H. Kösters
We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble essentially continues to hold for a general real symmetric Wigner matrix.
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We study the fluctuations associated to the a.s. convergence, established by Belinschi-Bercovici-Capitaine, of the outliers of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the non universality phenomenon previously established for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.
On the one hand, we prove that almost surely, for large dimension, there is no eigenvalue of a Hermitian polynomial in independent Wigner and deterministic matrices, in any interval lying at some distance from the supports of a sequence of deterministic probability measures, which is computed with the tools of free probability. On the other hand, we establish the strong asymptotic freeness of independent Wigner matrices and any family of deterministic matrices with strong limiting distribution.
We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions.
We prove that any non commutative polynomial of r independent copies of Wigner matrices converges a.s. towards the polynomial of r free semicircular variables in operator norm. This result extends a previous work of Haagerup and Thorbjornsen where GUE matrices are considered, as well as the classical asymptotic freeness for Wigner matrices (i.e. convergence of the moments) proved by Dykema. We also study the Wishart case.
We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised by Tao and Vu.
On the second moment of the determinant of random symmetric, Wigner, and Hermitian matrices
本文利用解析组合学技术研究随机 Hermitian 矩阵行列式的二阶矩，并将对称矩阵与 Wigner 矩阵的已有结果统一推广。
We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.
We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function of the characteristic polynomial.
The log-Characteristic Polynomial of Generalized Wigner Matrices is Log-Correlated
本研究证明大维广义Wigner矩阵特征多项式的对数收敛于对数相关场，通过扩展已有三步论证方法，结合多类已有结果，还建立了谱边缘的Wegner估计与微观局部律。
We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by $ 2 \*σ+ o(N^{-6/11+ε}), $ where $σ^2 $ is the variance of the matrix entries and $ε$ is an arbitrary small positive number. Our bound improves the earlier results by Z.Füredi and J.Komlós (1981), and the recent bound obtained by Van Vu (2005).
Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices
本文首次在首阶意义上确定了Wigner矩阵和β-系综特征多项式的最大值，并在Gaussian-divisible情形下证明其紧性普适性，从而回答了谱的最优刚性问题，同时回答了Fyodorov--Hiary--Keating猜想。
In the present paper we give two alternate proofs of the well known theorem that the empirical distribution of the appropriately normalized roots of the $n^{th}$ monic Hermite polynomial $H_n$ converges weakly to the semicircle law, which is also the weak limit of the empirical distribution of appropriately normalized eigenvalues of a Wigner matrix. In the first proof -- based on the recursion satisfied by the Hermite polynomials -- we show that the generating function of the moments of roots of $H_n$ is convergent and it satisfies a fixed point equation, which is also satisfied by $c(z^2)$, w…
It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions a…
Asymptotic freeness through unitaries generated by polynomials of Wigner matrices
本文研究确定性矩阵与独立Wigner矩阵自伴多项式处函数乘积的确定性近似与波动控制，并证明由Wigner矩阵多项式生成的量子演化在长时间下产生渐近自由性。
For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form "no eigenvalues outside the support of the limiting eigenvalue distribution." We build on ideas of Haagerup-Schultz-Thorbjørnsen on the one hand and Bai-Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Ins…
In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semic…
We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two cha…
Limiting Spectral Distributions (LSD) of real symmetric patterned matrices have been well-studied. In this article, we consider skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs of several common matrices. For the skew-symmetric Wigner, skew-symmetric Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are the same as those in the symmetric cases. For the skew-symmetric Hankel and the skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the existence of the LSDs for the triangular versions of these matrices. We then int…
In this short note we derive concentration inequalities for the empirical absolute moments of square symmetric matrices with independent symmetrically distributed +/-1 entries. Most of the previous results of this type are limited to functions with bounded Lipschitz constant, and therefore exclude the moments from consideration. Based on the fine analysis of the distribution of the largest eigenvalues developed by Soshnikov and Sinai, we extend the measure concentration tools of Talagrand to encompass power functions.
We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $β$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^α}$, for some constant $c>0$ and with $α\in (0,2)$, the case of Gaussian Wigner matrices, and the case of $β$-ensembles associated with a convex potential with polynomial growth.
We consider $N\times N$ random matrices of the form $H = W + V$ where $W$ is a real symmetric Wigner matrix and $V$ 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 rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution $F_1$ in the limit of large $N$. Our proofs also apply to the complex Hermitian setting, i.e., …
We consider large-dimensional Hermitian or symmetric random matrices of the form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a real diagonal matrix whose entries are independent of $M$. For a large class of diagonal matrices $V$, we prove that the fluctuations of linear spectral statistics of $W$ for $C^{2}_{c}$ test function can be decomposed into that of $M$ and of $V$, and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.
Norm Convergence Rate for Multivariate Quadratic Polynomials of Wigner Matrices
本文研究多个独立Wigner矩阵的Hermitian非交换二次多项式，证明除可约情形外其极限谱密度边缘有平方根增长，并建立最优局部律，从而得到算子范数以$N^{-2/3+o(1)}$速率收敛到确定性极限，指数最优，并分类了可约情形的边缘行为。
We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these class…
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of $X_N$ for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on the characteristic function method in [47], local laws for the Green function of $X_N$ in [3, 46, 51] and analytic subordination properties of the…
The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, has attracted much attention. The $2k$th moment of the limit equals the number of non-crossing pair-partitions of the set $\{1, 2, \ldots, 2k\}$. There are several extensions of this result in the literature. In this paper we consider a unifying extension which also yields additional results. Suppose $W_n$ is an $n\times n$ symmetric matrix where the entries are independently distributed. We show that under suitable assumpti…
A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived and the eigenvalues are localised. Three instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product $HX$ of a fixed diagonal matrix $H$ and the Wigner matrix $X$ and a special matrix polynomial. The results are illustrated with various examples and numerical simulations.
