Shambhu Nath Maurya, Koushik Saha
In this article, we study the fluctuation of linear eigenvalue statistics of symmetric circulant matrices $(SC_n)$ with independent entries which satisfy some moment conditions. We show that $\frac{1}{\sqrt{n}} \Tr ϕ(SC_n)$ obey the central limit theorem (CLT) type result, where $ϕ$ is a nice test function.
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Fluctuation of linear eigenvalue statistics of reverse circulant matrices with independent entries
该文研究满足矩条件的独立元素反向循环矩阵的线性特征值统计量涨落，证明其经标准化后服从中心极限定理型结果，丰富了随机矩阵理论相关结论。
We show that the linear statistics of eigenvalues of circulant matrix obey the Gaussian central limit theorem for a large class of input sequences.
In this paper we discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to $\infty $. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.
In this article we study the fluctuation of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant and Hankel matrices. We show that the linear spectral statistics of these matrices converges to the Gaussian distribution in total variation norm when the matrices are constructed using i.i.d. normal random variables. We also calculate the limiting variance of the linear spectral statistics for circulant, symmetric circulant and reverse circulant matrices.
This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.
Patterned matrices with random walk entries
研究连续时间随机游走条目模式化随机矩阵的非交换概率极限，发现Wigner矩阵收敛到自由布朗运动，循环矩阵收敛到布朗运动，iid和椭圆矩阵收敛到圆和椭圆变量。
Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread -- the $2k$th moment of the limit equals a weighted sum over different types of pair-partitions of the set $\{1, 2, \ldots, 2k\}$ and are universal. Some results are also known for the sparse case. In this paper we generalise these results by relaxing…
This paper studies the asymptotic behavior of eigenvalues of random abelian G-circulant matrices, that is, matrices whose structure is related to a finite abelian group G in a way that naturally generalizes the relationship between circulant matrices and cyclic groups. It is shown that, under mild conditions, when the size of the group G goes to infinity, the spectral measures of such random matrices approach a deterministic limit. Depending on some aspects of the structure of the groups, whether the matrices are constrained to be Hermitian, and a few details of the distributions of the matrix…
Limiting spectral distribution of Toeplitz and Hankel matrices with dependent entries
本文研究具有相依元素的对称 Toeplitz 和 Hankel 矩阵的极限谱分布，基于矩方法给出了 LSD 矩序列的显式表达式，并推广至反循环和对称循环矩阵。
A Class of Long Range Circulant Random Walks on $\mathbb{Z}_q^d$
研究在循环群直积 $\mathbb{Z}_q^d$ 上的一类长程随机游走，通过循环矩阵更新步长，分析其结构、转移函数的谱展开，并推广到环面过程，利用多元Krawtchouk多项式得到分组模型的中心极限定理。
Simple Extension of Halász's Result
本文受伯努利分解启发，给出 Halász 反集中结果的简单推广，并应用于随机循环矩阵，得到其奇异概率的上界，且允许元素非同分布。
Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments and $p/n \to y\neq 0$, then the limiting spectral distribution (LSD) of $\frac{1}{n}S$ converges to a Mar$\check{\text{c}}$enko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of $S$ in probability or almost surely, and in particular, many of the above results follow as special cases. At the …
Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices
本文研究元素为独立 Brown 运动的 RC_n(t) 与 SC_n(t) 矩阵，证明多项式检验函数下其线性特征值统计量的时间相关涨落随 n→∞ 具有过程收敛性。
Norms of Randomized Circulant Matrices
该研究针对非齐次独立元素随机矩阵的算子范数双侧界问题，给出Rademacher矩阵下界并提出相关猜想，验证了随机n×n循环矩阵下猜想在相差log log n因子时成立，且特定假设下可消除该双重对数项。
This article deals with the limiting spectral distribution and joint convergence of reverse circulant and symmetric circulant matrices with independent entries. These results are already proved in articles Bose and Sen (2008) \cite{bose_sen_LSD_EJP}, and Bose, Hazra and Saha (2011) \cite{bose_saha_patter_JC_annals}, respectively. But this article provides alternate proofs of these results which are shorter than the existing proofs. Our method is mainly based on moment method.
This paper deals with symmetric random matrices whose upper diagonal entries are obtained from a linear random field with heavy tailed noise. It is shown that the maximum eigenvalue and the spectral radius of such a random matrix with dependent entries converge to the Frechét distribution after appropriate scaling. This extends a seminal result of Soshnikov(2004) when the tail index is strictly less than one.
Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one…
Suppose $X_p$ is a real $p \times n$ matrix with independent entries and consider the (unscaled) sample covariance matrix $S_p=X_pX_p^T$. The Marchenko-Pastur law was discovered as the limit of the bulk distribution of the sample covariance matrix in 1967. There have been extensions of this result in several directions. In this paper, we consider an extension that handles several of the existing ones as well as generates new results. We show that under suitable assumptions on the entries of $X_p$, the limiting spectral distribution exists in probability or almost surely. The moments are descri…
Extreme eigenvalues of Laplacian random matrices with Gaussian entries
研究高斯元素拉普拉斯随机矩阵最大特征值的极限分布，发现其极端特征值呈泊松统计，最大特征值经平移缩放后收敛于冈贝尔分布，且其中心化项与最大对角元素不同。
We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_n$ be a Hermitian random matrix of size $n\times n$ that can be split into independent blocks of the size at most $d_n=o(n^2)$. We prove that under some mild conditions on the distribution of the entries of $X_n$, the empirical spectral measure of $X_n$ concentrates around its mean. The main theorem is a strengthening of a recent result by Kemp and Zimmerman, where the size of blocks grows as $o(\log n)$. As an application, we are able to upgrade th…
In this article, we consider random Wigner matrices, that is symmetric matrices such that the subdiagonal entries of Xn are independent, centered, and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This …
We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric …
Semicircular law with a few independent entries in a random matrix
本文针对随机矩阵领域的半圆律问题，打破Wigner矩阵需O(n²)个独立随机变量的固有认知，构造出仅含O(n)个独立随机变量、经验谱分布可任意接近半圆律的矩阵。
This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices $\frac1N(D_n\circ X_n)(D_n\circ X_n)^*$, studied in Girko 2001. Here, $X_n=(x_{ij})$ is an $n\times N$ random matrix consisting of independent complex standardized random variables, $D_n=(d_{ij})$, $n\times N$, has nonnegative entries, and $\circ$ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of $X_n$ and $D_n$ which are different from those in Girko (2001), which include a Lindeberg condition on the entries of $D_n\circ X_n$, as well as a bound on …
We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distri…
We study linear eigenvalue statistics of band Hankel matrices with Brownian motion entries. We prove that, the centred, normalized linear eigenvalue statistics of band Hankel matrices obey a central limit theorem (CLT) type result. We also discuss the convergence of linear eigenvalue statistics of band Hankel matrices with independent entries for odd power monomials. Our method is based on trace formula, moment method and some results of process convergence.
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.
We consider $n\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $κ_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\to c\in[0,\infty)$ with the same variance as for Gaussian …
Largest eigenvalue of positive mean Gaussian matrices
本文研究正均值相关高斯对称随机矩阵最大特征值的波动，在协方差核绝对可和条件下证明其中心化后依分布收敛于正态分布，推广了独立条目Wigner矩阵的已知结果。
We study the statistics of the largest eigenvalues of $p \times p$ sample covariance matrices $Σ_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p \times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-α}$, $α>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{μ+1}$, $0 \leq μ\leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $α<2(1+μ^{-1})$ and converge to a constant in the case $α>2(1+μ^{-1})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the …
Precise asymptotics for the norm of large random rectangular Toeplitz matrices
该研究分析了独立元素的大型矩形随机Toeplitz和circulant矩阵的谱范数渐近行为，推导了对应极限结果，通过蒙特卡洛实验验证有限样本性能，并在高斯情形下开展高阶渐近分析提出极限分布猜想。
We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<α<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.
