Semicircle Law Research Articles

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.

Random reverse-cyclic matrices and screened harmonic oscillator

We have calculated the joint probability distribution function for random reverse-cyclic matrices and shown that it is related to an N-body exactly solvable model. We refer to this well-known model potential as a screened harmonic oscillator. The connection enables us to obtain all the correlations among the particle positions moving in a screened harmonic potential. The density of nontrivial eigenvalues of this ensemble is found to be of the Wigner form and admits a hole at the origin, in contrast to the semicircle law of the Gaussian orthogonal ensemble of random matrices. The spacing distributions assume different forms ranging from Gaussian-like to Wigner.

Renormalization-group potential for quantum Hall effects

The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated situations where the spin or other "flavors" of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number $a$ and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when $a$ takes certain values this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG $\beta$-function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of $a$ is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the $\beta$-function given by a saddle point of the RG potential. The data are consistent with $a \in \mathbb{R}$, which together with the symmetry leads to a generalized semi-circle law.

Limiting spectral distribution of block matrices with Toeplitz block structure

We study two specific symmetric random block Toeplitz (of dimension k×k) matrices, where the blocks (of size n×n) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by nk) when (a) k is fixed and n→∞ (b) n is fixed and k→∞ (c) n and k go to ∞ simultaneously. Further, the LSDs obtained in (a) and (b) coincide with those in (c) when n or respectively k tends to infinity. This limit in (c) is the semicircle law in Case (i). In Case (ii), the limit is related to the limit of the random symmetric Toeplitz matrix as obtained by Bryc et al. (2006) and Hammond and Miller (2005).

Phase transitions for random states and a semicircle law for the partial transpose

For a system of N identical particles in a random pure state, there is a threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each typically share entanglement if k > k_0, and typically do not share entanglement if k < k_0. By "random" we mean here "uniformly distributed on the sphere of the corresponding Hilbert space." The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if k < k_1 := N/4 - 1/2, and typically in a non-PPT state if k > k_1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.

Local Semicircle Law at the Spectral Edge for Gaussian β-Ensembles

We study the local semicircle law for Gaussian β-ensembles at the edge of the spectrum. We prove that at the almost optimal level of \({n^{-2/3+\epsilon}}\), the local semicircle law holds for all β ≥ 1 at the edge. The proof of the main theorem relies on the calculation of the moments of the tridiagonal model of Gaussian β-ensembles up to the p n -moment, where \({p_n = O(n^{2/3-\epsilon})}\). The result is analogous to the result of Sinai and Soshnikov (Funct Anal Appl 32(2), 1998) for Wigner matrices, but the combinatorics involved in the calculations are different.

Universality of local spectral statistics of random matrices

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with potentialVVand inverse temperatureβ=1,2,4\beta = 1, 2, 4, corresponding to the orthogonal, unitary and symplectic ensembles. Forβ∉{1,2,4}\beta \notin \{1, 2, 4\}, there is no natural random matrix ensemble behind this model, but the statistical physics interpretation of the log-gas is still valid for allβ&gt;0\beta &gt; 0. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent ofVV. In this article, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Furthermore, we review the solution of Dyson’s conjecture on the local relaxation time of the Dyson Brownian motion. Related questions such as delocalization of eigenvectors and local version of Wigner’s semicircle law will also be discussed.

Semicircle law of Tyler’s [formula omitted]-estimator for scatter

This paper analyzes the spectral properties of Tyler’s M-estimator for scatter Tn,d. It is shown that if a multivariate sample stems from a generalized spherically distributed population and the sample size n and the dimension d both go to infinity while d/n→0, then the empirical spectral distribution of n/d(Tn,d−Id), Id being the identity, converges in probability to the semicircle law. In contrast to that of the sample covariance matrix, this convergence does not necessarily require the sample vectors to be componentwise independent. Further, moments of the generalized spherical population do not have to exist.

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.

