On the neighbor spacing of eigenvalues of unitary matrices

Nearest Neighbor Distances on a Circle: Multidimensional Case

We study the distances, called spacings, between pairs of neighboring energy levels for the quantum harmonic oscillator. Specifically, we consider all energy levels falling between E and E+1, and study how the spacings between these levels change for various choices of E, particularly when E goes to infinity. Primarily, we study the case in which the spring constant is a badly approximable vector. We first give the proof by Boshernitzan-Dyson that the number of distinct spacings has a uniform bound independent of E. Then, if the spring constant has components forming a basis of an algebraic nu…

Consecutive level spacings in the chiral Gaussian unitary ensemble: From the hard and soft edge to the bulk

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…

Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles

The ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra. It does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system. The Wigner-surmise-like results for the ratio distribution are known for the invariant classes of Gaussian random matrices. However, for the crossover ensembles, which are useful in modeling systems with partially broken symmetries, corresponding results have remained unavailable so far. In this work, we d…

圆酉系综的相邻能级间距分布及其比值：有限尺寸修正与 Riemann $ζ$ 零点

Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann $ζ$ zeros

该研究用 Jánossy 密度与 Tracy-Widom 方程计算 CUE 相邻本征相位间距及比值分布，揭示有限尺寸修正的阶次抵消，并解释 Riemann zeta 零点偏差的缩放规律。

高斯酉系综连续能级间距的分布及其比值：从头推导

Distributions of consecutive level spacings of Gaussian unitary ensemble and their ratio: ab initio derivation

本文基于 Tracy-Widom 方法和 Janossy 密度，从头推导了高斯酉系综在 $N\to\infty$ 时连续能级间距及其比值的解析分布，并通过与黎曼 $ζ$ 函数零点对比验证了其在刻画量子混沌性方面的有效性。

Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos

We introduce a complex-plane generalization of the consecutive level-spacing distribution, used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of complex-valued ratios between nearest- and next-to-nearest neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra. This is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in additio…

随机矩阵能级间距分布超出最近邻的推测

Surmise for random matrices' level spacing distributions beyond nearest-neighbors

针对随机矩阵第$k$近邻能级间距分布，提出一种基于已知方差的修正推测，在保持Wigner推测简单性的同时提高了精度，为多体量子混沌提供了精细探针。

Finite size effects for spacing distributions in random matrix theory: circular ensembles and Riemann zeros

According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1-ξ$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that…

Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles

The density function for the joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles is found in terms of a Painlevé II transcendent and its associated isomonodromic system. As a corollary, the density function for the spacing between these two eigenvalues is similarly characterized.The particular solution of Painlevé II that arises is a double shifted Bäcklund transformation of the Hasting-McLeod solution, which applies in the case of the distribution of the largest eigenvalue at the soft edge. Our deductions are made by employing the hard-to-soft edge tra…

Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form $a(s) e^{-b(s)}$ for $a$ simply related to a Painlevé transcendent and $b$ its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).

Higher order spacing ratios in random matrix theory and complex quantum systems

The distribution of the ratios of nearest neighbor level spacings has become a popular indicator of spectral fluctuations in complex quantum systems like interacting many-body localized and thermalization phases, quantum chaotic systems, and also in atomic and nuclear physics. In contrast to the level spacing distribution, which requires the cumbersome and at times ambiguous unfolding procedure, the ratios of spacings do not require unfolding and are easier to compute. In this work, for the class of Wigner-Dyson random matrices with nearest neighbor spacing ratios $r$ distributed as $P_β(r)$ f…

Circular unitary ensemble with highly oscillatory potential

We study the effect of highly oscillatory potentials to the eigenvalues of a random matrix. Consider the circular unitary ensembles with an external potential which is periodic with the period comparable to the average spacing of the eigenvalues. We show that in this case the density of states is periodic and does not converge in the large matrix limit, but the local correlation functions converge to some simple combinations of the sine kernel and the potential. We evaluate the correlation functions exactly and also asymptotically.

Multiplying unitary random matrices - universality and spectral properties

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 $t_c$.

The complex elliptic Ginibre ensemble at weak non-Hermiticity: bulk spacing distributions

We show that the distribution of bulk spacings between pairs of adjacent eigenvalue real parts of a random matrix drawn from the complex elliptic Ginibre ensemble is asymptotically given by a generalization of the Gaudin-Mehta distribution, in the limit of weak non-Hermiticity. The same generalization is expressed in terms of an integro-differential Painlevé function and it is shown that the generalized Gaudin-Mehta distribution describes the crossover, with increasing degree of non-Hermiticity, from Gaudin-Mehta nearest-neighbor bulk statistics in the Gaussian Unitary Ensemble to Poisson gap …

Empirical Spacings of Unfolded Eigenvalues

We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that th…

非厄米随机矩阵中复间距比的精确概率分布

Exact probability distributions of complex spacing ratios in non-Hermitian random matrices

本文从精确联合特征值概率密度出发，推导了非厄米随机矩阵高斯系综AI†和AII†中复间距比的精确分布，给出AII†类任意N的代数表达式及AI†类N=3的积分表示和渐近行为，并经数值验证。

A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in…

An invariant analytic orthonormalization procedure with an application to coherent states

We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators $A_j$, $j=1,2,...,n$, starting from a fixed normalized vector in $\Hil$ and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of {\em coherent-like} vectors can be produced and in which condition over the lattice spacing this can be done.

Random matrix theory for mixed regular-chaotic dynamics in the super-extensive regime

We apply Tsallis's q-indexed nonextensive entropy to formulate a random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. We consider the super-extensive regime of q < 1. We obtain analytical expressions for the level-spacing distributions, which are strictly valid for 2 \times 2 random-matrix ensembles, as usually done in the standard RMT. We compare the results with spacing distributions, numerically calculated for random matrix ensembles describing a harmonic oscillator perturbed by Gaussian orthogonal and unitary ensembles.

Universal spectral correlations in ensembles of random normal matrices

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

探索连续β系综：用于随机矩阵谱统计的Python实现

Exploring continuous beta-ensembles: A Python implementation for random matrix spectral statistics

该研究开发了用于采样随机矩阵理论中β系综的开源Python包，实现了相关构造，提供谱统计分析工具，并通过数值实验验证了β作为连续拟合参数等内容。

Invariant sums of random matrices and the onset of level repulsion

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex hermitian matrices $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The di…

Wigner surmise for mixed symmetry classes in random matrix theory

We consider the nearest-neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes, or between integrable and non-integrable systems. We derive analytical formulas for the spacing distributions of 2x2 or 4x4 matrices and show numerically that they provide very good approximations for those of random matrices with large dimension. This generalizes the Wigner surmise, which is valid for pure ensembles that are recovered as limits of the mixed ensembles. We show how the coupling parameters of small and large matrices must be matched depending …