We give a proof of universality in the bulk of spectrum of unitary matrix models, assuming that the potential is globally $C^{2}$ and locally $C^{3}$ function. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. We do not use asymptotics of orthogonal polynomials. We obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation.
We consider the deformed Gaussian Ensemble $H_n=M_n+H^{(0)}_n$ in which $H_n^{(0)}$ is a diagonal Hermitian matrix and $M_n$ is the Gaussian Unitary Ensemble (GUE) random matrix. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ (both non-random and random) converges weakly to a measure $N^{(0)}$ with a bounded support we prove universality of the local eigenvalue statistics in the bulk of the limiting spectrum of $H_n$.
We consider the deformed Gaussian ensemble $H_n=H_n^{(0)}+M_n$ in which $H_n^{(0)}$ is a hermitian matrix (possibly random) and $M_n$ is the Gaussian unitary random matrix (GUE) independent of $H_n^{(0)}$. Assuming that the Normalized Counting Measure of $H_n^{(0)}$ converges weakly (in probability if random) to a non-random measure $N^{(0)}$ with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of $H_n$.
Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.
The unitary evolution maps in closed chaotic quantum graphs are known to have universal spectral correlations, as predicted by random matrix theory. In chaotic graphs with absorption the quantum maps become non-unitary. We show that their spectral statistics exhibit universality at the "soft" edges of the spectrum. The same spectral behavior is observed in many classical non-unitary ensembles of random matrices with rotationally invariant measures.
This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich matrix model ("matrix Airy function", as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral ("matrix Bessel function") known in the literature as generalized Kontsevich matrix model.
In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $θ=π$ for a specific critical evolution time $t_c$.
We consider the deformed Laguerre Ensemble $H_n=\dfrac{1}{m}Σ_n^{1/2}A_{m,n}A_{m,n}^*Σ_n^{1/2}$ in which $Σ_n$ is a positive hermitian matrix (possibly random) and $A_{m,n}$ is a $n\times m$ complex Gaussian random matrix (independent of $Σ_n$), $\dfrac{m}{n}\to c>1$. Assuming that the Normalized Counting Measure of $Σ_n$ converges weakly (in probability) to a non-random measure $N^{(0)}$ with a bounded support we prove the universality of the local eigenvalue statistics in the bulk of the limiting spectrum of $H_n$.
The eigenvalue correlations of random matrices from the Jacobi Unitary Ensemble have a known asymptotic behavior as their size tends to infinity. In the bulk of the spectrum the behavior is described in terms of the sine kernel, and at the edge in terms of the Bessel kernel. We will prove that this behavior persists for the Modified Jacobi Unitary Ensemble. This generalization of the Jacobi Unitary Ensemble is associated with the modified Jacobi weight w(x)=(1-x)^α(1+x)^βh(x) where the extra factor h is assumed to be real analytic and strictly positive on [-1,1]. We use the connection with the…
We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give …
We prove the bulk universality of the $β$-ensembles with non-convex regular analytic potentials for any $β>0$. This removes the convexity assumption appeared in our earlier work. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a "convexified measure" so that the local statistics are preserved under this convexification.
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 h…
We prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum. Our starting point is the result of \cite{S:08} on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble.
Unitary random matrix ensembles Z_{n,N}^{-1} (\det M)^alpha exp(-N Tr V(M)) dM defined on positive definite matrices M, where alpha > -1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N tend to infinity such that n/N - 1 = O(n^{-2/3}). For each value of alpha > -1 we find a one-parameter family of limiting kernels …
We prove universality at the edge of the spectrum for unitary (beta=2), orthogonal (beta=1) and symplectic (beta=4) ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2, 1.3 below. For a proof of universality in the bulk of the spectrum, for the same class of weights, for unitary ensembles see [DKMVZ2], and for orthogonal and symplectic ensembles see [DG]. Our starting point in the unitary case is [DKMVZ2], and fo…
The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy--Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of th…
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 $pN \gg 1$ 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.
We consider $β$ matrix models with real analytic potentials. Assuming that the corresponding equilibrium density $ρ$ has a one-interval support (without loss of generality $σ=[-2,2]$), we study the transformation of the correlation functions after the change of variables $λ_i\toζ(λ_i)$ with $ζ(λ)$ chosen from the equation $ζ'(λ)ρ(ζ(λ))=ρ_{sc}(λ)$, where $ρ_{sc}(λ)$ is the standard semicircle density. This gives us the "deformed" $β$-model which has an additional "interaction" term. Standard transformation with the Gaussian integral allows us to show that the "deformed" $β$-model may be reduced…
We consider the non-Hermitian analogue of the celebrated Wigner-Dyson-Mehta bulk universality phenomenon, i.e. that in the bulk the local eigenvalue statistics of a large random matrix with independent, identically distributed centred entries are universal, in particular they asymptotically coincide with those of the Ginibre ensemble in the corresponding symmetry class. In this paper we reduce this problem to understanding a certain microscopic regime for the Hermitized resolvent in Girko's formula by showing that all other regimes are negligible.
