Out-of-time-ordered correlators for Wigner matrices

We characterize the possible limiting 2nd order distributions of certain independent complex Wigner and deterministic matrices thanks to Voiculescu's notions of operator-valued freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's notion of 2nd order freeness gives a universal rule, in terms of marginal 1st and 2nd order distribution. We adapt and reformulate this notion for operator-valued random variables in a 2nd order probability space. The Wigner matrices are assumed to be permutation invariant with null pseudo variance and the deterministic matrices to satisfy a restrictive property.

In this paper, we shall develop certain inequalities to bound the difference between distributions in terms of their Stieltjes transforms. Using these inequalities, convergence rates of expected spectral distributions of large dimensional Wigner and sample covariance matrices are established. The paper is organized into two parts. This is the first part, which is devoted to establishing the basic inequalities and a convergence rate for Wigner matrices.

Fluctuations de la loi empirique de grandes matrices aléatoires

A Wigner matrix is a symmetric (or Hermitian in the complex case) random matrix. Wigner matrices play an important role in nuclear physics and mathematical physics. The reader is referred to Mehta [212] for applications of Wigner matrices to these areas. Here we mention that they also have a strong statistical meaning. Consider the limit of a normalizedWishart matrix. Suppose that x1, …, x n are iid samples drawn from a p-dimensional multivariate normal population N(μ, I p ). Then, the sample covariance matrix is defined as $$ S_n = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {(x_i - \bar x)} (x_i - \bar x)',$$ where \(\overline{x}=\frac{1}{n}\sum\nolimits_{i=1}^{n}{x_i}.\) When n tends to infinity \( S_n \rightarrow I_p \) and \(\sqrt {n} (S_n - I_p) \rightarrow \sqrt {p} {W_p}\) It can be seen that the entries above the main diagonal of \(\sqrt {p} {W_p}\) are iid N(0, 1) and the entries on the diagonal are iid N(0, 2). This matrix is called the (standard) Gaussian matrix or Wigner matrix.

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, have attracted much attention. The [Formula: see text]th moment of the limit equals the number of non-crossing pair-partitions of the set [Formula: see text]. There are several extensions of this result in the literature. In this paper, we consider a unifying extension which also yields additional results. Suppose [Formula: see text] is an [Formula: see text] symmetric matrix where the entries are independently distributed. We show that under suitable assumptions on the entries, the limiting spectral distribution exists in probability or almost surely. The moments of the limit can be described through a set of partitions which in general is larger than the set of non-crossing pair-partitions. This set gives rise to interesting enumerative combinatorial problems. Several existing limiting spectral distribution results follow from our results. These include results on the standard Wigner matrix, the adjacency matrix of a sparse homogeneous Erdős–Rényi graph, heavy tailed Wigner matrix, some banded Wigner matrices, and Wigner matrices with variance profile. Some new results on these models and their extensions also follow from our main results.

This thesis focuses on several fundamental classes of random matrices with independent entries - Wigner matrices, random band matrices, and adjacency matrices of sparse random graphs. The results fall roughly into two two classes: local laws, which provide strong control of the eigenvalue density down to small spectral scales, and the study of mesoscopic linear eigenvalue statistics.

We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and β-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov–Hiary–Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices.Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.

Following E. Wigner’s original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H+xA with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble.

We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.

The limiting distributions of large heavy Wigner and arbitrary random matrices

Wigner matrices, the moments of roots of Hermite polynomials and the semicircle law

Let (sigma _N^{(i)})_{i in I} be a family of symmetric permutations of the entries of a Wigner matrix {mathbf {W}}_N. We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices ({mathbf {W}}_N^{sigma _N^{(i)}})_{i in I} in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of ({mathbf {W}}_N^{sigma _N^{(i)}})_{i in I}. In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions for traffic independence as well as sufficient conditions for free independence.

Nous étudions l’universalité des statistiques locales du spectre des matrices de Wigner hermitiennes divisibles par une gaussienne. Ces matrices aléatoires sont obtenues en ajoutant à une matrice de Wigner hermitienne avec des coefficients indépendants une matrice du GUE indépendante. Nous montrons que la classe d’universalité de la loi de Tracy–Widom pour les valeurs propres extrêmes est vérifiée sous la condition optimale d’une borne uniforme sur le quatrième moment des coefficients de la matrice. De plus, nous démontrons l’universalité des fluctuations dans l’intérieur du spectre dès lors que le second moment est fini.

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincar\'{e} inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semicircle compact support as the size $N$ becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of $W_N$. On the other hand, when $A_N$ is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of $W_N$.

Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the well-known Dyck paths. This combinatorial expression of the pre-Wigner matrix elements turns out to be particularly simple.