Wigner Matrices Research Articles

Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime

Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime

Independence of linear spectral statistics and the point process at the edge of Wigner matrices

Let [Formula: see text] be a Wigner matrix of dimension [Formula: see text] with eigenvalues [Formula: see text] and [Formula: see text] be an analytic function on [Formula: see text] with polynomial growth. It is known that Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] converges in distribution to a normal random variable with mean [Formula: see text] and a finite variance depending on [Formula: see text]. On the other hand, it is also known that [Formula: see text] converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that [Formula: see text] and Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300 ]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

Norm convergence rate for multivariate quadratic polynomials of Wigner matrices

We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root growth at its edges and prove an optimal local law around these edges. Combining these two results, we establish that, as the dimension N of the matrices grows to infinity, the operator norm of such polynomials q converges to a deterministic limit with a rate of convergence of N−2/3+o(1). Here, the exponent in the rate of convergence is optimal. For the specific reducible cases, we also provide a classification of all possible edge behaviors.

Spectral large deviations of sparse random matrices

AbstractEigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean . Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph equipped with independent and identically distributed (i.i.d.) edge‐weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case , that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph without edge‐weights and with Gaussian edge‐weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate with , where the regimes and correspond to tails heavier and lighter than the Gaussian tail, respectively. While in many natural settings the large deviations behavior is expected to depend crucially on the entry distribution, we establish a surprising and rare universal behavior showing that this is not the case when . In contrast, in the case, the large deviation rate function is no longer universal and is given by the solution to a variational problem, the description of which involves a generalization of the Motzkin–Straus theorem, a classical result from spectral graph theory. As a byproduct of our large deviation results, we also establish the law of large numbers behavior for the largest eigenvalue, which also seems to be new and difficult to obtain using existing methods. In particular, we show that the typical value of the largest eigenvalue exhibits a phase transition at , that is, corresponding to the Gaussian distribution.

Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.

Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to N, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in [9]. More precisely given P a self-adjoint non-commutative polynomial and YN a d-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator P(YN) yields asymptotic freeness for large times.

Optimal lower bound on eigenvector overlaps for non-Hermitian random matrices

We consider large non-Hermitian N×N matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements. We show that already a small noise of variance 1/N completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence. In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by Deutsch [34] and proven for Wigner matrices in [23], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.

Photon quantum kinetic equations and collective modes in an axion background

We develop a quantum kinetic theory for photons in the presence of an axion background and in the collisioness limit. In deriving the classical regime of our quantum kinetic equations, we observe that they capture well-known features of axion electrodynamics. By projecting the Wigner function onto a polarization basis, relating the Wigner matrix function with the Stokes parameters, we establish the dispersion relations and transport equations for each polarization space component. Additionally, we investigate how the axion background affects the dispersion relations of photon collective modes within an electron-positron plasma at equilibrium temperature T. While the plasmon remains unaffected, we find that the axion background breaks the degeneracy of transverse collective modes at order egaγT(∂a), where e represents the electron charge, gaγ denotes the photon-axion coupling, and ∂a represents the scale associated with variations in the axion field. Published by the American Physical Society 2024

A spectral based goodness-of-fit test for stochastic block models

Community detection is a fundamental problem in complex network data analysis. Though many methods have been proposed, most existing methods require the number of communities to be the known parameter, which is not in practice. In this paper, we propose a novel goodness-of-fit test for the stochastic block model. The test statistic is based on the linear spectral of the adjacency matrix. Under the null hypothesis, we prove that the linear spectral statistic converges in distribution to N(0,1). The proof uses some recent results in generalized Wigner matrices to prove the main theorem. Numerical experiments and real world data examples illustrate that our proposed linear spectral statistic has good performance.

Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices

Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices

Eigenstate thermalization hypothesis for generalized Wigner matrices

Eigenstate thermalization hypothesis for generalized Wigner matrices

Out-of-time-ordered correlators for Wigner matrices

Out-of-time-ordered correlators for Wigner matrices

Quantitative CLT for linear eigenvalue statistics of Wigner matrices

Quantitative CLT for linear eigenvalue statistics of Wigner matrices

Finite rank perturbations of heavy-tailed Wigner matrices

One-rank perturbations of Wigner matrices have been closely studied: let [Formula: see text] with [Formula: see text] symmetric, [Formula: see text] i.i.d. with centered standard normal distributions, and [Formula: see text] It is well known [Formula: see text] the largest eigenvalue of [Formula: see text] has a phase transition at [Formula: see text]: when [Formula: see text] [Formula: see text] whereas for [Formula: see text] [Formula: see text] Under more general conditions, the limiting behavior of [Formula: see text] appropriately normalized, has also been established: it is normal if [Formula: see text] or the convolution of the law of [Formula: see text] and a Gaussian distribution if [Formula: see text] is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions [Formula: see text] heavy-tailed with index [Formula: see text] the fluctuations are shown to be universal and dependent on [Formula: see text] but not on [Formula: see text] whereas a subfamily of the edge case [Formula: see text] displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether [Formula: see text] is localized, each presenting a continuous phase transition at [Formula: see text] respectively. These results build on our previous work which analyzes the asymptotic behavior of [Formula: see text] in the aforementioned subfamily.

