Hermitian Wigner Matrices Research Articles

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.

Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices

We consider N by N deformed Wigner random matrices of the form XN=HN+AN, where HN is a real symmetric or complex Hermitian Wigner matrix and AN is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of XN for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on studying the characteristic function of the linear statistics (Landon and Sosoe (2018)) by using the cumulant expansion method, along with local laws for the Green function of XN (Ann. Probab. 48 (2020) 963–1001; Probab. Theory Related Fields 169 (2017) 257–352; J. Math. Phys. 54 (2013) 103504) and analytic subordination properties of the free additive convolution (Dallaporta and Fevrier (2019); Random Matrices Theory Appl. 9 (2020) 2050011). We also prove the analogous results for high-dimensional sample covariance matrices.

Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N × N complex Deformed Wigner matrix M N. M N is defined as follows: M N = W N / √ N +A N where W N is an N ×N Hermitian Wigner matrix whose entries have a law µ satisfying a Poincare inequality and the matrix A N is a block diagonal matrix, with an eigenvalue θ of multiplicity one, generating an outlier in the spectrum of M N. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of M N onto a unit eigenvector corresponding to θ are not universal. Indeed, we take away a fit approximation of its limit from this norm and prove the convergence to zero as N goes to ∞ of the Levy-Prohorov distance between this rescaled quantity and the convolution of µ and a centered Gaussian distribution (whose variance may depend depend upon N and may not converge).

Correlated random matrices: Band rigidity and edge universality

We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models.

Bulk universality for deformed Wigner matrices

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.

CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions

In this paper, we study the Hermitian Wigner matrix $W_n=(x_{ij})_{1\le i,j\le n}$ with independent (up to symmetry) mean zero variance one entries. Under some Lindeberg type condition on the fourth moments of the entries, we establish a central limit theorem for the linear eigenvalue statistics of $W_n$. Our result extends the previous results on this topic to a more general case without the assumption ${\bf E}\,x_{ij}^2=0$ for $1\le i<j\le n$. Instead, we only assume that the real part and imaginary part of the upper-diagonal entry are uncorrelated. More precisely, we require ${\bf E}\,x_{ij}^2$ to be real and homogeneous for all $1\le i<j\le n$. The limiting normal distribution of the central limit theorem is shown to depend on the parameter ${\bf E}\,x_{ij}^2\in[-1,1]$.

Fixed Energy Universality for Generalized Wigner Matrices

AbstractWe prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc.

Edge universality for deformed Wigner matrices

We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy–Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix.

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}{\pi}e^{-|z|^2/2-|w|^2/2+z\bar{w}}$ in the limit $n\to\infty$. In this paper, we show that this asymptotic law is universal among all random $n\times n$ matrices $M_n$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles. These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants $\log|\det(M_n-z_0)|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname {tr}(M_n-z_0)^{-1}$, in order to exploit Girko's Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation. With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

We consider random matrices of the form $$H = W + \lambda V, \lambda \in {\mathbb {R}}^+$$ , where $$W$$ is a real symmetric or complex Hermitian Wigner matrix of size $$N$$ and $$V$$ is a real bounded diagonal random matrix of size $$N$$ with i.i.d. entries that are independent of $$W$$ . We assume subexponential decay of the distribution of the matrix entries of $$W$$ and we choose $$\lambda \sim 1$$ , so that the eigenvalues of $$W$$ and $$\lambda V$$ are typically of the same order. Further, we assume that the density of the entries of $$V$$ is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $$\lambda _+\in {\mathbb {R}}^+$$ such that the largest eigenvalues of $$H$$ are in the limit of large $$N$$ determined by the order statistics of $$V$$ for $$\lambda >\lambda _+$$ . In particular, the largest eigenvalue of $$H$$ has a Weibull distribution in the limit $$N\rightarrow \infty $$ if $$\lambda >\lambda _+$$ . Moreover, for $$N$$ sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $$\lambda >\lambda _+$$ , while they are completely delocalized for $$\lambda <\lambda _+$$ . Similar results hold for the lowest eigenvalues.