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν ij with a subexponential decay. Let \({\sigma_{ij}^2}\) be the variance for the probability measure ν ij with the normalization property that \({\sum_{i} \sigma^2_{ij} = 1}\) for all j. Under essentially the only condition that \({c\le N \sigma_{ij}^2 \le c^{-1}}\) for some constant c > 0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M −1.

Hyperbolic disordered ensembles of random matrices

Using the recently introduced simple procedure of dividing Gaussian matrices by a positive random variable, a family of random matrices is generated characterized by a behavior ruled by the generalized hyperbolic distribution. The spectral density evolves from the semicircle law to a Gaussian-like behavior while concomitantly, the local fluctuations show a transition from the Wigner-Dyson to the Poisson statistics. Long range statistics such as number variance exhibit large fluctuations typical of nonergodic ensembles.

The Semicircle Law for Matrices with Independent Diagonals

We investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real-valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law.

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.

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.

Noncrossing Partitions, Catalan Words, and the Semicircle Law

As is well known, the joint limit distribution of independent Wigner matrices is free with the marginals being semicircular. This freeness is intimately tied to noncrossing pair partitions or, equivalently what are known as Catalan words, each of which contributes one to the limit moments.

Quantum automorphisms of twisted group algebras and free hypergeometric laws

We prove that we have an isomorphism of type A a u t ( C σ [ G ] ) ≃ A a u t ( C [ G ] ) σ A_{aut}(\mathbb C_\sigma [G])\simeq A_{aut}(\mathbb C[G])^\sigma , for any finite group G G , and any 2 2 -cocycle σ \sigma on G G . In the particular case G = Z n 2 G=\mathbb Z_n^2 , this leads to a Haar measure-preserving identification between the subalgebra of A o ( n ) A_o(n) generated by the variables u i j 2 u_{ij}^2 and the subalgebra of A s ( n 2 ) A_s(n^2) generated by the variables X i j = ∑ a , b = 1 n p i a , j b X_{ij}=\sum _{a,b=1}^np_{ia,jb} . Since u i j u_{ij} is “free hyperspherical” and X i j X_{ij} is “free hypergeometric”, we obtain in this way a new free probability formula, which at n = ∞ n=\infty corresponds to the well-known relation between the semicircle law and the free Poisson law.

AdS/QHE: towards a holographic description of quantum Hall experiments

Transitions among quantum Hall plateaux share a suite of remarkable experimental features, such as semicircle laws and duality relations, whose accuracy and robustness are difficult to explain directly in terms of the detailed dynamics of the microscopic electrons. They would naturally follow if the low-energy transport properties were governed by an emergent discrete duality group relating the different plateaux, but no explicit examples of interacting systems having such a group are known. Recent progress using the AdS/CFT correspondence has identified examples with similar duality groups, but without the dc ohmic conductivity characteristic of quantum Hall experiments. We use this to propose a simple holographic model for low-energy quantum Hall systems, with a nonzero dc conductivity that automatically exhibits all of the observed consequences of duality, including the existence of the plateaux and the semicircle transitions between them. The model can be regarded as a strongly coupled analogue of the old ‘composite boson’ picture of quantum Hall systems. Non-universal features of the model can be used to test whether it describes actual materials, and we comment on some of these in our proposed model. In particular, the model indicates the value for low-temperature scaling exponents for transitions among quantum Hall plateaux, in agreement with the measured value 0.42±0.01.

Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulas of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where m is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that $\gamma_{T}^{(m)}$ $(\gamma_{H}^{(m)})$ converges weakly to the semicircle law as m→∞.

The energy of random graphs

In 1970s, Gutman introduced the concept of the energy E ( G ) for a simple graph G , which is defined as the sum of the absolute values of the eigenvalues of G . This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equalities of those bounds. We however obtain an exact estimate of the energy for almost all graphs by Wigner’s semi-circle law, which generalizes a result of Nikiforov. We further investigate the energy of random multipartite graphs by considering a generalization of Wigner matrix, and obtain some estimates of the energy for random multipartite graphs.