The smallest singular value of large random rectangular Toeplitz and circulant matrices
本文研究大型随机矩形Toeplitz与循环矩阵的最小奇异值，证明其最小特征值收敛于零，并给出收敛速率的下界（快于多对数但慢于多项式）及矩形循环矩阵的多项式上界。
We consider random matrices of the form $H = W + λV$, $λ\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\ entries that are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $λ\sim 1$, so that the eigenvalues of $W$ and $λV$ are typically of the same order. Further, we assume that the density of the entries of $V$ is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $λ_+\…
Fluctuations of Eigenvalues for Generalized Patterned Gaussian Random Matrices
本文研究介于 Wigner 模型与 Toeplitz、Hankel、循环等模式化矩阵之间的非齐次高斯随机矩阵，利用组合方法和非渐近理论证明其线性特征值统计量的波动近似高斯。
In this text, we consider an N by N random matrix X such that all but o(N) rows of X have W non identically zero entries, the other rows having lass than $W$ entries (such as, for example, standard or cyclic band matrices). We always suppose that 1 << W << N. We first prove that if the entries are independent, centered, have variance one, satisfy a certain tail upper-bound condition and W >> (\log N)^{6(1+α)}, where αis a positive parameter depending on the distribution of the entries, then the largest eigenvalue of X/\sqrt{W} converges to the upper bound of its limit spectral distribution, th…
We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic calculus with respect to the Young's integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H in (1/2,1), we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian …
In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semic…
Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period…
In this paper, we study the condition number of a random Toeplitz matrix. Since a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategy to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding, we can break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. Among our results, we show the condition …
We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid entries with variance 1/N. Under mild assumptions, as N grows, the empirical distribution of the eigenvalues of A+Y converges weakly to a limit probability measure βon the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where th…
Bulk universality for complex eigenvalues of real non-symmetric random matrices with i.i.d. entries
本文研究了具有独立同分布实元素的非 Hermitian 矩阵系综，证明其在远离实轴的体中的 $k$ 点相关函数收敛于一个 universal 极限。
Spectrum of Random Matrices with Exploding Moments
研究具有爆炸矩矩阵模型的线性特征值统计的中心极限定理，采用渐近 Wick 公式证明椭圆、中心对称、循环和互相关块矩阵的 CLT。
We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n\to \i…
We establish a large-deviations principle for the largest eigenvalue of a generalized sample covariance matrix, meaning a matrix proportional to $Z^T ΓZ$, where $Z$ has i.i.d. real or complex entries and $Γ$ is not necessarily the identity. We treat the classical case when $Z$ is Gaussian and $Γ$ is positive definite, but we also cover two orthogonal extensions: Either the entries of $Z$ can instead be sharp sub-Gaussian, a class including Rademacher and uniform distributions, where we find the same rate function as for the Gaussian model; or $Γ$ can have negative eigenvalues if $Z$ remains Ga…
This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval $[-π,π]$. Various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases. The rate of growth of the maximum eigenvalue is examined and shown to be no greater than $O(\log N)$ and no slower than $O(\log N/\log\log N)$ where $N$ is the dimension of the matrix. …
In this paper we study ensembles of random symmetric matrices $\X_n = {X_{ij}}_{i,j = 1}^n$ with dependent entries such that $\E X_{ij} = 0$, $\E X_{ij}^2 = σ_{ij}^2$, where $σ_{ij}$ may be different numbers. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg condition holds we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law.
The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.
In this paper we consider Wigner random matrices -- symmetric n by n random matrices whose entries are independent identically distributed real random variables. We prove that the probability distribution of one or several eigenvalues close to the center of the spectrum does not depend on the probability distribution of the entries of the matrix and is the same as for the Gaussian Orthogonal Ensemble. We make only mild smoothness assumptions on the probability distribution of the entries and assume that the probability distribution of the entries decays polynomially with sufficiently large pow…
Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will have a real eigenvalue in a particular interval? The purpose of this paper is to resolve these questions, once they are made suitably precise, in the setting of 2x2 matrices.
Limiting spectral laws for sparse random circulant matrices
本文研究稀疏随机循环矩阵的极限谱定律，证明其经验谱分布期望弱收敛的充要条件为群元素阶分布弱收敛，且概率收敛等价于阶分布为狄拉克质量，同时给出行列式的高概率渐近形式。
Eigenvalue degeneracy in sparse random matrices
本文通过随机二分图匹配理论研究稀疏随机矩阵特征值退化问题，发现特征值向原点累积导致退化概率非零。
We consider sample covariance matrices of the form $\mathcal{Q}=(Σ^{1/2}X)(Σ^{1/2} X)^*$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $Σ$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $Σ$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is giv…
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled …
The condition number of a random banded Toeplitz matrix is typically large
本文研究了带状Toeplitz矩阵的条件数，发现带宽对称性决定其条件数好坏。
The real polynomial eigenvalue problem is well conditioned on the average
研究随机矩阵集合中多项式特征值的平均条件数，证明高斯矩阵定义的多项式特征值问题在平均意义上条件良好。
Consider a $N\times n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary gaussian random field. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $n\to \infty$ and $\frac Nn \to c \in (0,\infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs …
Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly $k$ real eigenvalues as a determina…
We investigate the spectral distribution of random matrix ensembles with correlated entries. We consider symmetric matrices with real valued entries and stochastically independent diagonals. Along the diagonals the entries may be correlated. We show that under sufficiently nice moment conditions the empirical eigenvalue distribution converges almost surely weakly to the semi-circle law.
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