Mirko D'Ovidio, Enzo Orsingher, Lyudmyla Sakhno
We study general models of random fields associated with non-local equations in time and space. We discuss the properties of the corresponding angular power spectrum and find asymptotic results in terms of random time changes.
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Coordinates changed random fields on the sphere
通过坐标变换和从属变换，在球面上构造了不定常的随机场，并研究了随机场的角功率谱。这些随机场中的一些自然地产生为具有由高斯随机场表示的随机初始条件的偏微分方程的解。
We review the Dudley integral for the Belyaev dichotomy applied to Gaussian processes on spheres, and discuss the approximate (or restricted) continuity of paths in the discontinuous case. We discuss also the spatio-temporal case, of sphere cross line. In the continuous case, we investigate the link between the smoothness of paths and the decay rate of the angular power spectrum, following Tauberian work of the first author, Malyarenko, and Lang and Schwab.
A Strong threshold for the size of random caps to cover a sphere
在这篇文章中，我们认为“$N$”球冠面积$4πp$均匀分布在一个单位球的表面上。本文给出了任意球帽覆盖单位球面的大小的强门限函数。我们已经证明，对于大的$N,$，如果$\frac{Np}{\log\:N} > 1/2$的球的表面是完全覆盖的$N$帽几乎肯定，如果$\frac{Np}{\log\:N} \leq 1/2$的一个分区的球的表面仍然是开放的$N$帽几乎肯定。
Poisson asymptotics for random projections of points on a high-dimensional sphere
将高维球体上的一组点投影到随机方向上。如果大多数点在适当的意义上彼此足够远，则投影在分布上局部接近泊松点过程。
In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generates a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of Feller Markov invariant measure.
Time Varying Isotropic Vector Random Fields on Spheres
对于球面上均方连续、时域上平稳的各向同性矢量随机场，本文导出了其协方差阵函数的一般形式，并给出了随机场的一种级数表示，其中包含了超球面多项式。级数表示是协方差阵函数的某种模仿者，但在普通球谐函数方面不同于谱表示，并且对于建模和仿真是有用的。一些半参数模型也说明。
Constructing the Brownian sphere from a continuum random unicycle
本文基于四边形剖分的Miermont双射构造了带两点距离偏置的布朗球，给出固定距离两点布朗球的新构造，还推导了双测地线布朗平面的新表示。
Time varying axially symmetric vector random fields on the sphere
本文给出了球面上均方连续轴对称矢量随机场的协方差阵结构的一般形式，并给出了纵向可逆场的级数表示。级数表示在某种程度上是协方差阵函数的模仿者，并且它们都具有比文献中提出的相关Legendre函数更简单的形式，并且对于建模和仿真是有用的。对于球上轴对称均方连续的时空矢量随机场，导出了协方差阵结构的一般形式，对于纵向可逆的时空矢量随机场，给出了其级数表示.
In this paper we define (empirical) quadratic variations for a Gaussian isotropic random field $f$ on a unit sphere as sums over equidistant increments on one single geodesic line on the surface of the sphere. We prove a noncentral limit theorem for a fixed Fourier component of such a field as well as quantitative central limit theorems in the increasing frequency regime. Based on these results we propose estimators of the angular power spectrum and study their properties. Moreover, we show a quantitative central limit theorem for quadratic variations over the field $f$ and construct an estima…
Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere
我们导出原点是R^n中随机游动极值点的概率的渐近性。我们证明，为了使概率近似为1/2，随机游动的步数应在e^{Cn/ log n}$和e^{Cn log n}之间。因此，我们得到一个界限？球面布朗运动的pi/2-覆盖时间
