Wigner Semicircle Research Articles

Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices

In this paper, we study the complex Wigner matrices $M_n=\frac{1}{\sqrt{n}}W_n$ whose eigenvalues are typically in the interval $[-2,2]$. Let $\lambda_1\leq \lambda_2...\leq\lambda_n$ be the ordered eigenvalues of $M_n$. Under the assumption of four matching moments with the Gaussian Unitary Ensemble(GUE), for test function $f$ 4-times continuously differentiable on an open interval including $[-2,2]$, we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold $u$ in the bulk of the Wigner semicircle law as $\mathcal{A}_n[f; u]=\sum_{l=1}^nf(\lambda_l)\mathbf{1}_{\{\lambda_l\leq u\}}$. And the second one is $\mathcal{B}_n[f; k]=\sum_{l=1}^{k}f(\lambda_l)$ with positive integer $k=k_n$ such that $k/n\rightarrow y\in (0,1)$ as $n$ tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_n[f; \lfloor nt\rfloor]$.

Fluctuations of deformed Wigner random matrices

Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, ..., yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn, α = n−1/2Xn + n−α/2(y1,..., yn), where 0 < α < 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn, α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation \(\mathbb{E}m_{n,\alpha } \left( z \right)\) and variance Var(mn, α(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein’s equation and its generalization which naturally leads to a certain recursive equation.

Distributions on unbounded moment spaces and random moment sequences

In this paper we define distributions on moment spaces corresponding to measures on the real line with an unbounded support. We identify these distributions as limiting distributions of random moment vectors defined on compact moment spaces and as distributions corresponding to random spectral measures associated with the Jacobi, Laguerre and Hermite ensemble from random matrix theory. For random vectors on the unbounded moment spaces we prove a central limit theorem where the centering vectors correspond to the moments of the Marchenko-Pastur distribution and Wigner's semi-circle law.

Local semicircle law and Gaussian fluctuation for Hermite &amp;lt;italic&amp;gt;&amp;beta;&amp;lt;/italic&amp;gt; ensemble

Let β > 0 and consider an n -point process λ1, λ2,..., λn from Hermite β ensemble on the real line R. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper, we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than √log n , i.e., local semicircle law holds. And the number of positive states in (0,∞) is proved to uctuate normally around its mean n /2 with variance like log n / π 2 β . The proofs rely largely on the way invented by Valko and Virag of counting states in any interval and the classical martingale argument.

Invariant Beta Ensembles and the Gauss-Wigner Crossover

We define a new diffusive matrix model converging toward the β-Dyson Brownian motion for all β is an element of [0,2] that provides an explicit construction of beta ensembles of random matrices that is invariant under the orthogonal or unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semicircle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semicircle.

Entanglement between two subsystems, the Wigner semicircle and extreme-value statistics

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.

On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach

In this article we employ certain techniques in divided differences to relate the generalized Stieltjes transform of the distribution of a randomly weighted average of independent random variables X1,…,Xm to the generalized Stieltjes transforms of the distribution functions F1,…,Fm; Xi∼Fi,i=1,…,m. The random weights are assumed to be cuts of [0,1] by m−1 ordered statistics of independent and identically uniformly distributed random variables U1,…,Un on [0,1]; m≤n. Soltani and Homei (2009) treated the case m=n using the Schwartz distribution theory. We identified fairly large classes of randomly weighted average distributions by their generalized Stieltjes transforms; in particular including the uniform, Wigner and certain power semicircle distributions.

Fuzzy spaces and new random matrix ensembles

We analyze the expectation value of observables in a scalar theory on the fuzzy two sphere, represented as a generalized hermitian matrix model. We calculate explicitly the form of the expectation values in the large-N limit and demonstrate that, for any single kind of field (matrix), the distribution of its eigenvalues is still a Wigner semicircle but with a renormalized radius. For observables involving more than one type of matrix we obtain a new distribution corresponding to correlated Wigner semicircles.

