我们构造了一个层次的循环方程不变的圆形系综。这些都是有效的一般类的潜力和任意逆温度$ {\rm Re}\,β>0 $和本征值$ N $的数量。使用匹配参数的预解函数的线性统计$ f(ζ)=(ζ+z)/(ζ-z) $在一个特定的渐近制度，全球政权，我们系统地开发相应的大$ N $扩展和应用此解决方案的戴森圆形系综。目前，我们可以计算第二预解函数在此扩展的10个订单，也是它的一般傅立叶系数或矩$ m_{k} $的等效长度。领先的大$ N $，大$ k $，$ k/N $固定形式的时刻，可以涉及到小波数膨胀的结构功能的散装，缩放戴森圆形合奏，从早期的工作。从矩展开式出发，我们推测了低$ k $矩的某些精确部分分式形式。对于所有上述结果，我们已经与$ β= 1,2,4 $、一般$ N $和甚至阳性$ β$、$ N=2,3 $的完全可溶的情况进行了比较。
英文摘要：
We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,β>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(ζ)=(ζ+z)/(ζ-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ β= 1,2,4 $, general $ N $ and even, positive $ β$, $ N=2,3 $.
We study the linear statistics of the circular $β$-ensemble with a Stein's method argument, where the exchangeable pair is generated through circular Dyson Brownian motion. This generalizes previous results obtained in such a way for the CUE and provides a novel approach for studying linear statistics of $β$-ensembles. This approach allows studying simultaneously a collection of linear statistics whose number grows with the dimension of the ensemble. Also this approach requires estimating only low order moments of the linear statistics.
We study limiting distribution of pair counting statistics of the form $ \sum_{1\leq i\neq j\leq N} f(L_N\*(θ_i-θ_j))$ for the circular $β$-ensemble (C$β$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $β=2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.
Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $β$ generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices $β= 1,2$ and 4, and also to the spectral density in the corresponding $β$-ensemble w…
We consider the circular unitary ensemble with a Fisher-Hartwig singularity of both jump type and root type at $z=1$. A rescaling of the ensemble at the Fisher-Hartwig singularity leads to the confluent hypergeometric kernel. By studying the asymptotics of the Toeplitz determinants, we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the Hamiltonian of the coupled Painlevé V system in dimension four. This leads to a Painlevé-type representation o…
This paper gives a rigorous proof of a conjectured statistical self-similarity property of the eigenvalues random matrices from the Circular Unitary Ensemble. We consider on the one hand the eigenvalues of an $n \times n$ CUE matrix, and on the other hand those eigenvalues $e^{iϕ}$ of an $mn \times mn$ CUE matrix with $|ϕ| \le π/ m$, rescaled to fill the unit circle. We show that for a large range of mesoscopic scales, these collections of points are statistically indistinguishable for large $n$. The proof is based on a comparison theorem for determinantal point processes which may be of indep…
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…
In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the lim…
We consider the symplectic induced Ginibre process, which is a Pfaffian point process on the plane. Let $N$ be the number of points. We focus on the almost-circular regime where most of the points lie in a thin annulus $\mathcal{S}_{N}$ of width $O(\frac{1}{N})$ as $N \to \infty$. Our main results are the scaling limits of all correlation functions near the real axis, and also away from the real axis. Near the real axis, the limiting correlation functions are Pfaffians with a new correlation kernel, which interpolates the limiting kernels in the bulk of the symplectic Ginibre ensemble and of t…
The problem of calculating the scaled limit of the joint moments of the characteristic polynomial, and the derivative of the characteristic polynomial, for matrices from the unitary group with Haar measure first arose in studies relating to the Riemann zeta function in the thesis of Hughes. Subsequently, Winn showed that these joint moments can equivalently be written as the moments for the distribution of the trace in the Cauchy unitary ensemble, and furthermore relate to certain hypergeometric functions based on Schur polynomials, which enabled explicit computations. We give a $β$-generalisa…
The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $β$-Ensemble (C$β$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$β$…
We apply the operation of random independent thinning on the eigenvalues of $n\times n$ Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as $n\to\infty$.
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…
Gessel-Type Expansion for the Circular $β$-Ensemble and Central Limit Theorem for the Sine-$β$ Process for $β\le 2$
Institute for System Programming of the Russian Academy of Sciences（俄罗斯科学院系统编程研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）
We derive the anomalous conformal Ward identities for ${\cal N}=4$ SYM Wilson loops on polygon-like contours with edges formed by circular arcs. With a suitable choice of parameterisation they are very similarly to those for local correlation functions. Their solutions have a conformally covariant factor depending on the distances of the corners times a conformally invariant remainder factor depending, besides on cross ratios of the corners, on the cusp angles and angles parameterising the torsion of the contours.
