On the Gaussian Random Matrix Ensembles with Additional Symmetry Conditions

On Universality of Bulk Local Regime of the Deformed Gaussian Unitary Ensemble

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 $H_n$.

On Resolvent Identities in Gaussian Ensembles at the Edge of the Spectrum

We obtain the recursive identities for the joint moments of the traces of the powers of the resolvent for Gaussian ensembles of random matrices at the soft and hard edges of the spectrum. We also discuss the possible ways to extend these results to the non-Gaussian case.

Spectral fluctuation properties of constrained unitary ensembles of Gaussian-distributed random matrices

We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as N(Q) is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary ensemble of random matrices may be destroyed) all spectral fluctuation measures have the same form as for the Gaussian unitary ensemble.

Finite n Largest Eigenvalue Probability Distribution Function of Gaussian Ensembles

In this paper we focus on the finite n probability distribution function of the largest eigenvalue in the classical Gaussian Ensemble of n by n matrices (GEn). We derive the finite n largest eigenvalue probability distribution function for the Gaussian Orthogonal and Symplectic Ensembles and also prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of Gaussian Symplectic Ensemble. The correction terms to the limiting probability distribution are expressed in terms of the same Painleve II functions appearing in the Tracy-Widom distribution.

Level density for deformations of the Gaussian orthogonal ensemble

Formulas are derived for the average level density of deformed, or transition, Gaussian orthogonal random matrix ensembles. After some general considerations about Gaussian ensembles we derive formulas for the average level density for (i) the transition from the Gaussian orthogonal ensemble (GOE) to the Poisson ensemble and (ii) the transition from the GOE to $m$ GOEs.

Matrix Kernels for the Gaussian Orthogonal and Symplectic Ensembles

We derive the limiting matrix kernels for the the Gaussian Orthogonal and Symplectic ensembles scaled at the edge, with proofs of convergence in the operator norms that assure convergence of the determinants.

Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials …

Statistical properties of eigenvalues of an ensemble of pseudo-Hermitian Gaussian matrices

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.

Correlations of the characteristic polynomials in the Gaussian Unitary Ensemble or a singular Hankel determinant

We obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the Gaussian Unitary Ensemble of n\times n matrices. Our results can also be interpreted as asymptotics of the determinant of a Hankel matrix whose symbol is supported on the real line and possesses power-like (Fisher-Hartwig) singularities.

On the similarity between Nakagami-m Fading distribution and the Gaussian ensembles of random matrix theory

We report the similarity between the Nakagami-m fading distribution and the three Gaussian ensembles of random matrix theory. We provide a brief review of random matrix theory and wireless fading. We show that the Nakagami-m distribution serves as mapping between the three ensembles. The statistics of the wireless fading amplitude, as modeled by Nakagami-m distribution, provide a rare example of a classically chaotic system that exhibits a transition between the Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic ensembles of random matrix theory.

Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble

We study the level statistics for two classes of 1-dimensional random Schrödinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result \cite{KN} imply the coincidence of the limits of circular and Gaussian beta ensembles.

Recovery of the fidelity amplitude for the Gaussian ensembles

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude f_epsilon(tau)=< psi(0)| exp(2 pi i H_epsilon tau) exp(-2 pi i H_0 tau)| psi(0) > are obtained, where H_epsilon=H_0+(sqrt{epsilon}/(2 pi) )*V, and H_0 and H_epsilon are taken from the Gaussian unitary ensemble (GUE) or the Gaussian orthogonal ensemble (GOE), respectively. As long as the perturbation strength is small compared to the mean level spacing, a Gaussian decay of the fidelity amplitude is observed, whereas for stronger perturbations a change to a single-exponential decay takes place, in acc…

来自随机矩阵圆系综的高斯图解法

Gaussian diagrammatics from Circular Ensembles of random matrices

该研究发现随机矩阵三个圆系综内的隐藏高斯系综，给出矩计算的新图解规则，明确不同β下矩阵类型与维数设定，并应用于子矩阵迹的矩计算。

Statistics of eigenvectors in the deformed Gaussian unitary ensemble of random matrices