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there are some operations which to date have no RMT analogue. Our motivation is to find an RMT analogue of Rankin-Selberg convolution, which constructs a new $L$-functions from an input pair. We report one such attempt; while it does not appear to model convolution, it does create new ensembles with properties hybridizing those of its constituents. For definiteness …
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…
This paper is a continuation of our paper "Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices", J. Stat. Phys. (134), 147--159 (2009), in which we proved the Central Limit Theorem for the matrix elements of differential functions of the real symmetric random Gaussian matrices (GOE). Here we consider the real symmetric random Wigner matrices having independent (modulo symmetry conditions) but not necessarily Gaussian entries. We show that in this case the matrix elements of sufficiently smooth functions of these random matrices have in general another limiting law …
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 $N$.
We prove a CLT for spectra of submatrices of real symmetric and Hermitian Wigner matrices. We show that if in the standard normalization the fourth moment of the off-digonal entries is GOE/GUE-like then the limiting Gaussian process can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields.
We consider an ensemble of nxn real symmetric random matrices A whose entries are determined by independent identically distributed random variables that have symmetric probability distribution. Assuming that the moment 12+2delta of these random variables exists, we prove that the probability distribution of the spectral norm of A rescaled to n^{-2/3} is bounded by a universal expression. The proof is based on the completed and modified version of the approach proposed and developed by Ya. Sinai and A. Soshnikov to study high moments of Wigner random matrices.
We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit $n\to +\infty$. As a corollary, we show that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases.
We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.
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.
Testing for latent structure via the Wilcoxon--Wigner random matrix of normalized rank statistics
本文提出基于Wilcoxon-Wigner随机矩阵的统计检验方法，用于大规模数据中潜在结构的检测，具有计算高效和鲁棒性强的特点。
Fluctuation Moments for Regular Functions of Wigner Matrices
该研究计算Wigner矩阵一般Sobolev函数与确定性矩阵乘积的混合涨落矩的确定性近似，重现已有多项式情形结果并验证相关组合结构的普适性，还刻画了泛函中心极限定理的方差。
The "visible" Wigner matrix
该研究提出“可见”Wigner矩阵（i,j互素时元素为零），利用初等数论互素模式结果证明其极限谱分布存在，给出矩的素数无穷乘积显式表达，还研究了互补的“不可见”Wigner矩阵。
Beyond normal fluctuations in local laws for Wigner matrices
该研究针对Wigner矩阵的经验特征值分布Stieltjes变换，确定其全局与局部律中心极限定理的非高斯修正，发现实情形收敛速率慢于复情形，通过改进zigzag策略分析新扩展累积量层级完成证明。
In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider $N \times N$ symmetric Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, whose upper right entries $x_{ij}$ $(1\le i< j\le N)$ are $i.i.d.$ random variables with distribution $μ$ and diagonal entries $x_{ii}$ $(1\le i\le N)$ are $i.i.d.$ random variables with distribution $\wt μ$. The means of $μ$ and $\wt μ$ are zero, the variance of $μ$ is 1, and the variance of $\wt μ$ is finite. We prove that Tracy-Widom law holds if and only if $\lim_{s\to \infty}s^4\p(|x_{12}| \ge s)=0$. The s…
We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices $W$ and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem [Voiculescu 1991] from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to $\exp(\mathrm{i} tW)$ for large $t$, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalis…
In the current paper we consider a Wigner matrix and consider an analytic function of polynomial growth on a set containing the support of the semicircular law in its interior. We prove that the linear spectral statistics corresponding to the function and the point process at the edge of the Wigner matrix are asymptotically independent when the entries of the Wigner matrix are sub-Gaussian. The main ingredient of the proof is based on a recent paper by Banerjee [6]. 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.
We study the spectral properties of the dilute Wigner random real symmetric n-dimensional matrices H such that the entries H(i,j) take zero value with probability 1-p/n. We prove that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the dilute Wigner ensemble in the limit of infinite n and p. In the second part of the paper we study the leading term of the correlation function of the resolvent G(z) of H with large enough Im z in the limit of infinite n and p such that 3/5 log n <log p < log n. We show that this leading term, when conside…
We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the a…
We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size $N$ converge to the Tracy--Widom laws at a rate $O(N^{-1/3+ω})$, as $N$ tends to infinity. For Wigner matrices this improves the previous rate $O(N^{-2/9+ω})$ obtained by Bourgade [5] for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős, Yau and Yin [19] to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow indu…
We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number $k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we show that $x_k$ is normally distributed in the limit. We also consider the joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar conditions on the indices. The result is an $m$-dimensional normal dist…
We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers (analogous to the recently proven local semicircle law for Wigner matrices) under fairly weak conditions on the underlying measure $μ$. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.
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