Ginibre Ensemble Research Articles

On the universality of the non-singularity of general Ginibre and Wigner random matrices

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.

Universality for Products of Random Matrices I: Ginibre and Truncated Unitary Cases

Recently, the joint probability density functions of complex eigenvalues for products of independent complex Ginibre matrices have been explicitly derived as determinantal point processes. We express truncated series coming from the correlation kernels as multivariate integrals with singularity and investigate saddle point method for such a type of integrals. As an application, we prove that the eigenvalue correlation functions have the same scaling limits as those of the single complex Ginibre ensemble, both in the bulk and at the edge of the spectrum. We also prove that the similar results hold true for products of independent truncated unitary matrices.

Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble**Dedicated to Professor R J Baxter on the occasion of his birthday.

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled limit of the annihilation process . Furthermore, deleting each particle at random in the rescaled limit of the coalescence process , a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to , where s denotes the gap size and denotes the Riemann zeta function. It is shown how this can be rigorously established using an asymptotic formula for matrix Fredholm operators. A determinant formula is derived for the gap probability in the finite N case, and this is used to illustrate the asymptotic formulas against numerical computations.

Circular law for random matrices with unconditional log-concave distribution

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

Universal spectral correlations in ensembles of random normal matrices

We consider non-Gaussian ensembles of random normal matrices with the constraint that the ensembles are invariant under unitary transformations. We show that the level density of eigenvalues exhibits disk or ring structure in the complex plane. We also show that the n-eigenvalue correlation and the spacing distribution are universal and identical to that of complex (Gaussian) Ginibre ensemble. Our results are confirmed by Monte Carlo calculations. We verify the universality for dissipative quantum kicked rotor systems.

The spherical ensemble and uniform distribution of points on the sphere

The spherical ensemble is a well-studied determinantal process with a fixed number of points on $\mathbb{S}^2$. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of anarrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).

A rate of convergence for the circular law for the complex Ginibre ensemble

We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For $1 \le p \le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.

From Boltzmann to random matrices and beyond

These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.

A note on mixed matrix moments for the complex Ginibre ensemble

We consider the mixed matrix moments for the complex Ginibre ensemble. These are well-known. We consider the relation to the expected overlap functions of Chalker and Mehlig. This leads to new asymptotic problems for the overlap. We obtain some results, but we also state some remaining open problems.

Determinantal processes and completeness of random exponentials: the critical case

For a locally finite point set \(\Lambda \subset \mathbb {R}\), consider the collection of exponential functions given by \(\mathcal {E}_{\Lambda }:=\{ e^{i \lambda x } : \lambda \in \Lambda \}\). We examine the question whether \(\mathcal {E}_{\Lambda }\) spans the Hilbert space \(L^2[-\pi ,\pi ]\), when \(\Lambda \) is random. For several point processes of interest, this belongs to a certain critical case of the corresponding question for deterministic \(\Lambda \), about which little is known. For \(\Lambda \) the continuum sine kernel process, obtained as the bulk limit of GUE eigenvalues, we establish that \(\mathcal {E}_{\Lambda }\) is indeed complete almost surely. We also answer an analogous question on \(\mathbb {C}\) for the Ginibre ensemble, arising as weak limits of the spectra of certain non-Hermitian Gaussian random matrices. In fact we establish completeness for any “rigid” determinantal point process in a general setting. In addition, we partially answer two questions of Lyons and Steif about stationary determinantal processes on \(\mathbb {Z}^d\).

Permanental processes from products of complex and quaternionic induced Ginibre ensembles

We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalizing the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disk centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disk of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.

Universal distribution of Lyapunov exponents for products of Ginibre matrices

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices, also called the Ginibre ensemble, we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices, the distribution of each Lyapunov exponent is normal, and we compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for 2 × 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. We thereby rederive Newmanʼs triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law, and study the commutativity of the two limits and on the global scale and on the local scale. We show as a mathematical byproduct that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.

Dysonian dynamics of the Ginibre ensemble.

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a nontrivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of a new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the nonanalytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics that we observe for the Ginibre ensemble is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of $n\times n$ matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as $n\rightarrow\infty$. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.

Spectral statistics in directed complex networks and universality of the Ginibre ensemble

Spectra of the adjacency matrices of directed complex networks are analyzed by using non-Hermitian random matrix theory. Both the short-range and long-range correlations in the eigenvalues are calculated numerically for directed model complex networks and real-world networks. The results are compared with predictions of Ginibre’s ensemble. The spectral density ρ(λ), the nearest neighbor spacing distribution p(s) and the level-number variance Σ2(L) show good agreement with Ginibre’s ensemble when the adjacency matrices of directed complex networks are in the strongly non-Hermitian regime.

Universal microscopic correlation functions for products of truncated unitary matrices

We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly distributed on the group U(N + L). Such matrices are called truncated unitary matrices or random contractions. We first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed N, L, and M, given by a standard determinantal point process in the complex plane. The weight however is non-standard and can be expressed in terms of the Meijer G-function. The explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large N (and L) limits at fixed M. At strong non-unitarity, with L/N finite, the eigenvalues condense on a domain inside the unit circle. At the edge and in the bulk we find the same universal microscopic kernel as for a single complex non-Hermitian matrix from the Ginibre ensemble. At the origin we find the same new universality classes labeled by M as for the product of M matrices from the Ginibre ensemble. Keeping a fixed size of truncation, L, when N goes to infinity leads to weak non-unitarity, with most eigenvalues on the unit circle as for unitary matrices. Here we find a new microscopic edge kernel that generalizes the known results for M = 1. We briefly comment on the case when each product matrix results from a truncation of different size Lj.

The Ginibre evolution in the large-N limit

We analyse statistics of the real eigenvalues of gl(N, R)-valued Brownian motion (the Ginibre evolution) in the limit of large N. In particular, we calculate the limiting two-time correlation function of spin variables associated with real eigenvalues of the Ginibre evolution. We also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by Forrester and Nagao [“Eigenvalue statistics of the real Ginibre ensemble,” Phys. Rev. Lett. 99(5), 050603 (2007)] and Borodin and Sinclair [“The Ginibre ensemble of real random matrices and its scaling limits,” Commun. Math. Phys. 291(1), 177–224 (2009)].

Remarks on a Constrained Optimization Problem for the Ginibre Ensemble

We study the limiting distribution of the eigenvalues of the Ginibre ensemble conditioned on the event that a certain proportion lie in a given region of the complex plane. Using an equivalent formulation as an obstacle problem, we describe the optimal distribution and some of its monotonicity properties.

A Note on the Second Order Universality at the Edge of Coulomb Gases on the Plane

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.