CLT for linear spectral statistics of Hermitian Wigner matrices with general moment conditions

Изучается эрмитова матрица Вигнера $W_n=(x_{ij})_{1\leq i,j\leq n}$ с независимыми (с точностью до симметрии) внедиагональными элементами, имеющими нулевое среднее и единичную дисперсию. При некотором условии типа Линдеберга на четвертые моменты элементов матрицы доказана центральная предельная теорема для линейных статистик собственных значений матрицы $W_n$. Наш результат распространяет предыдущие результаты подобного рода на более общий случай, когда отсутствует ограничение $\bf E x^2_{ij}=0$ для $1\leq i &lt; j \leq n$. Вместо этого мы предполагаем только, что вещественная и мнимая части наддиагональных элементов некоррелированы. Более точно, мы требуем, чтобы математические ожидания $\bf E x^2_{ij}$ были вещественными и однородными для всех $1\leq i &lt; j \leq n$. Показано, что предельное нормальное распределение в центральной предельной теореме зависит от параметра $\bf E x^2_{ij}\in [-1,1]$.

A necessary and sufficient condition for edge universality of Wigner matrices

In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider N×N symmetric Wigner matrices H with Hij=N−1/2xij whose upper-right entries xij (1≤i<j≤N) are independent and identically distributed (i.i.d.) random variables with distribution ν and diagonal entries xii (1≤i≤N) are i.i.d. random variables with distribution ν̃. The means of ν and ν̃ are zero, the variance of ν is 1, and the variance of ν̃ is finite. We prove that the Tracy–Widom law holds if and only if lim s→∞s4P(|x12|≥s)=0. The same criterion holds for Hermitian Wigner matrices.

CLT for Spectra of Submatrices of Wigner Random Matrices

We prove a CLT for spectra of submatrices of real symmetric and Hermitian Wigner matrices. We show that if in the standard normalization the fourth moment of the off-digonal entries is GOE/GUE-like then the limiting Gaussian process can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields.

EIGENVALUE VARIANCE BOUNDS FOR WIGNER AND COVARIANCE RANDOM MATRICES

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.

Universality for certain Hermitian Wigner matrices under weak moment conditions

Nous étudions l’universalité des statistiques locales du spectre des matrices de Wigner hermitiennes divisibles par une gaussienne. Ces matrices aléatoires sont obtenues en ajoutant à une matrice de Wigner hermitienne avec des coefficients indépendants une matrice du GUE indépendante. Nous montrons que la classe d’universalité de la loi de Tracy–Widom pour les valeurs propres extrêmes est vérifiée sous la condition optimale d’une borne uniforme sur le quatrième moment des coefficients de la matrice. De plus, nous démontrons l’universalité des fluctuations dans l’intérieur du spectre dès lors que le second moment est fini.

A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

Recently we proved that the eigenvalue correlation functions of a general class of random matrices converge, weakly with respect to the energy, to the corresponding ones of Gaussian matrices. Tao and Vu gave a proof that for the special case of Hermitian Wigner matrices the convergence can be strengthened to vague convergence at any fixed energy in the bulk. In this article we comment on this result in the context of the universality conjectures of Mehta. We show that this theorem is an immediate corollary of our earlier results. Indeed, a more general form of this theorem also follows directly from our previous work.

On the Correlation Function of the Characteristic Polynomials of the Hermitian Wigner Ensemble

We consider the asymptotics of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this for the GUE up to a factor, depending only on the forth moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$, i.e. that the higher moments of $Q$ do not contribute to the above limit.

Average density of states for Hermitian Wigner matrices

We consider ensembles of N × N Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on arbitrarily small intervals converges to the semicircle law, as N tends to infinity.

Asymptotics of characteristic polynomials of Wigner matrices at the edge of the spectrum

We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.

On the Second-Order Correlation Function of the Characteristic Polynomial of a Hermitian Wigner Matrix

We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function of the characteristic polynomial. Furthermore, we show that the second-order correlation function of the characteristic polynomial is closely related to that of the permanental polynomial.