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using the supersymmetry approach we calculate analytically the moments and the distribution function of the eigenvectors components for a generic matrix $W$. We show that specific choices of $W$ can modify significantly the nature of the eigenvectors changing them from extended to critical to localized. Our analytical results are supported by numerical simulations…

Gaussian unitary ensemble with boundary spectrum singularity and $σ$-form of the Painlevé II equation

We consider the Gaussian unitary ensemble perturbed by a Fisher-Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the $σ$-form of the Painlevé II equation. In addition, we…

The Wilson loop in the Gaussian Unitary Ensemble

Using the supersymmetric formalism we compute exactly at finite $N$ the expectation of the Wilson loop in the Gaussian Unitary Ensemble and derive an exact formula for the spectral density at finite $N$. We obtain the same result by a second method relying on enumerative combinatorics and show that it leads to a novel proof of the Harer-Zagier series formula.

高斯张量族的特征化

Characterization of Gaussian Tensor Ensembles

本文基于Maxwell定理，定义了实对称、Hermitian和自偶Hermitian张量的高斯正交、酉和辛张量族，通过一阶和二阶情况恢复经典向量和矩阵高斯族，并证明了Maxwell型定理，统一并扩展了向量和矩阵的已有结果。

On universality of local edge regime for the deformed Gaussian Unitary Ensemble

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 $H_n$.

On the Variance of the Index for the Gaussian Unitary Ensemble

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $τ$-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.

关于高斯β集合最优软边展开的注记

A Note on Optimal Soft Edge Expansions for the Gaussian $β$ Ensembles

本文回顾了随机矩阵理论中β集合相关函数和观测量最优渐进行展开的研究，介绍了当前正在进行的相关研究方向。

From Alternating Sign Matrices to the Gaussian Unitary Ensemble

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.

Painleve IV and degenerate Gaussian Unitary Ensembles

We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t, this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painleve IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as …

Gaussian concentration bound and Ensemble equivalence in generic quantum many-body systems including long-range interaction

This work explores fundamental statistical and thermodynamic properties of short-and long-range-interacting systems. The purpose of this study is twofold. Firstly, we rigorously prove that the probability distribution of arbitrary few-body observables is restricted by a Gaussian concentration bound (or Chernoff--Hoeffding inequality) above some threshold temperature. This bound is then derived for arbitrary Gibbs states of systems that include long-range interactions Secondly, we establish a quantitative relationship between the concentration bound of the Gibbs state and the equivalence of can…

k-重高斯随机矩阵系综 I：将结构引入随机矩阵

k-Fold Gaussian Random Matrix Ensembles I: Forcing Structure into Random Matrices

本文基于 Schur-Weyl 对偶性和 Harish-Chandra 积分方法，提出并精确求解了满足 k-重不变约束的高斯随机矩阵系综，揭示了特征值谱与纠缠谱的耦合关系。

复矩阵多参数高斯系综的平均谱密度

Average Spectral Density of Multiparametric Gaussian Ensembles of Complex Matrices

针对非厄米多参数高斯系综，理论分析其平均谱密度的多参数演化，发现同全局约束类系综的共有演化路径，并推导出非平衡区谱密度的复杂性参数化公式。

Spectral density asymptotics for Gaussian and Laguerre $β$-ensembles in the exponentially small region

The first two terms in the large $N$ asymptotic expansion of the $β$ moment of the characteristic polynomial for the Gaussian and Laguerre $β$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at…

The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Pólya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

稀疏实非对称矩阵高斯系综的谱分布

Spectral distribution of sparse Gaussian Ensembles of Real Asymmetric Matrices

本文采用复杂度参数方法分析多参数稀疏实非对称矩阵高斯系综，推导了实与复特征值的系综平均谱密度，为多种稀疏实非对称系综提供了统一的谱统计数学框架并揭示了其普遍性。

Level Repulsion in Constrained Gaussian Random-Matrix Ensembles

Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a sufficient condition for GUE-type level repulsion to persist in the presence of constraints. For special classes of constraints, we extend this approach to the orthogonal and to the symplectic ensembles. A generalized Fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones. We find that in the DGUEs, l…

