Semicircle Law Research Articles

Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices

It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. What is more, we allow for a certain amount of dependence among the matrix entries, in the spirit of a recent generalization of Wigner's theorem, due to Schenker and Schulz-Baldes. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.

The Polynomial Method for Random Matrices

We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marcenko–Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.

Modular symmetry, the semicircle law, and quantum Hall bilayers

There is considerable experimental evidence for the existence in Quantum Hall systems of an approximate emergent discrete symmetry, $\Gamma_0(2) \subset SL(2,Z)$. The evidence consists of the robustness of the tests of a suite a predictions concerning the transitions between the phases of the system as magnetic fields and temperatures are varied, which follow from the existence of the symmetry alone. These include the universality of and quantum numbers of the fixed points which occur in these transitions; selection rules governing which phases may be related by transitions; and the semi-circular trajectories in the Ohmic-Hall conductivity plane which are followed during the transitions. We explore the implications of this symmetry for Quantum Hall systems involving {\it two} charge-carrying fluids, and so obtain predictions both for bilayer systems and for single-layer systems for which the Landau levels have a spin degeneracy. We obtain similarly striking predictions which include the novel new phases which are seen in these systems, as well as a prediction for semicircle trajectories which are traversed by specific combinations of the bilayer conductivities as magnetic fields are varied at low temperatures.

Quantum Hall effect in graphene: Emergent modular symmetry and the semicircle law

Low-energy transport measurements in quantum Hall systems have been argued to be governed by emergent modular symmetries whose predictions are robust against many of the detailed microscopic dynamics. We propose the recently observed quantum Hall effect in graphene as a test of these ideas, and identify to this end a class of predictions for graphene which would follow from the same modular arguments. We are led to a suite of predictions for high-mobility samples that differs from those obtained for the conventional quantum Hall effect in semiconductors, including predictions for the locations of the quantum Hall plateaus, predictions for the positions of critical points on transitions between plateaus, a selection rule for which plateaus can be connected by low-temperature transitions and a semicircle law for conductivities traversed during these transitions. Many of these predictions appear to provide a good description of graphene measurements performed with intermediate-strength magnetic fields.

Random sorting networks

A sorting network is a shortest path from 12 ⋯ n to n ⋯ 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the space–time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.

Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

Abstract It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n −2/3+v ).

Semicircle Law for Hadamard Products

In this paper, assuming $p/n\rightarrow 0$ as $n\rightarrow\infty$, we will prove the weak and strong convergence to the semicircle law of the empirical spectral distribution of the Hadamard product of a normalized sample covariance matrix and a sparsing matrix, which is of the form $A_p=\frac{1}{\sqrt{np}}(X_{m,n}X_{m,n}^*-\sigma^2nI_m)\circ D_{m}$, where the matrices $X_{m,n}$ and $D_m$ are independent and the entries of $X_{m,n}$ $(m\times n)$ are independent, the matrix $D_{m}$ $(m\times m)$ is Hermitian with independent entries above and on the diagonal, $p$ is the sum of the second moments of the row (and column) entries of $D_m$, and “$\circ$” denotes the Hadamard product of matrices.

The Rate of Convergence in Probability for Spectra of the GUE

It is shown that the Kolmogorov distance between the spectral distribution function of an $n\times n$ matrix from the Gaussian unitary ensemble (GUE) and the distribution function of the semicircle law is of stochastic order $O_p((\log n)/n^{2/3})$.

The spectral laws of Hermitian block-matrices with large random blocks

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k$-weakly dependent entries need not to be the semicircle law in the limit.

Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

We give a short proof of an extension of the free Talagrand transportation cost inequality to the semicircular which was originally proved in \cite{BV}. The proof is based on a convexity argument and is in the spirit of the original Talagrand's approach for the classical counterpart from \cite{T}. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of $\beta$ ensembles, which, as an application of Talagrand inequality gives in particular yet another proof of the convergence of the eigenvalue distribution to the semicircle law.

Limit Theorems for Spectra of Random Matrices with Martingale Structure

We study classical ensembles of real symmetric random matrices introduced by Eugene Wigner. We discuss Stein’s method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure of high dimensional matrices. The method is based on a differential equation for the density of the semicircle law.

An experimental study on Γ(2) modular symmetry in the quantum Hall system with a small spin splitting

Magnetic-field-induced phase transitions were studied with a two-dimensional electronAlGaAs/GaAs system. The temperature-driven flow diagram shows features of theΓ(2) modular symmetry, which includes distorted flowlines and a shifted critical point. Thedeviation of the critical conductivities is attributed to a small but resolved spin splitting,which reduces the symmetry in Landau quantization (Dolan 2000 Phys. Rev. B 62 10278).Universal scaling is found under the reduction of the modular symmetry. It is alsoshown that the Hall conductivity can still be governed by the scaling law whenthe semicircle law and the scaling on the longitudinal conductivity are invalid.

Large Deviations of Extreme Eigenvalues of Random Matrices

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (N x N) random matrix are positive (negative) decreases for large N as approximately exp[-betatheta(0)N2] where the parameter beta characterizes the ensemble and the exponent theta(0)=(ln3)/4=0.274 653... is universal. We also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number zeta, thus generalizing the celebrated Wigner semicircle law. The density of states generically exhibits an inverse square-root singularity at zeta.

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions \((p_1,p_2,\dots)\), with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k th moment of p N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.

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.

On the convergence of the spectral empirical process of Wigner matrices

It is well known that the spectral distribution Fn of a Wigner matrix converges to Wigner's semicircle law. We consider the empirical process indexed by a set of functions analytic on an open domain of the complex plane including the support of the semicircle law. Under fourth-moment conditions, we prove that this empirical process converges to a Gaussian process. Explicit formulae for the mean function and the covariance function of the limit process are provided.

The rate of convergence for spectra of GUE and LUE matrix ensembles

Abstract We obtain optimal bounds of order O(n −1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

Properties of dense partially random graphs

We study the properties of random graphs where for each vertex a neighborhood has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbors or not, as happens in small-world graphs (SWG's). But we consider the case where the average degree of each node is of order of the size of the graph (unlike SWG's, which are sparse). This allows us to calculate the mean distance and clustering, which are qualitatively similar (although not in such a dramatic scale range) to the case of SWG's. We also obtain analytically the distribution of eigenvalues of the corresponding adjacency matrices. This distribution is discrete for large eigenvalues and continuous for small eigenvalues. The continuous part of the distribution follows a semicircle law, whose width is proportional to the "disorder" of the graph, whereas the discrete part is simply a rescaling of the spectrum of the substrate. We apply our results to the calculation of the mixing rate and the synchronizability threshold.

Löwner's Equation from a Noncommutative Probability Perspective

Using concepts of noncommutative probability we show that the Lowner's evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Lowner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Lowner evolution “spreads out” the distribution and that it gives rise to a Markov process.

A study on the universality of the magnetic-field-induced phase transitions in the two-dimensional electron system in an AlGaAs/GaAs heterostructure

Plateau–plateau (P–P) and insulator–quantum Hall conductor (I–QH) transitions are observed in the two-dimensional electron system in an AlGaAs/GaAs heterostructure. At high fields, the critical conductivities are not of the expected universal values and the temperature-dependence of the width of the P–P transition does not follow the universal scaling. However, the semicircle law still holds, and universal scaling behavior was found in the P–P transition after mapping it to the I–QH transition by the Landau-level addition transformation. We pointed out that in order to get a correct critical exponent, it is essential that the scaling analysis must be performed near the critical point. And with proper analysis, we found that the P–P transition and the insulator quantum Hall conductor are of the same universal class.