We introduce a new family of random compact metric spaces $\mathcal{S}_α$ for $α\in(1,2)$, which we call stable shredded spheres. They are constructed from excursions of $α$-stable Lévy processes on $[0,1]$ possessing no negative jumps. Informally, viewing the graph of the Lévy excursion in the plane, each jump of the process is "cut open" and replaced by a circle and then all points on the graph at equal height which are not separated by a jump are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Fra…
We show that a point process of hard spheres exhibits long-range orientational order. This process is designed to be a random perturbation of a three-dimensional lattice that satisfies a specific rigidity property, examples include the FCC and HCP lattices. We also define two-dimensional near-lattice processes by local, geometry dependent hard disk conditions. Earlier results about existence of long-range orientational order carry over and we obtain the existence of infinite-volume measures on two-dimensional point configurations that turn out to follow the orientation of a fixed triangular la…
Random geometry on the sphere
介绍并研究了二维随机几何的一个万有模型。为此，我们开始从一个离散的图形绘制的领域，这是随机选择均匀在某一类图具有给定的大小$n$，例如类的所有三角形的领域与$n$的面孔。我们配备的顶点集的图与通常的图的距离重新缩放的因素$n^{-1/4}$。然后，我们证明了由此产生的随机度量空间收敛的分布为$n\to\infty$，在Gromov-Hausdorff意义上，向一个有限的随机紧度量空间称为布朗映射，这是普遍的意义上说，它不依赖于最初选择的类图。布朗映射与球面同胚，但其豪斯多夫维数等于$4$。我们得到了布朗映射中测地线结构的详细信息。我们还提出了称为布朗平面的布朗映射的无限体积变体，它作为均匀无限平面四边形的标度极限而出现。最后，我们讨论了一些开放的问题。这项研究的部分动机是在二维量子引力的物理理论中使用随机几何。
Time dependent random fields on spherical non-homogeneous surfaces
本文在非均匀球上引入了一类各向同性的不定常随时随机场，它由时变的ν \in（0，1]阶球布朗运动表示，利用它可以捕获宇宙学中的某些各向异性.这个过程是一个时变旋转扩散（TRD）或随机解的方程涉及球面拉普拉斯算子和时间分数阶导数的ν。TRD是非均匀球体上的扩散，因此，由TRD给出的球面坐标表示非均匀球体的坐标，通过该非均匀球体的坐标来索引同位素随机场。因此，本文提出的不定常随时间变化的随机场是通过合成实现的，可以看作是随机变化球面上的各向同性随机场。
Number of Edges in Random Intersection Graph on Surface of a Sphere
在这篇文章中，我们认为“$N$”球冠面积$4πp$均匀分布在一个单位球的表面上。我们研究了由这些帽构造的随机交图$G_N$。证明了对于$p = \frac{c}{N^{\al}},\:c >0$和$\al >2,$，图$G_N$的边数服从Poisson分布.我们还得到了关于$G_N$中孤立点个数的强律结果：对于$p = \frac{c}{N^{\al}},\:c >0$对于$\al < 1,$，几乎必然在$G_N$中没有孤立点，即，在$G_N$中至少存在$N/2$边并且对于$\al >3,$，$G_N$中的每个顶点是孤立的即，边集合$\cE_N.$中没有边
Grey Random Fields on the Sphere
本文提出灰色球随机场（GSRFs），将高斯随机场作为特例，通过多元从属子驱动的高斯混合构造，推导了其统计性质，证明其为重尾等待时间的连续时间随机游走的缩放极限，拓宽了球面上可处理的各向同性随机场类别。
The snake in the Brownian sphere
本文构造连续 CVS 双射的逆，将 Brown 蛇定义为 Brown 球的可测函数，并处理其定向问题。
Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation
研究了单位球面$\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$上独立均匀分布的$N$随机点关于表面积测度的几何性质，并给出了几个相关的证明.特别是，我们推导出渐近（$N \to \infty$）的预期时刻的半径与小平面的凸包的$N$随机点$\mathbb{S}^{d}$的球冠。我们提供了这些球冠的缩放半径的渐近分布和这些半径（覆盖半径）的最大值的期望值的示意图。数值证据包括支持这些acquitures。此外，利用Cai等人的成对角度极值定律，我们推导出$\mathbb{S}^{d}$上随机点的期望分离的精确渐近性。
Covering a Sphere with Four Random Circular Caps
设p（w）表示四个随机圆帽覆盖单位球面S^2的概率，圆帽的角半径为70度<w<90度。p（w）的精确表达式是未知的。当w> 84 deg时，我们给出了p（w）的非平凡下界;当w<84 deg时，对不等式p（w）>=0的改进是不可行的。一个对偶问题，涉及随机内接井心四面体也检查。
Phase Synchronization in Random Geometric Graphs on the 2D Sphere
针对二维球面上随机几何图的Kuramoto模型同步问题，本文证明节点数趋于无穷时，若初始条件收敛于光滑函数则系统高概率同步，并推测单连通闭黎曼流形有类似行为、非单连通流形全局同步失效。
Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case $d=3$, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.
