find that the asymptotic expressions give accurate predictions of the empirical performance.

Quantitative Tracy–Widom laws for the largest eigenvalue of generalized Wigner matrices

Quantitative Tracy–Widom laws for the largest eigenvalue of generalized Wigner matrices

Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let X be an M×N random matrix consisting of independent M-variate elliptically distributed column vectors x1,…,xN with general population covariance matrix Σ. In the literature, the quantity XX∗ is referred to as the sample covariance matrix after scaling, where X∗ is the transpose of X. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of XX∗ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that x1,…,xN follow as M,N→∞ with M∕N→ϕ0>0 if the weak fourth moment of the radius of x1 exists. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.

Tracy-Widom Distribution for Heterogeneous Gram Matrices With Applications in Signal Detection

Detection of the number of signals corrupted by high-dimensional noise is a fundamental problem in signal processing and statistics. This paper focuses on a general setting where the high-dimensional noise has an unknown complicated heterogeneous variance structure. We propose a sequential test which utilizes the edge singular values (i.e., the largest few singular values) of the data matrix. It also naturally leads to a consistent sequential testing estimate of the number of signals. We describe the asymptotic distribution of the test statistic in terms of the Tracy-Widom distribution. The test is shown to be accurate and have full power against the alternative, both theoretically and numerically. The theoretical analysis relies on establishing the Tracy-Widom law for a large class of Gram type random matrices with non-zero means and completely arbitrary variance profiles, which can be of independent interest.

Edge statistics of large dimensional deformed rectangular matrices

We consider the edge statistics of large dimensional deformed rectangular matrices of the form Yt=Y+tX, where Y is a p×n deterministic signal matrix whose rank is comparable to n, X is a p×n random noise matrix with i.i.d. entries of mean zero and variance n−1, and t>0 gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case t=1, the spectral statistics of this model have been studied to a certain extent in the literature (Dozier and Silverstein, 2007[17,18]; Vallet et al., 2012). In this paper, we study the singular value and singular vector statistics of Yt around the right-most edge of the spectrum in the harder regime n−1/3≪t≪1. This regime is harder than the t=1 case, because on the one hand, the edge behavior of the empirical spectral distribution (ESD) of YY⊤ has a strong effect on the edge statistics of YtYt⊤ for a “small” t≪1, while on the other hand, the edge eigenvalue behavior of YtYt⊤ is not merely a perturbation of that of YY⊤ for a “large” t≫n−1/3. Under certain regularity assumptions on Y, we prove the Tracy–Widom law for the edge eigenvalues, the eigenvalue rigidity, and eigenvector delocalization for the matrices YtYt⊤ and Yt⊤Yt. These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD of YtYt⊤ and establish optimal local laws on its resolvent. These results are of independent interest, and can be used as important inputs for many other problems regarding the spectral statistics of Yt.

Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices

We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.

Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.

Sample canonical correlation coefficients of high-dimensional random vectors: Local law and Tracy–Widom limit

Consider two random vectors [Formula: see text] and [Formula: see text], where the entries of [Formula: see text] and [Formula: see text] are i.i.d. random variables with mean zero and variance one, and [Formula: see text] and [Formula: see text] are respectively, [Formula: see text] and [Formula: see text] deterministic population covariance matrices. With [Formula: see text] independent samples of [Formula: see text], we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with [Formula: see text] and [Formula: see text] as [Formula: see text], we prove that the largest sample canonical correlation coefficient converges to the Tracy–Widom distribution as long as we have [Formula: see text] and [Formula: see text], which we believe to be a sharp moment condition. This extends the result in [19], which established the Tracy–Widom limit under the assumption that all moments exist for the entries of [Formula: see text] and [Formula: see text]. Our proof is based on a new linearization method, which reduces the problem to the study of a [Formula: see text] random matrix [Formula: see text]. In particular, we shall prove an optimal local law on its inverse [Formula: see text], called resolvent. This local law is the main tool for both the proof of the Tracy–Widom law in this paper, and the study in [26, 27] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.

Topological Metric Detects Hidden Order in Disordered Media.