Rigidity of eigenvalues of generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H with independent entries, where the distribution of the ( i , j ) matrix element is given by the probability measure ν i j with zero expectation and with variance σ i j 2 . We assume that the variances satisfy the normalization condition ∑ i σ i j 2 = 1 for all j and that there is a positive constant c such that c ⩽ N σ i j 2 ⩽ c − 1 . We further assume that the probability distributions ν i j have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of H is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order ( N η ) − 1 where η is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If γ j = γ j , N denotes the classical location of the j-th eigenvalue under the semicircle law ordered in increasing order, then the j-th eigenvalue λ j is close to γ j in the sense that for some positive constants C, c P ( ∃ j : | λ j − γ j | ⩾ ( log N ) C log log N [ min ( j , N − j + 1 ) ] − 1 / 3 N − 2 / 3 ) ⩽ C exp [ − ( log N ) c log log N ] for N large enough. (2) The proof of Dysonʼs conjecture (Dyson, 1962 [15]) which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order N − 1 up to logarithmic corrections. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large N limit provided that the second moments of the two ensembles are identical.

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law.

Product of Ginibre matrices: Fuss-Catalan and Raney distributions

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.

Higher-dimensional conformal field theories in the Coulomb branch

We use the AdS/CFT correspondence to study flows of N=4 SYM to non-conformal theories. The dual geometries can be seen as sourced by a Wigner's semicircle distribution of D3 branes. We consider two cases, the first case corresponds to a point in the Coulomb branch and the theory flows to a six-dimensional conformal field theory. In the second case a mass is introduced for a hypermultiplet and the theory flows to a five-dimensional conformal field theory. We argue from the gravity and the field theory side that the low energy theories correspond to the (2,0) theory in six dimensions and to a theory with exceptional global symmetry E1 in five dimensions.

Fixed trace β-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density

In the present paper, fixed trace β-Hermite ensembles generalizing the fixed trace Gaussian ensembles are considered. For all β, we prove the Wigner semicircle law for these ensembles by using two different methods: one is the moment equivalence method with the help of the matrix model for general β, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for β-HE implies the same limit for fixed trace β-Hermite ensembles. Consequently, explicit limit can be given for fixed trace Gaussian orthogonal, unitary, and symplectic ensembles. Furthermore, for even β, analogous to β-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.

Bulk universality for Wigner matrices

AbstractWe consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

Wegner Estimate and Level Repulsion for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales η ≫ N −1 . This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

We consider N×N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. We study the connection between eigenvalue statistics on microscopic energy scales η≪1 and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order η∼log N/N. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales η≫N−2/3. We show that most eigenvectors are fully delocalized in the sense that their ℓp-norms are comparable with N1/p−1/2 for p≥2, and we obtain the weaker bound N2/3(1/p−1/2) for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.

Eigenvalue separation in some random matrix models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s&amp;gt;1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogs, an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider $N\times N$ Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$ implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.

Cavity approach to the spectral density of sparse symmetric random matrices

The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally treelike, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, showing excellent agreement.

The energy of graphs and matrices

Given a complex m × n matrix A, we index its singular values as σ 1 ( A ) ⩾ σ 2 ( A ) ⩾ ⋯ and call the value E ( A ) = σ 1 ( A ) + σ 2 ( A ) + ⋯ the energy of A, thereby extending the concept of graph energy, introduced by Gutman. Let 2 ⩽ m ⩽ n , A be an m × n nonnegative matrix with maximum entry α, and ‖ A ‖ 1 ⩾ n α . Extending previous results of Koolen and Moulton for graphs, we prove that E ( A ) ⩽ ‖ A ‖ 1 m n + ( m − 1 ) ( ‖ A ‖ 2 2 − ‖ A ‖ 1 2 m n ) ⩽ α n ( m + m ) 2 . Furthermore, if A is any nonconstant matrix, then E ( A ) ⩾ σ 1 ( A ) + ‖ A ‖ 2 2 − σ 1 2 ( A ) σ 2 ( A ) . Finally, we note that Wigner's semicircle law implies that E ( G ) = ( 4 3 π + o ( 1 ) ) n 3 / 2 for almost all graphs G.