Observation of a Superradiant Phase Transition with Emergent Cat States.

Superradiant phase transitions (SPTs) are important for understanding light-matter interactions at the quantum level, and play a central role in criticality-enhanced quantum sensing. So far, SPTs have been observed in driven-dissipative systems, but the emergent light fields did not show any nonclassical characteristic due to the presence of strong dissipation. Here we report an experimental demonstration of the SPT featuring the emergence of a highly nonclassical photonic field, realized with a resonator coupled to a superconducting qubit, implementing the quantum Rabi model. We fully characterize the light-matter state by Wigner matrix tomography. The measured matrix elements exhibit quantum interference intrinsic of a photonic mesoscopic superposition, and reveal light-matter entanglement.

Geometric spin Hall effect and polarization-dependent transformations in the oblique section of a paraxial light beam

The spatial structure of light beams is usually considered in the transverse cross sections but supplementary analysis of the field pattern across an oblique plane may disclose additional details of the internal beam structure and energy flow distributions. Their manifestations are known as ‘geometric spin Hall effect of light’ (gSHEL). We analyze the ‘practical’ gSHEL scheme in which the light energy distribution is registered by a detector whose input plane is inclined with respect to the propagation axis. Based on the vector beam model and using the formalism of optical Wigner matrices, we find that the oblique-plane energy distribution differs from that observed in the transverse cross section. This difference is associated with the azimuthal energy circulation and the orbital angular momentum (AM) of the beam; it can be expressed as the lateral shift of the mean-weighted beam position (beam centroid). The similar effect can be observed in elliptically polarized beams without orbital AM: there, the oblique-section projection reveals a specific asymmetry induced by the spin AM in the longitudinal field components of such beams. The polarization-induced oblique-section beam shift is rather weak in paraxial approximation but can be observable if the light-detecting procedure is selectively sensitive to the longitudinal optical-field component.

On the concavity of the TAP free energy in the SK model

We analyze the Hessian of the Thouless–Anderson–Palmer (TAP) free energy for the Sherrington–Kirkpatrick model, below the de Almeida–Thouless line, evaluated in Bolthausen’s approximate solutions of the TAP equations. We show that the empirical spectral distribution weakly converges to a measure with negative support below the AT line, and that the support includes zero on the AT line. In this “macroscopic” sense, we show that TAP free energy is concave in the order parameter of the theory, i.e. the random spin-magnetizations. This proves a spectral interpretation of the AT line. We also find different magnetizations than Bolthausen’s approximate solutions at which the Hessian of the TAP free energy has positive outlier eigenvalues. In particular, when the magnetizations are assumed to be independent of the disorder, we prove that Plefka’s second condition is equivalent to all eigenvalues being negative. On this occasion, we extend the convergence result of Capitaine et al. (Electron. J. Probab. 16, no. 64, 2011) for the largest eigenvalue of perturbed complex Wigner matrices to the GOE.

Geometric deep learning and equivariant neural networks

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds mathcal {M} using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces mathcal {M}=G/K, which are instead equivariant with respect to the global symmetry G on mathcal {M}. Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case mathcal {M}=S^2=textrm{SO}(3)/textrm{SO}(2). Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G=textrm{SO}(3), illustrating the power of representation theory for deep learning.

Low-temperature dynamics for confined $$p=2$$ soft spin in the quenched regime

This paper aims to address the low-temperature dynamics issue for the $p=2$ spin dynamics with confining potential, focusing especially on quartic and sextic cases. The dynamics are described by a Langevin equation for a real vector $q_i$ of size $N$, where disorder is materialized by a Wigner matrix and we especially investigate the self consistent evolution equation for effective potential arising from self averaging of the square length $a(t)\equiv \sum_i q_i^2(t)/N$ for large $N$. We first focus on the static case, assuming the system reached some equilibrium point, and we then investigate the way the system reach this point dynamically. This allows to identify a critical temperature, above which the relaxation toward equilibrium follows an exponential law but below which it has infinite time life and corresponds to a power law decay.

The Effect of Anisotropic Gaussian Schell-Model Sources in Generalized Phase Space Stokes Parameters

Phase-space transforms describe spatial and angular information about light sources where one example is the Wigner functions in wave optics. Stokes parameters, on the other hand, supply information about the polarization of light beams. The generalized phase space Stokes parameters of 2D stochastic electromagnetic beams are already developed. In this article, the application of anisotropic light sources in generalized phase space Stokes parameters is theoretically investigated and numerically analyzed. There are several different ways of studying electromagnetic light beams depending on the spatial domain. But, most measure of the polarization of random light fields is carried out within the Stokes parameters context. In this account we study the electromagnetism, Stokes parameters, phase space, and the anisotropy properties of random light beams at once. We find here that when an anisotropy introduced in phase space then the cross terms of the Wigner matrix depart from the diagonal terms, which is not the same in configuration space. As a result, anisotropy has a different effect in Phase space, i.e. an anisotropic source introduces a phase and a variance change only in the cross terms of Wigner matrix. This is due to the use of anisotropy in the shifted kernel of Wigner transform.