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\indent Over the last two decades, randomization has emerged as a leading tool for pursuing efficiency in numerical linear algebra. Its benefits can be explained in part by smoothed analysis, where an algorithm that fails spectacularly in certain cases may be unlikely to do so on random -- or randomly perturbed -- inputs. This observation implies a simple framework for developing accurate and efficient randomized algorithms:\ apply a random perturbation and run an existing method, whose worst-case error (or run-time) can be avoided with high probability. \\\indent Recent work of Banks, Garza-Vargas, Kulkarni, and Srivastava applied this framework to the standard eigenvalue problem, developing a randomized algorithm that (with high probability) diagonalizes a matrix in nearly matrix multiplication time [Foundations of Computational Math 2022]. Central to their work is the phenomenon of \textit{pseudospectral shattering}, in which a small Gaussian perturbation regularizes the pseudospectrum of a matrix, with high probability breaking it into disjoint components and allowing classical, divide-and-conquer eigensolvers to run successfully. Prior to their work, no way of accessing the benefits of divide-and-conquer’s natural parallelization was known in general. \\ \indent In this thesis, we extend the work of Banks et al.\ to the generalized eigenvalue problem -- e.g., $Av = \lambda Bv$ for matrices $A,B \in {\mathbb C}^{n \times n}$. Our main contributions can be summarized as follows. \begin{enumerate}[itemsep=0em] \item First, we show that pseudospectral shattering generalizes directly:\ randomly perturbing $A$ and $B$ has a similar regularizing effect on the pseudospectra of the corresponding matrix pencil $(A,B)$ with high probability. \item Building on pseudospectral shattering, we construct and analyze a fast, randomized, divide-and-conquer algorithm for diagonalizing $(A,B)$, which begins by randomly perturbing the inputs. \item Finally, we demonstrate that both

We discuss the cumulant approach to spectral properties of large random matrices. In particular, we study in detail the joint cumulants of high traces of large unitary random matrices and prove Gaussian fluctuation for pair-counting statistics with non-smooth test functions.

A combinatorial proof of Wigner’s Semicircle Law for the Gaussian Unitary Ensemble (GUE) is presented in this report. The distribution of eigenvalues of different samples of general Wigner matrices is shown to converge to the semicircle distribution, with the aid of histograms created in Python. The type of convergence that is shown is that of the averaged moments of the eigenvalue distribution of sample GUE matrices to the moments of the semicircle distribution, as the size of the matrices grow large. This is done by using a method known as the ‘method of moments’. The concepts of random matrices, Catalan numbers, mixed moments of standard Gaussian random variables, (non- crossing) pairings, Wick’s formula and permutation cycles are introduced in this method. The aim of this report is to provide a detailed proof of Wigner’s Semicircle Law in expectation, understandable for bachelor level mathematics students., Applied Mathematics

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments.

In this paper, a ridgelized Hotelling's T2 test is developed for a hypothesis on a large-dimensional mean vector under certain moment conditions. It generalizes the main result of Chen et al. [A regularized Hotelling's t2 test for pathway analysis in proteomic studies, J. Am. Stat. Assoc. 106(496) (2011) 1345-1360.] by relaxing their Gaussian assumption. This is achieved by establishing an exact four-moment theorem that is a simplified version of Tao and Vu's [Random matrices: universality of local statistics of eigenvalues, Ann. Probab. 40(3) (2012) 1285-1315] work. Simulation results demonstrate the superiority of the proposed test over the traditional Hotelling's T2 test and its several extensions in high-dimensional situations.

