Hermitian Random Matrices Research Articles

Level statistics of the one-dimensional dimerized Hubbard model

The statistical properties of level spacings provide valuable insights into the dynamical properties of a many-body quantum systems. We investigate the level statistics of the Fermi–Hubbard model with dimerized hopping amplitude and find that after taking into account translation, reflection, spin and η pairing symmetries to isolate irreducible blocks of the Hamiltonian, the level spacings in the limit of large system sizes follow the distribution expected for hermitian random matrices from the Gaussian orthogonal ensemble. We show this by analyzing the distribution of the ratios of consecutive level spacings in this system, its cumulative distribution and quantify the deviations of the distributions using their mean, standard deviation and skewness.

On the Fredholm determinant of the confluent hypergeometric kernel with discontinuities

We consider the determinantal point process with the confluent hypergeometric kernel. This process is a universal point process in random matrix theory and describes the distribution of eigenvalues of large random Hermitian matrices near the Fisher-Hartwig singularity. Applying the Riemann-Hilbert method, we study the generating function of this process on any given number of intervals. It can be expressed as the Fredholm determinant of the confluent hypergeometric kernel with n discontinuities. In this paper, we derive an integral representation for the determinant by using the Hamiltonian of the coupled Painlevé V system. By evaluating the total integral of the Hamiltonian, we obtain the asymptotics of the determinant as the n discontinuities tend to infinity up to and including the constant term. Here the constant term is expressed in terms of the Barnes G-function.

Universality of the cokernels of random 𝑝-adic Hermitian matrices

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers O \mathcal {O} of a quadratic extension K K of Q p \mathbb {Q}_p . For each positive integer n n , let X n X_n be a random n × n n \times n Hermitian matrix over O \mathcal {O} whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of X n X_n always converges to the same distribution which does not depend on the choices of X n X_n as n → ∞ n \rightarrow \infty and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [Probability theory for random groups arising in number theory, 2022] in the case of the ring of integers of a quadratic extension of Q p \mathbb {Q}_p .

Spacing ratio statistics of multiplex directed networks

Eigenvalues statistics of various many-body systems have been widely studied using the nearest neighbor spacing distribution under the random matrix theory framework. Here, we numerically analyze eigenvalue ratio statistics of multiplex networks consisting of directed Erdős-Rényi random networks layers represented as, first, weighted non-Hermitian random matrices and then weighted Hermitian random matrices. We report that the multiplexing strength rules the behavior of average spacing ratio statistics for multiplexing networks represented by the non-Hermitian and Hermitian matrices, respectively. Additionally, for both these representations of the directed multiplex networks, the multiplexing strength appears as a guiding parameter for the eigenvector delocalization of the entire system. These results could be important for driving dynamical processes in several real-world multilayer systems, particularly, understanding the significance of multiplexing in comprehending network properties.

The elliptic Ginibre ensemble: A unifying approach to local and global statistics for higher dimensions

The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature β = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions Rd and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions Cd, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For Cd with d = 1, this corresponds to non-interacting fermions in a rotating trap.

Airy point process via supersymmetric lifts

We study the local asymptotics at the edge for particle systems arising from: (i) eigenvalues of sums of unitarily invariant random Hermitian matrices and (ii) signatures corresponding to decompositions of tensor products of representations of the unitary group. Our method treats these two models in parallel, and is based on new formulas for observables described in terms of a special family of lifts, which we call supersymmetric lifts, of Schur functions and multivariate Bessel functions. We obtain explicit expressions for a class of supersymmetric lifts inspired by determinantal formulas for supersymmetric Schur functions due to \cite{MJ03}. Asymptotic analysis of these lifts enable us to probe the edge. We focus on several settings where the Airy point process arises.

Universality for Multiplicative Statistics of Hermitian Random Matrices and the Integro-Differential Painlevé II Equation

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the polynomial potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlevé II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.

Delocalization and Quantum Diffusion of Random Band Matrices in High Dimensions II: T-expansion

We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $n\in \mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Green's function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $d\ge 8$ in part I of this series.

Necessary and sufficient conditions for convergence to the semicircle distribution

We consider random Hermitian matrices with independent upper triangular entries. Wigner’s semicircle law says that under certain additional assumptions, the empirical spectral distribution converges to the semicircle distribution. We characterize convergence to semicircle in terms of the variances of the entries, under natural assumptions such as the Lindeberg condition. The result extends to certain matrices with entries having infinite second moments. As a corollary, another characterization of semicircle convergence is given in terms of convergence in distribution of the row sums to the standard normal distribution.

Universality for Random Matrices with Equi-spaced External Source: A Case Study of a Biorthogonal Ensemble

We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel–Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel–Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.

Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel–Appell polynomials

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

Hard edge statistics of products of Pólya ensembles and shifted GUE’s

Very recently, we have shown how the harmonic analysis approach can be modified to deal with products of general Hermitian and complex random matrices at finite matrix dimension. In the present work, we consider the particular product of a multiplicative Pólya ensemble on the complex square matrices and a Gaussian unitary ensemble (GUE) shifted by a constant multiplicative of the identity. The shift shall show that the limiting hard edge statistics of the product matrix is weakly dependent on the local spectral statistics of the GUE, but depends more on the global statistics via its Stieltjes transform (Green function). Under rather mild conditions for the Pólya ensemble, we prove formulas for the hard edge kernel of the singular value statistics of the Pólya ensemble alone and the product matrix to highlight their very close similarity. Due to these observations, we even propose a conjecture for the hard edge statistics of a multiplicative Pólya ensemble on the complex matrices and a polynomial ensemble on the Hermitian matrices.