Recent advances in microscopy techniques make it possible to study the growth, dynamics, and response of complex biophysical systems at single-cell resolution, from bacterial communities to tissues and organoids. In contrast to ordered crystals, it is less obvious how one can reliably distinguish two amorphous yet structurally different cellular materials. Here, we introduce a topological earth mover's (TEM) distance between disordered structures that compares local graph neighborhoods of the microscopic cell-centroid networks. Leveraging structural information contained in the neighborhood motif distributions, the TEM metric allows an interpretable reconstruction of equilibrium and nonequilibrium phase spaces and embedded pathways from static system snapshots alone. Applied to cell-resolution imaging data, the framework recovers time ordering without prior knowledge about the underlying dynamics, revealing that fly wing development solves a topological optimal transport problem. Extending our topological analysis to bacterial swarms, we find a universal neighborhood size distribution consistent with a Tracy-Widom law.

Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices

We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy–Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a central limit theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.

Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by VARMA

In this paper, we derive the Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by the vector autoregressive moving average model when the dimension is comparable to the sample size. This result is applied to make inference on the vector autoregressive moving average model. Simulations are conducted to demonstrate the finite sample performance of our inference.

Universality of dissipative self-assembly from quantum dots to human cells

An important goal of self-assembly research is to develop a general methodology applicable to almost any material, from the smallest to the largest scales, whereby qualitatively identical results are obtained independently of initial conditions, size, shape and function of the constituents. Here, we introduce a dissipative self-assembly methodology demonstrated on a diverse spectrum of materials, from simple, passive, identical quantum dots (a few hundred atoms) that experience extreme Brownian motion, to complex, active, non-identical human cells (~1017 atoms) with sophisticated internal dynamics. Autocatalytic growth curves of the self-assembled aggregates are shown to scale identically, and interface fluctuations of growing aggregates obey the universal Tracy–Widom law. Example applications for nanoscience and biotechnology are further provided. Biological systems are able to self-assemble in non-equilibrium conditions thanks to a continuous injection of energy. Here the authors present a tool to achieve non-equilibrium self-assembly of synthetic and biological constituents with sizes spanning three orders of magnitude.

Tracy–Widom limit for Kendall’s tau

In this paper, we study a high-dimensional random matrix model from nonparametric statistics called the Kendall rank correlation matrix, which is a natural multivariate extension of the Kendall rank correlation coefficient. We establish the Tracy–Widom law for its largest eigenvalue. It is the first Tracy–Widom law for a nonparametric random matrix model, and also the first Tracy–Widom law for a high-dimensional U-statistic.

Optimality Regions and Fluctuations for Bernoulli Last Passage Models

We study the sequence alignment problem and its independent version, the discrete Hammersley process with an exploration penalty. We obtain rigorous upper bounds for the number of optimality regions in both models near the soft edge. At zero penalty the independent model becomes an exactly solvable model and we identify cases for which the law of the last passage time converges to a Tracy-Widom law.

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal Q = TX(TX)^{*}$, where the sample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with mean zero and variance $N^{-1}$, and $T$ is an $M_1 \times M_2$ deterministic matrix satisfying $T^* T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal Q$ when $M:=\min\{M_1,M_2\}$ and $N$ tends to infinity with $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. Under mild assumptions of $T$, we prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal Q$ if and only if $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(| \sqrt{N} x_{ij}| \geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.

On the domain of attraction of a Tracy–Widom law with applications to testing multiple largest roots

The greatest root statistic arises as a test statistic in several multivariate analysis settings. Suppose there is a global null hypothesis H0 that consists of m different independent sub null hypotheses, i.e., H0=H01∩⋯∩H0m, and suppose the greatest root statistic is used as the test statistic for each sub null hypothesis. Such problems may arise when conducting a batch MANOVA or several batches of pairwise testing for equality of covariance matrices. Using the union-intersection testing approach, and by letting the problem dimension p→∞ faster than m→∞, we show that H0 can be tested using a Gumbel distribution to approximate the critical values. Although the theoretical results are asymptotic, simulation studies indicate that the approximation is accurate even for small to moderate dimensions. The results are general and can be applied in any setting where the greatest root statistic is used, not just for the two methods discussed for illustrative purposes.

Edge statistics for a class of repulsive particle systems

We study a class of interacting particle systems on $$\mathbb {R}$$ which was recently investigated by Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Gotze and Venker (Ann Probab 42(6):2207–2242, 2014. doi: 10.1214/13-AOP844 ) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi: 10.3842/SIGMA.2016.093 ; Schuler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schuler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).