We study the distribution of the supremum of the Airy process with m wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of N non-intersecting Brownian bridges as N -> oo, where the first N - m paths start and end at the origin and the remaining m go between arbitrary positions. The distribution provides a 2m-parameter deformation of the Tracy-Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths starting and ending at the origin.We provide several descriptions of this distribution function: (i) A Fredholm determinant formula; (ii) A formula in terms of Painleve II functions; (iii) A representation as a marginal of the KPZ fixed point with initial data given as the top path in a stationary system of reflected Brownian motions with drift; (iv) A characterization as the solution of a version of the Bloemendal-Virag PDE (Probab. Theory Related Fields 156 (2013) 795-825; Ann. Probab. 44 (2016) 2726-2769) for spiked Tracy-Widom distributions; (v) A representation as a solution of the KdV equation. We also discuss connections with a model of last passage percolation with boundary sources., QC 20230105

We present the probability of two large gaps (intervals without eigenvalues) in the bulk and also in the edge scaling limit of the Gaussian Unitary Ensemble of random matrices., QC 20230720

Homochirality, i.e. the dominance across all living matter of one enantiomer over the other among chiral molecules, is thought to be a key step in the emergence of life. Building on ideas put forward by Frank and many others, we proposed recently one such mechanism in Laurent et al. (Laurent, 2021 Proc. Natl Acad. Sci. USA 118, e2012741118. (doi:10.1073/pnas.2012741118)) based on the properties of large out of equilibrium chemical networks. We showed that in such networks, a phase transition towards a homochiral state is likely to occur as the number of chiral species in the system becomes large or as the amount of free energy injected into the system increases. This paper aims at clarifying some important points in that scenario, not covered by our previous work. We first analyse the various conventions used to measure chirality, introduce the notion of chiral symmetry of a network and study its implications regarding the relative chiral signs adopted by different groups of molecules. We then propose a generalization of Frank's model for large chemical networks, which we characterize completely using methods of random matrices. This analysis is extended to sparse networks, which shows that the emergence of homochirality is a robust transition., SCOPUS: ar.j, info:eu-repo/semantics/published

Let FN and F be the empirical and limiting spectral distributions of an N × N Wigner matrix. The Cramér-von Mises (CvM) statistic is a classical goodness-of-fit statistic that characterizes the distance between FN and F in L2-norm. In this paper, we consider a mesoscopic approximation of the CvM statistic for Wigner matrices, and derive its limiting distribution. In the Appendix, we also give the limiting distribution of the CvM statistic (without approximation) for the toy model CUE. © Institute of Mathematical Statistics, 2022.

We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq \frac{\varepsilon}{\sqrt{n}} \right) \leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0. $$ This extends earlier results of Rudelson and Vershynin, and Rebrova and Tikhomirov by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result of Livshyts, where the matrix was required to have i.i.d. rows. Our model covers "inhomogeneus" matrices allowing different variances of the entries, as long as the sum of the second moments is of order $O(n^2)$. In the past advances, the assumption of i.i.d. rows was required due to lack of Littlewood--Offord--type inequalities for weighted sums of non-i.i.d. random variables. Here, we overcome this problem by introducing the Randomized Least Common Denominator (RLCD) which allows to study anti-concentration properties of weighted sums of independent but not identically distributed variables. We construct efficient nets on the sphere with lattice structure, and show that the lattice points typically have large RLCD. This allows us to derive strong anti-concentration properties for the distance between a fixed column of $A$ and the linear span of the remaining columns, and prove the main result.

Let $\theta_1,...,\theta_N$ be the angles of the eigenvalues of a $N\times N$ matrix sampled from either $C\beta E$, $SO(N)$, or $Sp(N)$. In this dissertation, we study the limiting distribution of the "pair dependent" linear statistic $\left(\frac{1}{\sqrt{L_N}}\right)\sum_{1\leq i\neq j\leq N} f(L_N(\theta_i-\theta_j))$, where $f$ is a sufficiently smooth function and $L_N$ is a positive, non-decreasing sequence such that $1\leq L_N<< N$. When $L_N=1$ (global case), the limiting distribution is an infinite sum of independent random variables, exponential in the case of $C\beta E$ and chi-squared distributed in the cases of $SO(N)$ and $Sp(N)$. When $L_N\to \infty$ (mesoscopic case), we are able to prove central limit theorems for each of the mentioned random matrix ensembles.

Let X be a x random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of X in terms of large deviations for large n, with p being fixed or with . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent random variables, which enables us to establish an application in statistical inference.