Exploring the Structure of Anisotropic Random Fields on the Sphere
本文通过几何与谱方法，对球面随机场的各向同性与各向异性结构进行了分类，识别了可行与不可行情形，并建立了几何不变性与谱协方差矩阵的联系，为球面数据分析提供了模型与抽样指导。
A sequence $(X_N ) \subset \mathbb S^d$ of N-point sets from the d-dimensional sphere has QMC strength $s^*>d/2$ if it has worst-case error of optimal order, $N^{s/d}$, for Sobolev spaces of order $s$ for all $d/2 < s < s^*$ , and the order is not optimal for $s > s^*$ . In arXiv:1208.3267 conjectured values of the strength are given for some well known point families in $\mathbb S^2$ based on numerical results. We study the average QMC strength for some related random configurations.
Generating an equidistributed net on a unit n-sphere using random rotations
本文提出一种高概率成功的随机旋转算法，通过随机旋转生成的有限字作用于固定点来构造 n-球面上的 ε-网，并用于近似 Lipschitz 函数的球面积分。
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit…
We study the asymptotic behaviour of least squares estimators in regression models for long-range dependent random fields observed on spheres. The least squares estimator can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings.
Series expansions of isotropic Gaussian random fields on $\mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localised structure provide an alternative to the standard Karhunen-Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field…
On the isotropic constant of random polytopes with vertices on an $\ell_p$-sphere
研究了$\ell_p$ -单位球面上按锥概率测度分布的独立随机点的对称凸包，其中$\ell_p$ -单位球面上的随机点按$1\leq p < \infty$分布。我们证明了这些随机多面体具有一致绝对有界迷向常数的压倒性概率。这推广了D.阿隆索-古铁雷斯证明需要几种不同的工具，包括由于G锥测度的概率表示。Schechtman和J. Zinn的工作以及具有对数凹尾的独立随机变量和的矩估计。Gluskin和S.夸皮耶
A review of some recent results on random polynomials over R and over C
本文分为两个部分。第一部分回顾了随机多项式方程组实根期望数的一些最新结果。在第二部分中，我们处理一个不同的问题，即某些复杂的随机多项式的根的分布。我们讨论了最近的结果在这个方向上，这表明，相关的点在球（通过球极平面投影）是令人惊讶的非常适合相对于最小对数能量的领域。
Excursion probability of Gaussian random fields on sphere
设$X=\{X(x): x\in\mathbb{S}^N\}$是在$N$维单位球面$\mathbb{S}^N$上索引的实值中心高斯随机场.近似的偏移概率${\mathbb{P}}\{\sup_{x\in\mathbb{S}^N}X(x)\ge u\}$，作为$u\to\infty$，获得两种情况下：（i）$X$是局部各向同性和它的样本函数是非光滑的;（ii）$X$是各向同性和它的样本函数是两次可微的。对于情况（i），可以通过应用Piterbarg（Asymptotic Methods in the Theory of Gaussian Processes and Fields（1996）Amer. Math. Soc.）中的结果来研究偏移概率，Mikhaleva和Piterbarg（理论Probab。41（1997）367- 379）和Chan和Lai（Ann. Probab. 34（2006）80- 121）。证明了${\mathbb{P}}\{\sup_{x\in\mathbb {S}^N}X(x)\ge u\}$的渐近性类似于Pickands在欧氏空间上的近似，其中包含Pickands常数.对于情形（ii），我们应用期望的欧拉示性数方法来获得更精确的近似，使得误差是超指数小的。
Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions
研究高维单位球面上随机部分覆盖模型，通过建立中心极限定理，证明随机覆盖体积的波动渐近正态，并给出收敛速率的定量界。
In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere $\mathbb{S}^2$, i.e., spherical caps, focussing in particular on the excursion area. Precisely, we show that the asymptotic behavior of the excursion area is dominated by the so-called second-order chaos component, and we exploit this result to establish a Quantitative Central Limit Theorem, in the high energy limit. These results generalize …
Aliasing Effects for Samples of Spin Random Fields on the Sphere
本文研究了球面上自旋随机场离散采样产生的别名效应，分析了不同采样方案对角功率谱的影响，并证明带限场在足够节点数下无别名效应。
A random cell splitting scheme on the sphere
考虑了$2$维单位球的随机递归胞分裂方案，该方案是欧氏随机几何中STIT镶嵌过程的球面模拟。一阶矩计算的一个大阵列的组合和度量参数的诱导分裂镶嵌的鞅方法结合工具，从球面积分几何。研究结果进行了比较，在欧几里德的情况下，从而透明的影响的曲率的基础空间。此外，容量功能的计算和点过程，所产生的分裂镶嵌与一个固定的大圆相交的特点。
An Exponential Concentration Inequality for the Components of a Uniform Random Vector on the Sphere
