Bulk Universality Research Articles

Bulk universality for complex non-Hermitian matrices with independent and identically distributed entries

AbstractWe consider $$N\times N$$ N × N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed (iid) with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Almost-Hermitian random matrices and bandlimited point processes

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow “band” about the real line, of height proportional to tfrac{1}{N}, where N denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward’s equation and a property which we call “cross-section convergence”, which relates the large-N limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko–Pastur law for LUE. As an application of our approach, we prove the bulk universality of the almost-circular ensembles.

Quenched universality for deformed Wigner matrices

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.

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.

GOE statistics for Lévy matrices

We establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, N \times N random matrices \mathbf{H} whose upper triangular entries are independent, identically distributed \alpha -stable laws. First, if \alpha\in(1,2) and E\in\mathbb{R} is bounded away from 0, we show that every eigenvector of \mathbf{H} corresponding to an eigenvalue near E is completely delocalized and that the local spectral statistics of \mathbf{H} around E converge to those of the Gaussian Orthogonal Ensemble as N tends to \infty . Second, we show for almost all \alpha\in(0,2) , there exists a constant c(\alpha)>0 such that the same statements hold if |E|<c(\alpha) .

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 $$N\ll W^{4/3}$$ .

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969).

Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point $$x_0$$ , and their connection to the scaling limits of the Christoffel–Darboux kernel at that point. We show that in the case when the orthogonal polynomials are bounded at $$x_0$$ , a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions on the associated measure, bulk universality of the CD kernel is equivalent to the existence of a limit of a particular explicit form.

Determinantal Structure and Bulk Universality of Conditional Overlaps in the Complex Ginibre Ensemble

In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $k\geq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

Random Band Matrices in the Delocalized Phase, II: Generalized Resolvent Estimates

This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\ll W\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\left(z 1_{1\leq i \leq W}+\widetilde z 1_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:{\rm Im} z>0\}$ and $\widetilde z\in \mathbb C_+\cup \mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in \cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in \cite{PartIII}.

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

Using large N arguments, we propose a scheme for calculating the two-point eigenvector correlation function for non-normal random matrices in the large N limit. The setting generalizes the quaternionic extension of free probability to two-point functions. In the particular case of biunitarily invariant random matrices, we obtain a simple, general expression for the two-point eigenvector correlation function, which can be viewed as a further generalization of the single ring theorem. This construction has some striking similarities to the freeness of the second kind known for the Hermitian ensembles in large N. On the basis of several solved examples, we conjecture two kinds of microscopic universality of the eigenvectors — one in the bulk, and one at the rim. The form of the conjectured bulk universality agrees with the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, Phys. Rev. Lett. 81 (1998) 3367] in the case of the complex Ginibre ensemble.

Bulk universality for generalized Wigner matrices with few moments

In this paper we consider $$N \times N$$ real generalized Wigner matrices whose entries are only assumed to have finite $$(2 + \varepsilon )$$ th moment for some fixed, but arbitrarily small, $$\varepsilon > 0$$ . We show that the Stieltjes transforms $$m_N (z)$$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $$\eta = \mathfrak {I}(z)$$ is almost of order $$N^{-1}$$ . As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large N limit.

Fluctuation induced forces in critical films with disorder at their surfaces

We investigate the effect of quenched surface disorder on effective interactions between two planar surfaces immersed in fluids which are near criticality and belong to the Ising bulk universality class. We consider the case that, in the absence of random surface fields, the surfaces of the film belong to the surface universality class of the so-called ordinary transition. We find analytically that in the linear weak-coupling regime, i.e. upon including the mean-field contribution and Gaussian fluctuations, the presence of random surface fields with zero mean leads to an attractive, disorder-induced contribution to the critical Casimir interactions between the two confining surfaces. Our analytical, field-theoretic results are compared with corresponding Monte Carlo simulation data.

Bulk Universality for Random Lozenge Tilings Near Straight Boundaries and for Tensor Products

We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in Gelfand-Tsetlin bases of tensor products of representations of unitary groups. In a weaker form our theorem also applies to random domino tilings.

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Il a été démontré par Akemann, Ipsen et Kieburg que les valeurs singulières carrées de produits de $M$ matrices aléatoires rectangulaires composées de variables aléatoires complexes gaussiennes, suivent un processus ponctuel déterminantal, avec un noyau de corrélation qui admet une représentation en fonction des fonctions G de Meijer. Nous démontrons l’universalité des statistiques locales des valeurs singulières carrées, c’est-à-dire, l’universalité dans le bulk donné par le noyau sinus et l’universalité au bord donné par le noyau Airy. La preuve est basée sur l’analyse asymptotique de la représentation du noyau de corrélation comme une double intégrale de contour. Notre stratégie peut être généralisée à d’autres modèles de produits de matrices aléatoires introduits récemment, afin d’obtenir des résultats d’universalité similaires. Deux autres exemples sont considérés, le premier étant le produit de $M$ matrices de Ginibre et l’inverse de $K$ matrices de Ginibre, étudié par Forrester, le deuxième étant le produit de $M-1$ matrices de Ginibre avec une troncation d’une matrice unitaire, considéré par Kuijlaars et Stivigny.

Universality for general Wigner-type matrices

We consider the local eigenvalue distribution of large self-adjoint Ntimes N random matrices mathbf {H}=mathbf {H}^* with centered independent entries. In contrast to previous works the matrix of variances s_{i j}=mathbbm {E} |h_{i j}|^2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, mathbf {G}(z)=(mathbf {H}-z)^{-1}, converges to a diagonal matrix, mathrm {diag}(mathbf {m}(z)), where mathbf {m}(z)=(m_1(z),dots ,m_N(z)) solves the vector equation -1/m_i(z) = z,+,sum _j s_{i j} m_j(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.

Bulk universality for deformed Wigner matrices

We consider $N\times N$ random matrices of the form $H=W+V$ where $W$ is a real symmetric or complex Hermitian Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are independent of $W$. We assume subexponential decay for the matrix entries of $W$, and we choose $V$ so that the eigenvalues of $W$ and $V$ are typically of the same order. For a large class of diagonal matrices $V$, we show that the local statistics in the bulk of the spectrum are universal in the limit of large $N$.

Local Spectral Statistics of Gaussian Matrices with Correlated Entries.

We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries.

Bulk universality of sparse random matrices

We consider the adjacency matrix of the ensemble of Erdős-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these matrices exhibit bulk universality in the sense that both the averaged n-point correlation functions and distribution of a single eigenvalue gap coincide with those of the GOE. Our methods extend to a class of random matrices which includes sparse ensembles whose entries have different variances.