One-dimensional stochastic growth and Gaussian ensembles of random matrices

In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multi-matrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the l…

从艾里线系综到高斯自由场

From the Airy line ensemble to the Gaussian free field

研究艾里线系综高度函数全局涨落，借助行列式结构和最速下降分析扩展艾里核，证明重标度后中心化高度函数收敛到高斯自由场的显式拉回，在特定意义下成立。

On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.

Probability Distribution of Curvatures of Isosurfaces in Gaussian Random Fields

An expression for the joint probability distribution of the principal curvatures at an arbitrary point in the ensemble of isosurfaces defined on isotropic Gaussian random fields on Rn is derived. The result is obtained by deriving symmetry properties of the ensemble of second derivative matrices of isotropic Gaussian random fields akin to those of the Gaussian orthogonal ensemble.

Gaussian ensembles distributions from mixing quantum systems

In the context of the mixing dynamical systems we present a derivation of the Gaussian ensembles distributions from mixing quantum systems having a classical analog that is mixing. We find that mixing factorization property is satisfied for the mixing quantum systems expressed as a factorization of quantum mean values. For the case of the kicked rotator and in its fully chaotic regime, the factorization property links decoherence by dephasing with Gaussian ensembles in terms of the weak limit, interpreted as a decohered state. Moreover, a discussion about the connection between random matrix t…

Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion …

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(β= 1)$, complex ($β= 2)$ and real quaternion $(β= 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $β> 0$. In the Gaussian case a further construction valid for $β> 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. Th…

On an average over the Gaussian Unitary Ensemble

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble with weight exp(-z^2/2x^2 + t/x - x^2/2). We compute the leading order term of the partition function and of the coefficients of its Taylor expansion. Our results are valid in the range N^(-1/2) < z < N^(1/4). Such partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) that was introduced by Berry and Shukla (J. Phys. A: Math. Theor., Vol. 41 (2008), 385202, arXiv:0807.3474). It can also be interpreted as…

Fidelity susceptibility in Gaussian Random Ensembles

The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system size. The results are verified by a comparison with numerical data.

Optimal soft edge scaling variables for the Gaussian and Laguerre even $β$ ensembles

The $β$ ensembles are a class of eigenvalue probability densities which generalise the invariant ensembles of classical random matrix theory. In the case of the Gaussian and Laguerre weights, the corresponding eigenvalue densities are known in terms of certain $β$ dimensional integrals. We study the large $N$ asymptotics of the density with a soft edge scaling. In the Laguerre case, this is done with both the parameter $a$ fixed, and with $a$ proportional to $N$. It is found in all these cases that by appropriately centring the scaled variable, the leading correction term to the limiting densi…

Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $β$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion adapted for general $β$ Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.

Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond

These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random mat…

Scaling limits of correlations of characteristic polynomials for the Gaussian beta-ensemble with external source

We study the averaged product of characteristic polynomials of large random matrices in the Gaussian beta-ensemble perturbed by an external source of finite rank. We prove that at the edge of the spectrum, the limiting correlations involve two families of multivariate functions of Airy and Gaussian types. The precise form of the limiting correlations depends on the strength of the nonzero eigenvalues of the external source. A critical value for the latter is obtained and a phase transition phenomenon similar to that of arXiv:math/0403022 is established. The derivation of our results relies mai…

PDE's for the Gaussian ensemble with external source and the Pearcey distribution

The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues a and -a. As a first result, the probability that the eigenvalues of the ensemble belong to a set satisfies a fourth order PDE with quartic non-linearity; the variables being the eigenvalue a and the boundary points of the set. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap; i.e., when the support of the equilibrium measure for large size random ma…

The Distribution of the Largest Eigenvalue in the Gaussian Ensembles

The focus of this survey paper is on the distribution function for the largest eigenvalue in the finite N Gaussian ensembles (GOE,GUE,GSE) in the edge scaling limit of N->infinity. These limiting distribution functions are expressible in terms of a particular Painleve II function. Comparisons are made with finite N simulations as well as a discussion of the universality of these distribution functions.