本文证明了球面上均匀随机向量分量的经验CDF以指数速度集中于标准高斯CDF，并给出了具有显式函数和常数的有限样本统计保证。
On spectral representations of tensor random fields on the sphere
基于旋转群表示理论，研究了球面上张量随机场的表示。引入张量场的特定分量，并在弱各向同性和均方连续条件下，导出了它们的广义球函数谱分解.给出了分解的随机系数的性质，包括高斯性条件等重要问题。
In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from zero, the observed length will deviate from the actual measure by at most a fixed error for any subset, while in convex bodies we can always choose a subset for which the behavior would be close to a zero-one law, as the dimension grows. The result for the sphere is based on an analysis of the Radon transform. Using similar tools we analyze the variance of …
In this paper, aliasing effects are investigated for random fields defined on the d-dimensional sphere and reconstructed from discrete samples. First, we introduce the concept of an aliasing function on the sphere. The aliasing function allows one to identify explicitly the aliases of a given harmonic coefficient in the Fourier decomposition. Then, we exploit this tool to establish the aliases of the harmonic coefficients approximated by means of the quadrature procedure named spherical uniform sampling. Subsequently, we study the consequences of the aliasing errors in the approximation of the…
Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere
证明了均匀分布在所有有根2k-角化集上的随机平面映射的标度极限是a.s.与二维球面同胚。我们的方法依赖于研究某些随机测地线叠层的磁盘。
Corrected mean-field model for random sequential adsorption on random geometric graphs
在数学和物理学中一个著名的问题是用$d\geq 2$在$d$维欧几里得空间中建立一个可解的非重叠全等球随机顺序吸附模型。球体依次到达空间中均匀选择的位置，只有当与先前沉积的球体没有重叠时才被接受。由于空间相关性，表征接受球体的分数在很大程度上仍然是棘手的。我们研究这个分数采取一种新的方法，比较随机顺序吸附在欧几里得空间上的最近邻阻塞的序列聚类随机图。这个随机网络模型可以被认为是一个修正的平均场模型的相互作用图之间的尝试领域。利用泛函极限定理，我们刻画了可接受球的分数及其涨落。
Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles…
We study the regularity properties of Gaussian fields defined over spheres cross time. In particular, we consider two alternative spectral decompositions for a Gaussian field on $\mathbb{S}^d \times \mathbb{R}$. For each decomposition, we establish regularity properties through Sobolev and interpolation spaces. We then propose a simulation method and study its level of accuracy in the $L^2$ sense. The method turns to be both fast and efficient.
A Dimension-Independent Bound on the Wasserstein Contraction Rate of a Geodesic Random Walk on the Sphere
本文证明了欧几里得$d$-球面上测地线随机游走的Wasserstein收缩率具有与维度$d$无关的上界，为geodesic slice sampling和Gibbsian polar slice sampling等MCMC方法实现维度无关性能提供了理论依据。
Random triangulations of the d-sphere with minimum volume
本文研究随机旅行商问题的高维推广，确定n个顶点上同胚于二维球面的最小体积随机三角剖分的增长速率与集中性，并对高维情形在现有知识范围内给出相应结果，同时提出与均匀随机三角剖分相联系的广义随机几何框架。
Critical sets of random smooth functions on products of spheres
本文证明了一个Chern-Lashof型公式，该公式用于计算光滑流形$M$上光滑函数临界点的期望个数，该光滑流形是从一个具有高斯概率测度的有限维子空间$V\subset C^\infty(M)$中随机选取的.然后，我们使用这个公式这个公式来找到一个随机线性组合的大量本征函数的拉普拉斯算子的圆球，环面，或产品的两个圆球的临界点的预期数量的渐近。在$M=S^1$的情况下，我们证明了次数为$\leq ν$的三角多项式的临界点的数目是随机变量$Z_ν$，期望值为$E(Z_ν)\sim 2\sqrt{0.6}\,ν$，方差为$var(Z_ν)\sim cν$，即$ν\to \infty$，$c\approx 0.35$。
Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations
研究球面上各向同性高斯随机场的正则性、快速模拟方法及其在环境科学模型中的应用，探讨其与随机偏微分方程的关系。
Fractional Stochastic Partial Differential Equation for Random Tangent Fields on the Sphere
本文提出球面上随机切向量场的分数阶SPDE，用分数阶扩散、时间分数阶导数和向量值分数阶布朗运动刻画空间Lévy行为、时间间歇性与长程依赖，给出基于向量球谐函数的Karhunen-Loève解、Legendre张量核协方差结构及近似误差收敛速率。