Matrix decompositions are crucial for analysing fixed matrices. Also when the matrix entries become random such decompositions are invaluable. For instance, they can yield factorisations of the measure that simplifies the analytical problems drastically. In this thesis we use those formulae to explore and link several topics in the common theme of decompositions, invariances and harmonic analysis in random matrix theory. Several applications of those decompositions will be studied. In this thesis, three types of matrix decompositions are considered: the LU decomposition, the spectral decomposition and the QR decomposition. For the LU decomposition, we will study two topics of parametrising correlation matrices and parametrising compact groups, where the former is an application of random matrix theory in statistics and the latter is motivated by the study of harmonic analysis of random matrices. The LU decomposition is utilised in these two topics due to the convenience it provides in studying positive definite matrices and principal minors. For the spectral decomposition and the QR decomposition, both of which contain at least one element drawn from a compact matrix group, it is natural to impose invariances to the random matrix models by letting the compact matrix group to be distributed according to the Haar measure. Specially, the spectral decomposition can be applied in the context of harmonic analysis of random matrices, where a link between the historical content of spherical transform and sums and products of random matrices will be shown. We will firstly further develop the theory of sums and products of random matrices using the idea of spherical transforms. Two topics involving the randomised Horn problems and the cyclic P\'olya ensembles will be studied as the applications. In the study of the QR decomposition, we will consider random matrix ensembles that have only a one-sided invariance such that the probability density function factorises into a conve

author: Andre Gladbach, Masterarbeit Universität Innsbruck 2021

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others. Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker-Campbell-Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero. We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials. (C) 2021 The Authors. Published by Elsevier Inc.

We address the old and widely debated question of the spectrum statistics of integrable quantum systems, through the analysis of the paradigmatic Lieb-Liniger model. This quantum many-body model of one-dimensional interacting bosons allows for the rigorous determination of energy spectra via the Bethe ansatz approach and our interest is to reveal the characteristic properties of energy levels in dependence of the model parameters. Using both analytical and numerical studies we show that the properties of spectra strongly depend on whether the analysis is done for a full energy spectrum or for a single subset with fixed total momentum. We show that the Poisson distribution of spacing between nearest-neighbor energies can occur only for a set of energy levels with fixed total momentum, for neither too large nor too weak interaction strength, and for sufficiently high energy. By studying long-range correlations between energy levels, we found strong deviations from the predictions based on the assumption of pseudorandom character of the distribution of energy levels.

Biorthogonal ensembles are generalisations of classical orthogonal ensembles such as the Laguerre or the Hermite ensembles. Local fluctuation of these ensembles at the origin has been studied, and determinantal kernels in the limit are described by the Wright generalised Bessel functions. The limit kernels are one parameter deformations of the Bessel kernel and the sine kernel for the Laguerre weight and the Hermite weight, respectively. We study transitions from these generalised Bessel kernels to the sine kernel under appropriate scaling limits in common with classical kernels.

The two-time distribution gives the limiting joint distribution of the heights at two different times of a local 1D random growth model in the curved geometry. This distribution has been computed in a specific model but is expected to be universal in the KPZ universality class. Its marginals are the GUE Tracy-Widom distribution. In this paper we study two limits of the two-time distribution. The first, is the limit of long time separation when the quotient of the two times goes to infinity, and the second, is the short time limit when the quotient goes to zero., QC 20201215

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix., QC 20200617

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

We study the unique solution m of the Dyson equation-m(z)(-1) = z1 - a + S[m(z)]on a von Neumann algebra A with the constraint Im m >= 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving linear operator on A. We show that m is the Stieltjes transform of a compactly supported A-valued measure on I Under suitable assumptions, we establish that this measure has a uniformly 1/3-Holder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of m near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [8] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner type matrices [15, 19] We also extend the finite dimensional band mass formula from [8] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases.

Hermitian operators with exact zero modes subject to non-Hermitian perturbations are considered. Specific focus is on the distribution of the former zero eigenvalues of the Hermitian operators. The broadening of these zero modes is found to follow an elliptic Gaussian random matrix ensemble of fixed size, where the symmetry class of the perturbation determines the behavior of the modes. This distribution follows from a central limit theorem of matrices and is shown to be robust to deformations.