Global spectral fluctuations in the Gaussian Unitary Ensemble

We consider global fluctuations of the spectrum of the GUE. Using results on the linear statistics of such matrices as well as variance bounds on the eigenvalues, we show that under a suitable scaling, global fluctuations of the spectrum can be asymptotically described in terms of a logarithmically correlated Gaussian field. We also discuss briefly connections between different objects in RMT giving rise to log-correlated fields: the logarithm of the absolute value of the characteristic polynomial, the eigenvalue counting function, and the field of fluctuations of the eigenvalues around their …

Gaussian asymptotics of discrete $β$-ensembles

We introduce and study stochastic $N$-particle ensembles which are discretizations for general-$β$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work…

Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles

In this paper, we address a class of problems in unitary ensembles. Specifically, we study the probability that a gap symmetric about 0, i.e. $(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi unitary ensembles (JUE) (where in the JUE, we take the parameters $α=β$). By exploiting the even parity of the weight, a doubling of the interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE, shows that the gap probabilities maybe determined as the product of the smallest eigenvalue distributions of the LUE with parameter $α=-1/2,$ and $α=1/2$ and the (shift…

Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states

We investigate the problem of optimally reversing the action of an arbitrary quantum channel C which acts independently on each component of an ensemble of n identically prepared d-dimensional quantum systems. In the limit of large ensembles, we construct the optimal reversing channel R* which has to be applied at the output ensemble state, to retrieve a smaller ensemble of m systems prepared in the input state, with the highest possible rate m/n. The solution is found by mapping the problem into the optimal reversal of Gaussian channels on quantum-classical continuous variable systems, which …

Toda系统的散射与高斯β系综

Scattering of the Toda system and the Gaussian $β$-ensemble

该研究利用作用于对称实三对角矩阵的经典Toda流，给出了Dumitriu-Edelman三对角模型谱服从高斯β系综的辛证明，拓展了Toda系统散射与对数气体矩阵模型的相关研究。

The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles

We characterize the phenomenon of "crowding" near the largest eigenvalue $λ_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $β$-ensemble of random matrix theory, including in particular the Gaussian orthogonal ($β=1$), unitary ($β=2$) and symplectic ($β= 4$) ensembles. We focus on two distinct quantities: (i) the density of states (DOS) near $λ_{\max}$, $ρ_{\rm DOS}(r,N)$, which is the average density of eigenvalues located at a distance $r$ from $λ_{\max}$ (or the density of eigenvalues seen from $λ_{\max}$) and (ii) the probability density function of the gap between the fi…

Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure

We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andréief's integration formula, is used to compactly express the $r$-point correlation function of eigenvalues. This representation circumvents the complications encountered in the u…

玻色高斯系综的相当好测量

Pretty good measurement for bosonic Gaussian ensembles

本研究给出多模玻色态高斯系综相当好测量高斯性的新证明，得到其显式可高效计算的高斯描述与均方误差形式，还证明相当好仪器满足条件时输出为高斯态并给出描述，推动其实验实现。

On correlation functions of characteristic polynomials for chiral Gaussian Unitary Ensemble

We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a $N\times N$ random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding an Itzykson-Zuber type integral for matrices from the non-compact manifold ${\sf{Gl(n,{\mathcal{C}})/U(1)\times ...\times U(1)}}$ (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalising the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation fun…

Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble

Consider a $n \times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(Δ_{i,n},\ 1\leq i\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\mathcal N}_n(Δ_{i,n}), 1\leq i\leq p)$, where ${\mathcal N}_n(Δ)$ represents the number of eigenvalues within $Δ$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly cent…