We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in $\mathbb R^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and the expected volume of the spherical convex hull of $n$ random points sampled uniformly and independently from the $d$-dimensional half-sphere. In the case when $n=d+2$, we compute the probability that this spherical conve…
Needlet approximation for isotropic random fields on the sphere
本文利用球面Needlet对球面上各向同性随机场进行多尺度近似，证明了半离散Needlet分解在弱各向同性随机场上的均值和点wise收敛性，并构造了全离散Needlet近似并证明其收敛阶与半离散Needlet相同。
Gaussian approximation for non-linearity parameter estimation in perturbed random fields on the sphere
研究球面上受扰随机场非线性参数估计，通过将估计问题转化为普通最小二乘回归得出渐近矩，结合斯坦 - 马利瓦因技术得到定量高斯近似及收敛速率，数值实验验证相关规律和行为。
Random punctured hyperbolic surfaces & the Brownian sphere
研究按Weil-Petersson测度采样的n+1个穿孔0类随机双曲面Sₙ，证明度量经n⁻¹/⁴缩放后收敛于布朗球面，未缩放时局部收敛于特定无限体积双曲面，通过平面树编码等技术实现。
Fluctuations in the logarithmic energy for zeros of random polynomials on the sphere
针对Smale第七问题，本文研究了随机椭圆多项式根的球极平面投影构型的对数能量涨落，证明了中心极限定理，并表明能量在√n尺度上高度集中。
The typical cell of a Voronoi tessellation generated by $n+1$ uniformly distributed random points on the $d$-dimensional unit sphere $\mathbb S^d$ is studied. Its $f$-vector is identified in distribution with the $f$-vector of a beta' polytope generated by $n$ random points in $\mathbb R^d$. Explicit formulae for the expected $f$-vector are provided for any $d$ and the low-dimensional cases $d\in\{2,3,4\}$ are studied separately. This implies an explicit formula for the total number of $k$-dimensional faces in the spherical Voronoi tessellation as well.
Given a closed Riemann surface $Σ$ equipped with a volume form $ω$, we construct a natural probability measure on the space $\mathcal{M}_d(Σ)$ of degree $d$ branched coverings from $Σ$ to the Riemann sphere $\mathbb{C}\mathbb{P}^1.$ We prove a large deviations principle for the number of critical points in a given open set $U\subset Σ$: given any sequence $ε_d$ of positive numbers, the probability that the number of critical points of a branched covering deviates from $2d\cdot\textrm{Vol}(U)$ more than $ε_d$ is smaller than $\exp(-C_Uε^3_d d)$, for some positive constant $C_U$. In particular, …
Random Quadratic Form on a Sphere: Synchronization by Common Noise
本文研究了球面上随机二次型模型的同步现象，通过共同噪声机制解释了Transformer中聚类行为的形成，且不依赖自注意力机制。
Khinchin inequalities for uniforms on spheres with a deficit
本文改进了球面上均匀分布随机向量的矩比较不等式，给出了具有最优高维性能的缺陷项。
Short geodesics and small eigenvalues on random hyperbolic punctured spheres
本文研究了 $n\to\infty$ 时 Weil-Petersson 随机双曲穿孔球面上的短测地线与小特征值，证明闭测地线长度经缩放后收敛于 Poisson 点过程，并得出具有至少 $k=o(n)$ 个任意小特征值的概率趋于 $1$。
A real quaternion spherical ensemble of random matrices
人们可以确定一个三方分类的随机矩阵合奏到对应的平面，球和反球的几何普适性类。平面被确定为Ginibre型（iid）矩阵和反球与截断的酉矩阵。本文主要研究与球面相对应的系综：矩阵的形式为$\bY= \bA^{-1} \bB$，其中$\bA$和$\bB$是独立的$N\times N$矩阵，具有iid标准高斯实四元数元素。通过应用类似于那些用于类似的复杂和真正的球形系综，本征值jpdf和关联函数的计算技术。这就完成了使用传统Dyson指数$β=1,2,4$的球形矩阵的探索。我们发现，本征值密度（球面投影后）有一个耗尽的本征值沿环对应的实轴，与此环的反射对称。然而，在大矩阵维数的限制下，该特征值密度接近于相应的复系综的密度，该密度在球体上是均匀的。该结果与球面定律（类似于iid矩阵的圆形定律）一致，该定律指出，对于具有球面结构的矩阵$\bY= \bA^{-1} \bB$，其中$\bA$和$\bB$是独立的，iid矩阵的（立体投影的）本征值密度在球面上趋于均匀。
In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …
We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai--Komlós--Tusnády optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.
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