This thesis is concerned with the analysis of the statistical behavior of Gram matrices and with the design and analysis of high-dimensional robust covariance matrix estimators using random matrix theory. The first and main part of this thesis, divided into two subparts, is devoted to the design and analysis of a large class of robust estimators of covariance matrices known as Maronna’s M-estimators. Under the Big Data paradigm, classical estimators such as the sample covariance matrix typically perform poorly in the recurring scenario where the number of samples is of the same order as the number of variables. Traditional estimators also suffer from possible contamination of the sample by outliers. This motivates the use of more sophisticated, robust covariance matrix estimators, the statistical behavior of which shall be investigated. First, we characterize the asymptotic behavior of regularized robust Maronna’s M-estimators by utilizing random matrix theory tools. We show that, in the large-dimensional regime and in the absence of outliers, these estimators all exhibit the same general behavior, after consistent estimation of their optimal regularization parameters. We then compare the theoretical performance of these estimators in the presence of outliers, providing insight on which estimator is best under different outlier contamination models. Second, we utilize Maronna’s M-estimators to improve a classical version of linear discriminant analysis. We show that, in a large-dimensional regime, the use of these estimators provides the same asymptotic classification error as non-robust methods when data is free from outliers. Simulations on both synthetic and real data sets attest to the higher performance of our proposed robust method relative to more conventional methods. In the last part of this thesis, we study the statistical properties of a 2x2 Gram matrix with an arbitrary variance profile. The underpinning random matrix model fundamentally departs from cla

Knutson introduced two families of reverse juggling Markov chains (single and multispecies) motivated by the study of random semi-infinite matrices over Fq. We present natural generalizations of both chains by placing generic weights that still lead to simple combinatorial expressions for the stationary distribution. For permutations, this is a seemingly new multivariate generalization of the inversion polynomial., QC 20190904

This thesis presents new results on spectral statistics of different families of large random matrices. Our main tool is certain types of {\textit{local estimates}} of the resolvents (or the Green's functions) of the random matrices, which are generally referred to as {\textit{local laws}}. Utilizing the standard approach developed over the last decade \cite{Yau_book} combined with a comparison method developed recently in \cite{Anisotropic}, we are able to prove (almost) optimal local laws for various random matrix ensembles with correlated and heavy-tailed entries. With these local laws, we establish the following three results.We first study the largest eigenvalues for separable covariance matrices of the form $\mathcal Q :=A^{1/2}XBX^*A^{1/2}$. Here $X=(x_{ij})$ is an $n\times N$ random matrix, whose entries are $i.i.d.$ random variables with mean zero and variance $N^{-1}$; $A$ and $B$ are respectively $n \times n$ and $N\times N$ deterministic non-negative definite symmetric (or Hermitian) matrices. Under a sharp fourth moment tail condition, we prove that the limiting distribution of the largest eigenvalues of $\mathcal Q$ is universal under an $N^{2/3}$ scaling, as long as ${n}/{N}$ converges to a finite $d \in (0, \infty)$ as $N\to \infty$. In particular, if $B=I$, then $\mathcal Q$ becomes the sample covariance matrix, which is one of the most fundamental objects of study in high-dimensional statistics. Our result provides the strongest edge universality result for large dimensional sample covariance matrices so far. Then we study the {\textit{eigenvector empirical spectral distribution}} (VESD)---an important tool in studying the limiting behavior of eigenvectors---for large separable covariance matrices. Under certain low moment assumptions, we prove an optimal convergence rate of the VESD to an anisotropic Mar{\v c}enko-Pastur law in the metric of Kolmogorov distance. Our results improve the suboptimal convergence rate in \cite{XYZ2013} under much more

In this paper we study the distribution of level crossings for the spectra of linear families A + lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some of their generalisations. Besides, we present intriguing numerical information about the monodromy distribution in case of linear families for the classical Gaussian ensembles of 3 x 3-matrices.