The $β$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi $β$ ensembles, and use these to systematically construct the explicit form of the $1/N$ expansion at low orders. This allows us to give the explicit form of corrections to the global density, and al…
We consider the macroscopic large N limit of the Circular beta-Ensemble at high temperature, and its weighted version as well, in the regime where the inverse temperature scales as beta/N for some parameter beta>0. More precisely, in the large N limit, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (beta=0) and the weighted logarithmic energy (beta=\infty). More precisely, we provide subGaussian concentration estimates in the W1 metric for the deviations of the empirical measure to this equili…
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.
We calculate Wilson loops in lowest order of perturbation theory for triangular contours whose edges are circular arcs. Based on a suitable disentanglement of the relations between metrical and conformal parameters of the contours, the result fits perfectly in the structure predicted by the anomalous conformal Ward identity. The conformal remainder function depends in the generic 4D case on three cusp and on three torsion angles. The restrictions on these angles imposed by the closing of the contour are discussed in detail and also for cases in 3D and 2D.
The conformal loop ensemble $\operatorname {CLE}_κ$ with parameter $8/3<κ<8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $κ$ and $ν$, we compute the almost-sure Hausdorff dimension of the set of points $z$ for which the number of CLE loops surrounding the disk of radius $\varepsilon$ centered at $z$ has asymptotic growth $ν\log (1/\varepsilon )$ as $\varepsilon \to0$. By extending these results to a setting in which the loops are given i.i.d. weights, we give a CLE-based treatment of the extremes of t…
The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE$_κ$ (both with simple and non-simple loops, i.e., for the whole range $κ\in (8/3, 8)$) how to derive the connection probabilities in conformal rectangles for a conditioned version of CLE$_κ$ which can be interpreted as a CLE$_κ$ with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a conformal square, we prove that the probability that the two wired sides hook up so that they create…
We investigate eigenvalue moments of matrices from Circular Orthogonal Ensemble multiplicatively perturbed by a permutation matrix. More precisely we investigate variance of the sum of the eigenvalues raised to power $k$, for arbitrary but fixed $k$ and in the limit of large matrix size. We find that when the permutation defining the perturbed ensemble has only long cycles, the answer is universal and approaches the corresponding moment of the Circular Unitary Ensemble with a particularly fast rate: the error is of order $1/N^3$ and the terms of orders $1/N$ and $1/N^2$ disappear due to cancel…
The Circular Unitary Ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios
本文证明了随机酉矩阵的特征多项式经过适当的尺度变换后，几乎必然收敛于一个随机解析函数，该函数的零点在实直线上形成一个具有正弦核的行列式点过程.我们的缩放是在所谓的“微观”水平上进行的，也就是说，我们认为特征多项式的点是$1/n$远。我们证明了这一点的框架中的虚拟等距规避的事实，即重新缩放的特征多项式甚至没有一阶矩，从而使经典的随机矩阵理论的技术难以应用。强收敛的结果，在此设置为我们提供了一个新的方法比：我们能够解决开放的问题的极限分布的特征多项式的比率评估点的形式$\exp(2 i πα/n)$和相关的对象（如对数导数）。我们还明确描述了依赖关系的特征多项式的对数在几个点上的微观尺度上进行评估。在数论方面，受Keating-Snaith哲学的启发，我们在随机过程水平上猜想了临界线上的黎曼ζ函数的值分布的一些新的极限定理。
In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the …
In the classical $β$-ensembles of random matrix theory, setting $β= 2 α/N$ and taking the $N \to \infty$ limit gives a statistical state depending on $α$. Using the loop equations for the classical $β$-ensembles, we study the corresponding eigenvalue density, its moments, covariances of monomial linear statistics, and the moments of the leading $1/N$ correction to the density. From earlier literature the limiting eigenvalue density is known to be related to classical functions. Our study gives a unifying mechanism underlying this fact, identifying in particular the Gauss hypergeometric differe…
Suppose that $Γ$ is a conformal loop ensemble (CLE$_κ$) with simple loops ($κ\in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf C}$ whose boundary is itself a type of CLE$_κ$ loop. Let $Υ$ be the carpet of $Γ$, i.e., the set of points in $D$ not surrounded by a loop of $Γ$. We prove that certain approximations to the chemical distance metric in $Υ$ are tight. More precisely, for each path $ω\colon [0,1] \to Υ$ and $ε> 0$ we let ${\mathfrak N}_ε(ω)$ be the Lebesgue measure of the $ε$-neighborhood of $ω$. For $z,w \in Υ$ we let ${\mathfrak d}_ε(z,w;Γ) = \inf_ω{\mathfrak N}_ε(ω)$ wh…
