Semicircle Law Research Articles

Zeros of exceptional Hermite polynomials

We study the zeros of exceptional Hermite polynomials associated with an even partition λ. We prove several conjectures regarding the asymptotic behaviour of both the regular (real) and the exceptional (complex) zeros. The real zeros are distributed as the zeros of usual Hermite polynomials and, after contracting by a factor 2n, we prove that they follow the semi-circle law. The non-real zeros tend to the zeros of the generalized Hermite polynomial Hλ, provided that these zeros are simple. It was conjectured by Veselov that the zeros of generalized Hermite polynomials are always simple, except possibly for the zero at the origin, but this conjecture remains open.

Remarks on absolute continuity in the context of free probability and random matrices

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a probability measure for its free multiplicative convolution with the semicircle law to be absolutely continuous.

Limiting spectral distribution of random birth-death <italic>Q</italic> matrices

This article studies the limiting spectral distributions of random birth-death Q matrices. Under the strictly stationary ergodic condition, we prove that the empirical spectral distribution converges weakly to a non- random probability distribution. Furthermore, in the situations without strictly stationary ergodic condition, we study a class of random birth-death Q matrices, corresponding to generalizations of the Beta-Hermite ensembles, and establish the existences as well as convolution formulations for their limiting spectral distributions. In particular, for the random birth-death Q matrices corresponding to the Beta-Hermite ensembles, the limiting spectral distribution is the convolution of the semi-circle law and Dirac measure δ -2.

Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time.

We consider the system of N one-dimensional free fermions confined by a harmonic well V(x)=mω(2)x(2)/2 at finite inverse temperature β=1/T. The average density of fermions ρ(N)(x,T) at position x is derived. For N≫1 and β∼O(1/N), ρ(N)(x,T) is given by a scaling function interpolating between a Gaussian at high temperature, for β≪1/N, and the Wigner semicircle law at low temperature, for β≫N(-1). In the latter regime, we unveil a scaling limit, for βℏω=bN(-1/3), where the fluctuations close to the edge of the support, at x∼±√[2ℏN/(mω)], are described by a limiting kernel K(b)(ff)(s,s') that depends continuously on b and is a generalization of the Airy kernel, found in the Gaussian unitary ensemble of random matrices. Remarkably, exactly the same kernel K(b)(ff)(s,s') arises in the exact solution of the Kardar-Parisi-Zhang equation in 1+1 dimensions at finite time t, with the correspondence t=b(3).

Semicircle Law for a Matrix Ensemble with Dependent Entries

We study ensembles of random symmetric matrices whose entries exhibit certain correlations. Examples are distributions of Curie–Weiss type. We provide a criterion on the correlations ensuring the validity of Wigner’s semicircle law for the eigenvalue distribution measure. In case of Curie–Weiss distributions, this criterion applies above the critical temperature (i.e., \(\beta \,<\,1\)). We also investigate the largest eigenvalue of certain ensembles of Curie–Weiss type and find a transition in its behavior at the critical temperature.

Limit Theorems for Two Classes of Random Matrices with Dependent Entries

In this paper we study random symmetric matrices with dependent entries. Suppose that all entries have zero mean and finite variances, which can be different. Assuming that the average of normalized sums of variances in each row converges to one and the Lindeberg condition holds true, we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law. The result can be generalized to the class of covariance matrices with dependent entries. In this case expected empirical spectral distribution function converges to the Marchenko--Pastur law.

On the universality of spectral limit for random matrices with martingale differences entries

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko–Pastur law and other convergence results. In the stationary case, we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko–Pastur one. Our results are well adapted to study several examples including nonlinear autoregressive conditional heteroscedastic random fields of infinite order.

Cooperative effects and disorder: A scaling analysis of the spectrum of the effective atomic Hamiltonian

We study numerically the spectrum of the non-Hermitian effective Hamiltonian that describes the dipolar interaction of a gas of $N\ensuremath{\gg}1$ atoms with the radiation field. We analyze the interplay between cooperative effects and disorder for both scalar and vectorial radiation fields. We show that for dense gases, the resonance width distribution follows, both in the scalar and vectorial cases, a power law $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}4/3}$ that originates from cooperative effects between more than two atoms. This power law is different from the $P(\ensuremath{\Gamma})\ensuremath{\sim}{\ensuremath{\Gamma}}^{\ensuremath{-}1}$ behavior, which has been considered as a signature of Anderson localization of light in random systems. We show that in dilute clouds, the center of the energy distribution is described by Wigner's semicircle law in the scalar and vectorial cases. For dense gases, this law is replaced in the vectorial case by the Laplace distribution. Finally, we show that in the scalar case the degree of resonance overlap increases as a power law of the system size for dilute gases, but decays exponentially with the system size for dense clouds.

Limit Theorems for Two Classes of Random Matrices with Gaussian Entries

In this note, we consider ensembles of random symmetric matrices with Gaussian elements. Assume that $$ \mathbb{E} $$ X ij = 0 and $$ \mathbb{E}{X}_{ij}^2={\sigma}_{ij}^2 $$ We do not assume that all the σ ij are equal. Assuming that the average of the normalized sums of variances in each row converges to one and the Lindeberg condition holds, we prove that the empirical spectral distribution of eigenvalues converges to Wigner’s semicircle law. We also provide an analogue of this result for sample covariance matrices and prove the convergence to the Marchenko–Pastur law. Bibliography: 5 titles.