We derive bulk asymptotics of skew-orthogonal polynomials (sop) $π^{\bt}_{m}$, $β=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there exists an $ε> 0$, such that $ε\leq (m/N)\leq λ_{\rm cr}-ε$, where $λ_{\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptoti…
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 $η= \Im (z)$ 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…
We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary, orthogonal, and symplectic ensembles. The Riemann-Hilbert problem for orthogonal polynomials is one possible tool to derive universality in unitary random matrix ensembles. An overview is presented of the Deift/Zhou steepest descent analysis of the Riemann-Hilbert problem in the one-cut regular case. Non-standard universality classes that arise at singular point…
In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close to the center of the spectrum does not depend on the probability distribution of the entries of the matrix and is the same as for the Gaussian Orthogonal Ensemble. We make only mild smoothness assumptions on the probability distribution of the entries and assume that the probability distribution of the entries decays polynomially with sufficiently large pow…
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.
Basing on our recent results on the $1/n$-expansion in unitary invariant random matrix ensembles, known as matrix models, we prove that the local eigenvalue statistic, arising in a certain neighborhood of the edges of the support of the Density of States, is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of real analytic potentials and of supports, consisting of one or two disjoint intervals.
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of…
We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such weights the associated equilibrium measure is supported on a single interval. The precise statement of our results is given in Theorem 1.1 below. For a proof of the Universality Conjecture for unitary ensembles, for the same class of weights, see [DKMVZ2]. Our starting point is Widom's representation [W] of the orthogonal and symplectic correlation kernels …
We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also universality of local eigenvalue statistics in the bulk.
We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.
Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $ν_{ij}$ with a subexponential decay. Let $σ_{ij}^2$ be the variance for the probability measure $ν_{ij}$ with the normalization property that $\sum_{i} σ^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N σ_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal en…
We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Ampère operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and app…
The distributions of $ N $-particle systems of Gaussian unitary ensembles converge to Sine$_2$ point processes under bulk-scaling limits. These scalings are parameterized by a macro-position $ θ$ in the support of the semicircle distribution. The limits are always Sine$_{2}$ point processes and independent of the macro-position $ θ$ up to the dilations of determinantal kernels. We prove a dynamical counter part of this fact. We prove that the solution of the $ N $-particle systems given by stochastic differential equations (SDEs) converges to the solution of the infinite-dimensional Dyson mode…
We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. These conditions are verified, hence bulk spectral universality is proven, for a large class of Wigner-like matrices, including deformed Wigner ensembles and ensembles with non-stochastic variance matrices whose limiting densities differ from the Wigner semicircle law.
The winding number is a concept in complex analysis which has, in the presence of chiral symmetry, a physics interpretation as the topological index belonging to gapped phases of fermions. We study statistical properties of this topological quantity. To this end, we set up a random matrix model for a chiral unitary system with a parametric dependence. We analytically calculate the discrete probability distribution of the winding numbers, as well as the parametric correlations functions of the winding number density. Moreover, we address aspects of universality for the two-point function of the…
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the Christoffel-Darboux kernel holds for any point where the imaginary part of the m-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel-Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding m-function has normal limits at th…
We consider the ensemble of $n \times n$ Wigner hermitian matrices $H = (h_{\ell k})_{1 \leq \ell,k \leq n}$ that generalize the Gaussian unitary ensemble (GUE). The matrix elements $h_{k\ell} = \bar h_{\ell k}$ are given by $h_{\ell k} = n^{-1/2} (x_{\ell k} + \sqrt{-1} y_{\ell k})$, where $x_{\ell k}, y_{\ell k}$ for $1 \leq \ell < k \leq n$ are i.i.d. random variables with mean zero and variance 1/2, $y_{\ell\ell}=0$ and $x_{\ell \ell}$ have mean zero and variance 1. We assume the distribution of $x_{\ell k}, y_{\ell k}$ to have subexponential decay. In a recent paper, four of the authors r…
We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for…
We give a streamlined proof of a quantitative version of a result from [DG1] which is crucial for the proof of universality in the bulk [DG1] and also at the edge [DG2] for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the beta=1,2,4 partition functions for log gases.
We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.
We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications …
It is well-known that the partition function of the unitary ensembles of random matrices is given by a tau-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are tau-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy …
This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever a boundary of the domain has three adjacent straight segments inclined under 120 degrees to each other, the asymptotics of tilings near the middle segment is described by the GUE--corners process of random matrix theory. An important step in our argument is to show that fluctuations of the height function of random tilings on essentially arbitrary simply-connected domains of diameter $N$ have smaller magnitude than $N^{1/2}$.
We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random matrix ensembles with their counterparts for the Korteweg-de Vries equation, emphasizing the similarities between both subjects.
The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the $k$th and $(k+1)$st smallest eigenvalues in the chiral Gaussia…
We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble ${\rm CUE}(N)$ and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, $N\rightarrow\infty$, we derive a ${\it\, concise\,}$ parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the ${\rm Sine}_2$ determinantal random point field. Further, we discuss a universality of the predicted power spectrum law an…
