Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk

We study a class of tridiagonal matrix models, the q-roots of unity models, which includes the sign (q=2) and the clock (q=) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M ^k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.     

On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations     

On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.     

Random Incidence Matrices: Moments of the Spectral Density

We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of small eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.     

On the Level Spacing Distribution in Quantum Graphs

We derive a formula for the level spacing probability distribution in quantum graphs. We apply it to simple examples and we discuss its relation with previous work and its possible application in more general cases. Moreover, we derive an exact and explicit formula for the level spacing distribution of integrable quantum graphs.     

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectrum is considered both numerically and analytically using previous work of Edelman et al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion does not hold, e.g., real random matrices with Gaussian elements with a large positive mean and finite variance. PACS numbers: 05.45.−a, 05.45.Tp, 89.75.−k, 89.75.Fb     

On the Law of Multiplication of Random Matrices

We recover Voiculescu's results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n×n unitary matrices and the measure of the product of three n×n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure.     

CS-MRI 中评价随机欠采样矩阵的新方法

在压缩感知-磁共振成像(CS-MRI)中,随机欠采样矩阵与重建图像质量密切相关．而选取随机欠采样矩阵一般是通过计算点扩散函数(PSF),以可能产生的伪影的最大值为评价参数,评估欠采样对图像重建的影响,然而最大值只反应了伪影的最坏情况．该文引入了两种新的统计学评价参数平均值(MV)和标准差(SD),其中平均值评估了伪影的平均大小,标准差可以反映伪影的波动情况．该文分别使用这3种参数对小鼠和人体脑部MRI数据以不同的采样比率进行CS图像重建,实验结果表明,当采样比率不低于4倍稀疏度时,使用平均值获得了质量更优的重建图像．因此,通过稀疏度先验知识指导合理选取采样比率,并以平均值为评价参数选取随机欠采样矩阵,能够获得更优的CS-MRI重建图像．
     

Exact Tracer Diffusion Coefficient in the Asymmetric Random Average Process

We study tracer diffusion in the continuous-time asymmetric random average process which is an interacting particle system on generalizing the Hammersley process. From the equations of motion for the particle-position correlations we obtain the exact tracer diffusion coefficient which is in agreement with a recent heuristic result by Krug and Garcia.     

The Wigner Band Random Matrix Model: Studied from the View Point of a Generalization of Brillouin- Wigner Perturbation Theory

The Wigner band random matrix model is studied by making use of a generalization of Brillouin-Wigner perturbation theory. Energy eigenfunctions are shown to be divided into perturbative and nonperturbative parts. A relation between the average shape of eigenstates and that of the so-called local spectral density of states (LDOS) is derived by making use of some properties of energy eigenfunctions drawn from numerical results. Several perturbation strengths predicted by the perturbation theory are found to play important roles in the variation of the shape of the LDOS with perturbation strength.     

Perturbation of Transmission Matrices in Nonlinear Random Media

Random media with tailored optical properties are attracting burgeoning interest for applications in imaging, biophysics, energy, nanomedicine, spectroscopy, cryptography, and telecommunications. A key paradigm for devices based on this class of materials is the transmission matrix, the tensorial link between the input and the output signals, that describes in full their optical behavior. The transmission matrix has specific statistical properties, such as the existence of lossless channels, that can be used to transmit information, and are determined by the disorder distribution. In nonlinear materials, these channels may be modulated and the transmission matrix tuned accordingly. Here, the direct measurement of the nonlinear transmission matrix of complex materials is reported, exploiting the strong optothermal nonlinearity of scattering silica aerogel (SA). It is shown that the dephasing effects due to nonlinearity are both controllable and reversible, opening the road to applications based on the nonlinear response of random media.     