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure $\mathfrak{m}$, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with $\mathfrak{m}$ by the Markov--Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of $\mathfrak{m}$. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.

The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe)

We study the genus expansion on compact Riemann surfaces of the gravitational path integral {mathcal{Z}}_{mathrm{grav}}^{(m)} in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models ℳ2m − 1, 2. In the semiclassical limit, corresponding to large m, {mathcal{Z}}_{mathrm{grav}}^{(m)} admits a Euclidean saddle for genus h ≥ 2. Upon fixing the area of the metric, the path integral admits a round two-sphere saddle for h = 0. We show that the OPE coefficients for the minimal weight operators of ℳ2m − 1, 2 grow exponentially in m at large m. Employing the sewing formula, we use these OPE coefficients to obtain the large m limit of the partition function of ℳ2m − 1, 2 for genus h ≥ 2. Combining these results we arrive at a semiclassical expression for {mathcal{Z}}_{mathrm{grav}}^{(m)} . Conjecturally, {mathcal{Z}}_{mathrm{grav}}^{(m)} admits a completion in terms of an integral over large random Hermitian matrices, known as a multicritical matrix integral. This matrix integral is built from an even polynomial potential of order 2m. We obtain explicit expressions for the large m genus expansion of multicritical matrix integrals in the double scaling limit. We compute invariant quantities involving contributions at different genera, both from a matrix as well as a gravity perspective, and establish a link between the two pictures. Inspired by the proposal of Gibbons and Hawking relating the de Sitter entropy to a gravitational path integral, our setup paves a possible path toward a microscopic picture of a two-dimensional de Sitter universe.

Numerical Experimentation to obtain the Laguerre polynomial from average value of all characteristic polynomials of Hermitian Random Matrices

Although the behavior of all subatomic particles is inherently probabilistic, Schrodinger´s equation does not itself contain any probabilities. In this work the Authors reinterprets the Schrodinger Equation; to find in it the randomness that was hidden and that was overlooked by Schrodinger himself. From the generation of Hermitian random matrices and their corresponding characteristic polynomials, the Authors concludes that the radial part solution of the Schrodinger equation for the Hydrogen Atom, namely Laguerre Polynomial, is obtained from the average value of all characteristic polynomials. This is how in this work it is made clear that the deterministic method to obtain the Laguerre Polynomial through the Rodrigues Formula is equivalent to the probabilistic method proposed by the Authors.

Tracy–Widom method for Jánossy density and joint distribution of extremal eigenvalues of random matrices

Abstract The Jánossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The Jánossy density associated with an integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom’s criteria if $\mathbf{K}$ does, because of the structure that the map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy–Widom method [7] to Jánossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlevé equations employed in the preceding works. As illustrative examples we compute Jánossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices.

Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

How much can the eigenvalues of a random Hermitian matrix fluctuate?

The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations—or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in the setting of one-cut regular unitary invariant ensembles of random Hermitian matrices, the Gaussian unitary ensemble (GUE) being the prime example of such an ensemble. Our approach to this question combines extreme value theory of log-correlated stochastic processes, and in particular the theory of multiplicative chaos, with asymptotic analysis of large Hankel determinants with Fisher–Hartwig symbols of various types, such as merging jump singularities, size-dependent impurities, and jump singularities approaching the edge of the spectrum. In addition to optimal rigidity estimates, our approach sheds light on the fractal geometry of the eigenvalue counting function.

Random band matrices in the delocalized phase, III: averaging fluctuations

We consider a general class of symmetric or Hermitian random band matrices $$H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket ^d}$$ in any dimension $$d\ge 1$$ , where the entries are independent, centered random variables with variances $$s_{xy}=\mathbb {E}|h_{xy}|^2$$ . We assume that $$s_{xy}$$ vanishes if $$|x-y|$$ exceeds the band width W, and we are interested in the mesoscopic scale with $$1\ll W\ll N$$ . Define the generalized resolvent of H as $$G(H,Z):=(H - Z)^{-1}$$ , where Z is a deterministic diagonal matrix with entries $$Z_{xx}\in \mathbb {C}_+$$ for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions $$d\ge 2$$ . More precisely, for any fixed $$d\ge 2$$ , we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if $$N\le W^{1+\frac{d}{2}}$$ . This improves the corresponding results in He and Marcozzi (Diffusion profile for random band matrices: a short proof, 2018. arXiv:1804.09446 ) that imposed the assumption $$N\ll W^{1+\frac{d}{d+1}}$$ , and the results in Erdős and Knowles (Ann Henri Poincare12(7):1227–1319, 2011; Commun Math Phys 303(2): 509–554, 2011) that imposed the assumption $$N\ll W^{1+\frac{d}{6}}$$ . For 1D random band matrices, our fluctuation averaging result was used in Bourgade et al. (J Stat Phys 174:1189–1221, 2019; Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality, 2018. arXiv:1807.01559 ) to prove the delocalization conjecture and bulk universality for random band matrices with $$N\ll W^{4/3}$$ .

Spectral statistics of random Toeplitz matrices.

The spectral statistics of Hermitian random Toeplitz matrices with independent and identically distributed elements are investigated numerically. It is found that eigenvalue statistics of complex Toeplitz matrices are surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudointegrable billiards. The origin of intermediate behavior could be attributed to the fact that Fourier transformed random Toeplitz matrices have the same slow decay outside the main diagonal as critical random matrix ensembles. The statistical properties of the full spectrum of real random Toeplitz matrices are close to the Poisson distribution, but each of their constituent subspectra is again well described by the semi-Poisson distribution. The findings indicate that intermediate statistics in general and the semi-Poisson distribution in particular are more universal than considered before.