We consider linear spectral statistics of the form $\mathrm{tr} ( φ(H))$ for test functions $φ$ of low regularity and Wigner matrices $H$ with smooth entry distribution. We show that for functions $φ$ in the Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\varepsilon >0$.
We derive a Central Limit Theorem (CLT) for $\log \left\vert\det \left( W_{N}-E_{N}\right)\right\vert,$ where $W_{N}$ is a Wigner matrix, and $E_{N}$ is local to the edge of the semi-circle law. Precisely, $E_N=2+N^{-2/3}σ_N$ with $σ_N$ being either a constant (possibly negative), or a sequence of positive real numbers, slowly diverging to infinity so that $σ_N \ll \log^{2} N$. We also extend our CLT to cover spiked Wigner matrices. Our interest in the CLT is motivated by its applications to statistical testing in critically spiked models and to the fluctuations of the free energy in the spher…
We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following Tao-Vu and Erdös, Yau, et al. and a Littlewood-Paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the Hölder class $C^{1/2+ε}$ …
As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral statistics (LSS) of Wigner beta ensembles. We first establish a central limit theorem (CLT) for LSS of Wigner quaternion matrices, then give a general CLT for Wigner $β$ ensembles.
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.
We show that the global fluctuations of spectra of GOE and GUE matrices and their principal submatrices executing Dyson's Brownian motion are Gaussian in the limit of large matrix dimensions. For nested submatrices one obtains a limiting three-dimensional generalized Gaussian process; its restrictions to two-dimensional sections that are monotone in matrix sizes and time moments coincide with the two-dimensional Gaussian Free Field with zero boundary conditions. The proof is by moment convergence, and it extends to more general Wigner matrices and their stochastic evolution.
We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on the behavior of single eigenvalues of Wigner matrices and $β$-ensembles. Among the results we obtain are the Gaussian fluctuations of single eigenvalues for Wigner matrices (without an assumption of 4 matching moments) and classical $β$-ensembles ($β=1, 2, 4$), Gaussian fluctuations of the eigenvalue counting function, and an asymptotic expansion up to order…
We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.
In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is either $N^{-1/2+\varepsilon}$ or $N^{-1+\varepsilon}$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, …
We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $H$ and $\hat H$. In particular our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues of $H$ and $\hat H$, around their asymptotic shape, the Vershik-Kerov-Logan-Shepp curve. This result demonstrates yet another aspect of the close connect…
A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincare type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices.
Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures, and establish a CLT for the corresponding linear spectral statistics via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type matrices. Further, we show that for certain estimator of such renormalized adjacency matrices, which will be no longer Wigner-type but share long-range non-decaying weak correlations among the en…
We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.
We study the renormalized real sample covariance matrix $H=X^TX/\sqrt{MN}-\sqrt{M/N}$ with $N/M\rightarrow0$ as $N, M\rightarrow \infty$ in this paper. And we always assume $M=M(N)$. Here $X=[X_{jk}]_{M\times N}$ is an $M\times N$ real random matrix with i.i.d entries, and we assume $\mathbb{E}|X_{11}|^{5+δ}<\infty$ with some small positive $δ$. The Stieltjes transform $m_N(z)=N^{-1}Tr(H-z)^{-1}$ and the linear eigenvalue statistics of $H$ are considered. We mainly focus on the asymptotic expansion of $\mathbb{E}\{m_N(z)\}$ in this paper. Then for some fine test function, a central limit theor…
We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $ξ_j$, perturbed by predictable multiplicative factors $λ_j$ with values in intervals $[\underlineλ_j,\overlineλ_j]$. It is assumed that the sequences $\underlineλ_j$, $\overlineλ_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst" sequence $λ_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expec…
We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $α$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent r…
A well known conjecture of Wigner, Dyson, and Mehta asserts that the (appropriately normalized) $k$-point correlation functions of the eigenvalues of random $n \times n$ Wigner matrices in the bulk of the spectrum converge (in various senses) to the $k$-point correlation function of the Dyson sine process in the asymptotic limit $n \to \infty$. There has been much recent progress on this conjecture, in particular it has been established under a wide variety of decay, regularity, and moment hypotheses on the underlying atom distribution of the Wigner ensemble, and using various notions of conve…