In this thesis the beta log-gas probability density function is discussed. It is shown that there is a strong link between this density function and Jacobi matrices. A change of variables exercise shows that the distribution of eigenvalues is exactly like the quadratic beta log-gas. The change of variables gives the normalization constant for the quadratic beta log-gas. Finally, it is made likely that the Jacobi matrix adheres to Wigners semicircle law, and that the beta log-gas is limited by the semicircle distribution., Bachelor Project, Applied Mathematics

Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15% erasures, and asked whether there exist frames that are robust against erasures of more than 50%. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least 1 + δ0 times the dimension for some δ0> 0. This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.

We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval [0, lambda] as lambda -> infinity. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine process., QC 20181001

In this paper, we consider the maximum of the Sine(beta) counting process from its expectation. We show the leading order behavior is consistent with the predictions of log-correlated Gaussian fields, also consistent with work on the imaginary part of the log-characteristic polynomial of random matrices. We do this by a direct analysis of the stochastic sine equation, which gives a description of the continuum limit of the Prufer phases of a Gaussian beta-ensemble matrix., QC 20181001

Can the behavior of a random matrix be improved by modifying a small fraction of its entries? Consider a random matrix A with i.i.d. entries. We show that the operator norm of A can be reduced to the optimal order O(n) by zeroing out a small submatrix of A if and only if the entries have zero mean and finite variance. Moreover, we obtain an almost optimal dependence between the size of the removed submatrix and the resulting operator norm. Our approach utilizes the cut norm and Grothendieck–Pietsch factorization for matrices, and it combines the methods developed recently by C. Le and R. Vershynin and by E. Rebrova and K. Tikhomirov.

Random-effects models are a popular tool for analysing total narrow-sense heritability for quantitative phenotypes, on the basis of large-scale SNP data. Recently, there have been disputes over the validity of conclusions that may be drawn from such analysis. We derive some of the fundamental statistical properties of heritability estimates arising from these models, showing that the bias will generally be small. We show that that the score function may be manipulated into a form that facilitates intelligible interpretations of the results. We go on to use this score function to explore the behavior of the model when certain key assumptions of the model are not satisfied - shared environment, measurement error, and genetic effects that are confined to a small subset of sites. The variance and bias depend crucially on the variance of certain functionals of the singular values of the genotype matrix. A useful baseline is the singular value distribution associated with genotypes that are completely independent - that is, with no linkage and no relatedness - for a given number of individuals and sites. We calculate the corresponding variance and bias for this setting.

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

In this paper, the Stein-Haff identity is established for positive-definite and symmetric random matrices belonging to the exponential family. The identity is then applied to the matrix-variate gamma distribution, and an estimator that dominates the maximum likelihood estimator in terms of Stein's loss is obtained. Finally, a simulation study is conducted in order to support the theoretical results.

Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze convergentie beschrijft is Wigners halve cirkel stelling. Het doel van dit project is het begrijpen van deze stelling en zijn bewijs om het vervolgens op een voor andere studenten toegankelijke manier te reproduceren., Applied Mathematics

We consider the problem of accurately recovering a matrix B of size M by M, which represents a probability distribution over M^2 outcomes, given access to an observed matrix of "counts" generated by taking independent samples from the distribution B. How can structural properties of the underlying matrix B be leveraged to yield computationally efficient and information theoretically optimal reconstruction algorithms? When can accurate reconstruction be accomplished in the sparse data regime? This basic problem lies at the core of a number of questions that are currently being considered by different communities, including building recommendation systems and collaborative filtering in the sparse data regime, community detection in sparse random graphs, learning structured models such as topic models or hidden Markov models, and the efforts from the natural language processing community to compute "word embeddings". Many aspects of this problem---both in terms of learning and property testing/estimation and on both the algorithmic and information theoretic sides---remain open. Our results apply to the setting where B has a low rank structure. For this setting, we propose an efficient (and practically viable) algorithm that accurately recovers the underlying M by M matrix using O(M) samples} (where we assume the rank is a constant). This linear sample complexity is optimal, up to constant factors, in an extremely strong sense: even testing basic properties of the underlying matrix (such as whether it has rank 1 or 2) requires Omega(M) samples. Additionally, we provide an even stronger lower bound showing that distinguishing whether a sequence of observations were drawn from the uniform distribution over M observations versus being generated by a well-conditioned Hidden Markov Model with two hidden states requires Omega(M) observations, while our positive results for recovering B immediately imply that Omega(M) observations suffice to learn such an HMM. This lower bound

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 limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators., QC 20170712

We consider level crossing in a matrix family H = H-0 + lambda V where H-0 is a fixed N x N matrix and V belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of lambda, for which we obtain a number of exact, asymptotic and approximate formulas.

Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A UBU{*} converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate 1/N in the bulk of the spectrum. (C) 2017 Elsevier Inc. All rights reserved.

In this paper, we are concerned with the independence test for k high-dimensional sub-vectors of a normal vector, with fixed positive integer k. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the k sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013)[8] and Jiang, Bai and Zheng (2013)[7], is satisfactory. Real data analysis is also used to illustrate our method.

In this thesis we will brush through fundamental multivariate statistical theory and then present MIMO-systems briefly in order to later calculate the channel capacity of a MIMO system. After theory has been presented we will then look at different properties of the channel capacity and then investigate a supposed MIMO system dataset and use standardized methods to verify it’s model. The properties of the limiting channel capacity uses the Marčenko-Pastur law. Therefore we will present some fundamental theorems and definitions of limiting spectral distribution of Wishart and Wigner matrices, and some fundamental properties they hold.

In this paper, we give a general recursive formula for , where denotes a real Wishart matrix. Formulas for fixed n, p are presented as well as asymptotic versions when i.e. when the so called Kolmogorov condition holds. Finally, we show application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical illustration using implementation of the main result is also performed.

In this note we review recent results on existence and uniqueness of solutions of infinite-dimensional stochastic differential equations describing interacting Brownian motions on Rd.

In probability theory, time series analysis, and signal processing, many identification and estimation methods rely on covariance estimates as an intermediate statistics. Errors in estimated covariances propagate and degrade the quality of the estimation result. In particular, in large network systems where each system node of the network gather and pass on results, it is important to know the reliability of the information so that informed decisions can be made. In this work, we design confidence regions based on covariance estimates and study how these can be used for spectral estimation. In particular, we consider three different confidence regions based on sets of unitarily invariant matrices and bound the eigenvalue distribution based on three principles: uniform bounds; arithmetic and harmonic means; and the Marcenko-Pastur Law eigenvalue distribution for random matrices. Using these methodologies we robustly bound the energy in a selected frequency band, and compare the resulting spectral bound from the respective confidence regions., QC 20161104

This thesis presents new results concerning the spectral properties of certain families of large random matrices. The overarching goal is to extend some well-known results for matrices with independent and identically distributed (iid) entries to random matrices whose entries are either dependent or non-identically distributed. Particular attention is given to the adjacency matrix of a random regular digraph. Making use of the method of exchangeable pairs, we establish concentration bounds for codegrees and edge densities of induced subgraphs of a random regular digraph. We apply these bounds along with other coupling techniques to prove that the associated adjacency matrix is invertible with high probability, assuming a mild growth condition on the degree of the graph. We next prove lower tail estimates for the smallest singular value of matrices with independent but non-identically distributed entries, including matrices with most entries set to zero deterministically. In particular, we show that for small diagonal perturbations of centered random matrices with independent entries of bounded variance, the smallest singular value is of inverse-polynomial order with high probability. Finally, we prove that the circular law holds for a random matrix obtained from the adjacency matrix of a dense random regular digraph by multiplying each entry by a random sign.

In a series of papers and MPhil theses, researchers have studied the group randomness property associated with sequences based on linear codes over finite fields, mutually unbiased bases (MUBs) and approximately mutually unbiased bases (AMUBs). In particular, they found conditions such that the sequences satisfy the group randomness properties with respect to the Marchenko-Pastur distribution and the Wigner semicircle distribution in terms of the dual distance and compressed sensing properties. In this thesis, we discover similar conditions such that sequences based on a Z4-linear code possess the same properties with respect to the two distributions.