Power law scaling of the top Lyapunov exponent of a Product of Random Matrices

A sequence of i.i.d. matrix-valued random variables with probabilityp and with probability 1–p is considered. Leta() = a ₀ + O(), c() = c ₀ + O() lim ₀ b() = Oa ₀,c ₀, >0, andb()>0 for all >0. It is shown show that the top Lyapunov exponent of the matrix productX _n X _n-1...X ₁, = lim_n (1/n) n X _n X _n-1...X ₁ satisfies a power law with an exponent 1/2. That is, lim ₀(ln /ln ) = 1/2.     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one-and two-dimensional substrates focusing our analysis on the probability distribution function P(M,L) of the number M of maximal points (i.e., local “peaks”) of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one-dimensional ballistic growth process in the steady state can be mapped onto “rise-and-descent” sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. [G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004)], that different characteristics of “rise-and-descent” patterns in random permutations can be interpreted in terms of a certain continuous-space Hammersley-type process. For one-dimensional system we compute P(M,L) exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long-ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as L → ∞.     

Exact Mean-Field Solutions of the Asymmetric Random Average Process

We consider the asymmetric random average process (ARAP) with continuous mass variables and parallel discrete time dynamics studied recently by Krug/Garcia and Rajesh/Majumdar [both J. Statist. Phys. 99 (2000)]. The model is defined by an arbitrary state-independent fraction density function (r) with support on the unit interval. We examine the exactness of mean-field steady-state mass distributions in dependence of and identify as a conjecture based on high order calculations the class of density functions yielding product measure solutions. Additionally the exact form of the associated mass distributions P(m) is derived. Using these results we show examplary the exactness of the mean-field ansatz for monomial fraction densities (r)=(n–1) r ^n–2 with n2. For verification we calculate the mass distributions P(m) explicitly and prove directly that product measure holds. Furthermore we show that even in cases where the steady state is not given by a product measure very accurate approximants can be found in the class .     

散射颗粒分布对二维随机激光器腔内各模式的影响

采用时域有限差分法数值求解Maxwell方程组,分析了二维随机激光器中光波模式的频谱特性.计算表明,处于不同分布区域的散射颗粒对腔内不同模式的影响有很大差异,处于该模式空间分布区域的散射颗粒比区域外的颗粒对模式有更大的影响.这个特性是随机激光器所独有,并可以用来进行随机激光器的模式选择.     

随机介质背散射二维Mueller矩阵的数值模拟与分析

推导了随机介质背散射Mueller矩阵的直接计算公式,并运用矢量Monte Carlo方法进行了数值模拟.结果表明随机介质背散射二维Mueller矩阵方位关系随散射系数的减小而增强,而与微粒大小关系不大;Mueller矩阵元素绝对值的空间分布随径向呈近似指数规律衰减,矩阵元素的方位变化具有周期性.对称系统的二维Mueller矩阵的花样图中仅有7幅独立,其余9幅可通过对称、旋转变换得到.     

二维随机介质中模式的空间局域化分布

基于随机激光的时域理论,利用时域有限差分法(FDTD)数值求解麦克斯韦方程组和速率方程组,利用分区域计算法,分别计算了整体和局部泵浦二维随机介质时,不同泵浦强度下,不同区域的辐射谱,得出了随机介质中模式的空间局域化分布。计算结果显示,同样条件下不同区域的辐射谱是不同的,且随着激励源泵浦强度的变化,随机激光辐射始终集中在介质中某几个固定区域,但各个区域的随机激射效率不同,在区域之间存在模式的空间范围重叠。利用该方法,可以分析介质的随机分布对随机激射的影响,理论上可以为高激射效率的伪随机介质制备提供一定指导。     

Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and Other Determinantal Random Point Fields

We prove the Central Limit Theorem (CLT) for the number of eigenvalues near the spectrum edge for certain Hermitian ensembles of random matrices. To derive our results, we use a general theorem, essentially due to Costin and Lebowitz, concerning the Gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions. As another corollary of the Costin–Lebowitz Theorem we prove the CLT for the empirical distribution function of the eigenvalues of random matrices from classical compact groups.