Wigner integrals and the corresponding Wigner chaos were introduced by P. Biane and R. Speicher in 1998 as a non-commutative counterpart of classical Wiener-Itô integrals and the corresponding Wiener-Itô chaos, respectively, in free probability. In the classical case, a famous result of I. Shigekawa states that non-trivial elements in the finite Wiener-Itô chaos have an absolutely continuous distribution. We provide here a first contribution to such regularity questions for Wigner integrals by showing that the distribution of non-trivial elements in the finite Wigner chaos cannot have atoms. T…
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gas…
We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt n}\sum_{i=1}^{n}Z_{i}\Big)\Big)-{\mathbb{E}}\big(f(G)\big)\rightarrow 0 $$ where $Z_{i}$ are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), …
We study the central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum\limits_{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where $X_i \in \mathbb{R}^n$ and $n$ may scale with $d$. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if $n^{2p-1}\gg d$, then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a no…
We consider $n\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $κ_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\to c\in[0,\infty)$ with the same variance as for Gaussian …
We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of $β$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions. The method relies on a general normal approximation result of …
We consider ensembles of $N \times N$ 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 {\it arbitrarily} small intervals converges to the semicircle law, as $N$ tends to infinity.
We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation and thus considerably relaxes the regularity requirement imposed in [7,11].
In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum_{k=1}^n a_k f(k t)$, where $f$ is a $2π-$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, that are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance onl…
It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Giné-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $π$. The effect is diminishing for growing dimension b…
We consider a class of sparse random matrices, which includes the adjacency matrix of Erdős-Rényi graphs $\mathcal G(N,p)$ for $p \in [N^{\varepsilon-1},N^{-\varepsilon}]$. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization $N\sqrt{p}$. In addition, the eigenvalues fluctuate simultaneously: the correlation of two eigenvalues of the same/different sign is asymptotically 1/-1. We also prove CLTs for the eigenvalue …
Consider the multivariate Stein equation $Δf - x\cdot \nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\_h$ be the solution given by Barbour's generator approach. We prove that, when $h$ is $α$-Hölder ($0<α\leq1$), all derivatives of order $2$ of $f\_h$ are $α$-Hölder {\it up to a $\log$ factor}; in particular they are $β$-Hölder for all $β\in (0, α)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For $α=1$, the regularity we obtain is optimal, as shown by an example given by Raič \c…
We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previo…
The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley to prove a new generalized covariance inequality similar to Yoshihara's. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions. The validity of the bootstrap for U-statistics has until now only been establi…
We consider the ensemble of $n \times n$ Wigner hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{\ell k}$ are given by $h_{\ell k} = n^{-1/2} (x_{\ell k} + \sqrt{-1} y_{\ell k})$, where $x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random variables with mean zero and variance 1/2, $y_{\ell\ell}=0$ and $x_{\ell \ell}$ have mean zero and variance 1. We assume the distribution of $x_{\ell k}, y_{\ell k}$ to have subexponential decay. In a recent paper, four of the authors r…
Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1]\] to within an additive $\pm \eps$. For any constant $\eps > 0$ and $k \geq 1$ the running time of our algorithm is a fixed polynomial in $n$. This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assi…
Eugene Wigner's revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior: the statistics of energy gaps depend only on the basic symmetry type of the model. Simplified models of Wigner's thesis have recently become mathematically accessible. For mean field models represented by large random matrices with independent entries, the celebrated Wigner-Dyson-Gaudin-Mehta (WDGM) conjecture asserts that the local eigenvalue statistics are universal. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potential…
We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds f…