Minimax risk of matrix denoising by singular value thresholding

An unknown $m$ by $n$ matrix $X_0$ is to be estimated from noisy measurements $Y=X_0+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname {min}_X\|Y-X\|_F^2/2+\lambda\|X\|_*$, where $\|X\|_*$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_1$ penalization in the vector case. It has been empirically observed that if $X_0$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$. In a proportional growth framework where the rank $r_n$, number of rows $m_n$ and number of columns $n$ all tend to $\infty$ proportionally to each other ($r_n/m_n\rightarrow \rho$, $m_n/n\rightarrow \beta$), we evaluate the asymptotic minimax MSE $\mathcal {M}(\rho,\beta)=\lim_{m_n,n\rightarrow \infty}\inf_{\lambda}\sup_{\operatorname {rank}(X)\leq r_n}\operatorname {MSE}(X_0,\hat{X}_{\lambda})$. Our formulas involve incomplete moments of the quarter- and semi-circle laws ($\beta=1$, square case) and the Mar\v{c}enko-Pastur law ($\beta<1$, nonsquare case). For finite $m$ and $n$, we show that MSE increases as the nonzero singular values of $X_0$ grow larger. As a result, the finite-$n$ worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal $X_0$ is "infinitely strong." The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of $Y$. We also derive the minimax threshold, namely the value $\lambda^*(\rho)$, which is the optimal place to threshold the singular values. All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegative-definite matrices.

Phase Transition in the Density of States of Quantum Spin Glasses

We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of [6] that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for $p$-uniform hypergraphs that correspond to $p$-spin glass Hamiltonians acting on $n$ distinguishable spin-$1/2$ particles. At the critical threshold $p=n^{1/2}$ we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory.

Bounds for the Stieltjes transform and the density of states of Wigner matrices

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of [22,10,23].

A large deviation principle for Wigner matrices without Gaussian tails

We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha}}$ for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+\alpha /2}$ with a good rate function $J(\mu)$ that is finite only if $\mu$ is of the form $\mu=\mu_{\mathrm{sc}}\boxplus\nu$ for some probability measure $\nu$ on $\mathbb {R}$, where $\boxplus$ denotes the free convolution and $\mu_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(\mu_{\mathrm{sc}}\boxplus\nu)$ in terms of the $\alpha$th moment of $\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.

Modular anomaly equations in N $$ \mathcal{N} $$ =2* theories and their large-N limit

We propose a modular anomaly equation for the prepotential of the N=2* super Yang-Mills theory on R^4 with gauge group U(N) in the presence of an Omega-background. We then study the behaviour of the prepotential in a large-N limit, in which N goes to infinity with the gauge coupling constant kept fixed. In this regime instantons are not suppressed. We focus on two representative choices of gauge theory vacua, where the vacuum expectation values of the scalar fields are distributed either homogeneously or according to the Wigner semi-circle law. In both cases we derive an all-instanton exact formula for the prepotential. As an application, we show that the gauge theory partition function on S^4 at large N localises around a Wigner distribution for the vacuum expectation values leading to a very simple expression in which the instanton contribution becomes independent of the coupling constant.

Random Vector and Matrix Theories: A Renormalization Group Approach

The study of the statistical properties of random matrices of large size has a long history, dating back to Wigner, who suggested using Gaussian ensembles to give a statistical description of the spectrum of complex Hamiltonians and derived the famous semi-circle law, the work of Dyson and many others. Later, ’t Hooft noticed that, in $$SU(N)$$ non-Abelian gauge theories, tessalated surfaces can be associated to Feynman diagrams and that the large $$N$$ expansion corresponds to an expansion in successive topologies. Following this observation, some times later, it was realized that some ensembles of random matrices in the large N expansion and the so-called double scaling limit could be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has resulted in a tremendous expansion of random matrix theory, tackled with increasingly sophisticated mathematical methods and number of matrix models have been solved exactly. However, the somewhat paradoxical situation is that either models can be solved exactly or little can be said. Since the solved models display critical points and universal properties, it is tempting to use renormalization group ideas to determine universal properties, without solving models explicitly. Initiated by Brézin and Zinn-Justin, the approach has led to encouraging results, first for matrix integrals and then quantum mechanics with matrices, but has not yet become a universal tool as initially hoped. In particular, general quantum field theories with matrix fields require more detailed investigations. To better understand some of the encountered difficulties, we first apply analogous ideas to the simpler $$O(N)$$ symmetric vector models, models that can be solved quite generally in the large $$N$$ limit. Unlike other attempts, our method is a close extension of Brézin and Zinn-Justin. Discussing vector and matrix models with similar approximation scheme, we notice that in all cases (vector and matrix integrals, vector and matrix path integrals in the local approximation), at leading order, non-trivial fixed points satisfy the same universal algebraic equation, and this is the main result of this work. However, its precise meaning and role have still to be better understood.

The outliers of a deformed Wigner matrix

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1–47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107–133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.

Random weighted projections, random quadratic forms and random eigenvectors

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of a random $\pm 1$ matrix is of order $O( \sqrt { \log n/n})$; (3) An estimate on the threshold for the local semi-circle law which is tight up to a $\sqrt {\log n}$ factor.

Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

In this paper, we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension n. In these cases, the limit measure is given by the Marchenko–Pastur law and the semicircle law, respectively. For the weighted spectral measure, we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.

Universality of local eigenvalue statistics in random matrices with external source

Consider a random matrix of the form [Formula: see text], where Mn is a Wigner matrix and Dn is a real deterministic diagonal matrix (Dn is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of Wn for a general class of Wigner matrices Mn and diagonal matrices Dn. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

On the limit of extreme eigenvalues of large dimensional random quaternion matrices

Since E.P.Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case.