Vladislav Kargin
Suppose that $X_1,\...,X_n,\...$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let $Π_n=X_n\... X_1$. It is known that $n^{-1}\log |Π_n|$ converges to a nonrandom limit. We prove that under certain additional assumptions on matrices $X_i$ the speed of convergence to this limit does not decrease when the size of matrices, $N$, grows.
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Let $(ξ_j)_{j\ge1} $, be a non-stationary Markov chain with phase space $X$ and let $\mathfrak{g}_j:\,X\mapsto\mathrm{SL}(m,\mathbb{R})$ be a sequence of functions on $X$ with values in the unimodular group. Set $g_j=\mathfrak{g}_j(ξ_j)$ and denote by $S_n=g_n\ldots g_1$, the product of the matrices $g_j$. We provide sufficient conditions for exponential growth of the norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. This generalizes the classical theorem of Furstenberg on the exponential growth of products of independent identically distributed matrices as well as its ext…
Complete pivoting growth of butterfly matrices and butterfly Hadamard matrices
本文研究了完全消元下蝴蝶矩阵的增长问题，提出了一种能精确计算增长的密集随机蝴蝶矩阵，并扩展了之前关于蝴蝶矩阵使用部分消元和王消元的精确增长计算结果，同时提出了一种用蝴蝶矩阵构造随机Hadamard矩阵的新方法。
In this note, we prove Gaussian field convergence of fluctuations of eigenvalues of random normal matrices in the interior of a quantum droplet.
Growth factors of random butterfly matrices and the stability of avoiding pivoting
本文从理论和数值角度探讨随机蝴蝶矩阵预处理对高斯消元增长因子的影响，发现其比常见预条件子更能有效抑制大增长因子，并确定了某子类增长因子的完整分布。
A few remarks on the operator norm of random Toeplitz matrices
本文给出了算子范数或随机Toeplitz矩阵的几乎必然性质的一些结果，包括范数的大数定律，用其期望归一化（在独立同分布中）。case）。作为工具，我们提出了一些集中的经验过程的上确界的不等式，这是最近的结果Einmahl和李的改进。
In this short note we address a gaussian property of normal vectors in random non-Hermitian matrices. The approach uses a simple geometric and comparison technique.
We prove estimates for the expected value of operator norms of Gaussian random matrices with independent and mean-zero entries, acting as operators from $\ell^m_{p^*}$ to $\ell_q^n$, $1\leq p^* \leq 2 \leq q \leq \infty$.
We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element $a$, by the addition of a polynomially vanishing Gaussian Ginibre matrix, forces the empirical measure of eigenvalues to converge to the Brown measure of $a$.
The limit of the operator norm for random matrices with a variance profile
研究具有方差剖面的对称随机矩阵，证明其算子范数收敛于极限谱分布支撑集的最大值，仅需4阶矩或4+ε阶矩条件，并推广至多种随机矩阵类型。
Growth factors of orthogonal matrices and local behavior of Gaussian elimination with partial and complete pivoting
该研究聚焦部分与完全选主元高斯消元法的增长因子，分析二者在同一线性系统及正交矩阵等子类上的表现，并通过增长差异矩阵探究小增长策略在初始矩阵邻域的主导性，补充了相关算法的行为理论。
We investigate the spectral norms of symmetric $N \times N$ matrices from two pseudo-random ensembles. The first is the pseudo-Wigner ensemble introduced in "Pseudo-Wigner Matrices" by Soloveychik, Xiang and Tarokh and the second is its sample covariance-type analog defined in this work. Both ensembles are defined through the concept of $r$-independence by controlling the amount of randomness in the underlying matrices, and can be constructed from dual BCH codes. We show that when the measure of randomness $r$ grows as $N^ρ$, where $ρ\in (0,1]$ and $\varepsilon > 0$, the norm of the matrices i…
Deviation Inequalities for the Spectral Norm of Structured Random Matrices
本文针对非高斯结构化随机矩阵，利用典型过程上确界及强弱矩比较建立谱范数 p 阶矩最优界，并推出矩形矩阵最小奇异值的偏差不等式。
Operator $\ell^\infty \to \ell^\infty$ norm of products of random matrices
本文研究了随机矩阵乘积的 $\ell^\infty \to \ell^\infty$ 范数，证明了其渐近行为与矩阵大小和乘积数量的关系。
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。
This paper investigates an upper bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients. However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance are not equal or for matrix for which coefficients are not independent. This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator no…
We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.
Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices
本研究证明随机矩阵部分主元高斯消元法出现大增长因子的概率至少为逆拟多项式量级，首次给出非平凡下界，并挑战了平滑与平均情况分析中的标准猜想。
In this work, we consider symmetric random Toeplitz matrices $T_n$ generated by i.i.d. zero mean random variables ${X_k}$ satisfying the moment conditions: $E|X_k|^2=1$ and $\E|X_1|^n \le n^{\sqrt{n}}$ for all $n\ge 3$. We prove that the largest eigenvalue of $T_n$ scaled by $\sqrt{n log(n)}$ converges almost surely to $1$.
Spectral norm bound for the product of random Fourier-Walsh matrices
本文研究了归一化随机 Fourier-Walsh 矩阵乘积的谱范数界，在维数趋于无穷且期望矩阵乘积算子范数趋于零时，识别出一种有趣的多项式缩放机制。
Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of X_N, for which the operator norm of P(X_N, Y_N, Y_N^*) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y_N and of the polynomials P we get for a large class of matrices the "no eigenvalues outside a neighborhood of the limiting spectru…
Density of eigenvalues of random normal matrices
通过限制正规矩阵在有界集上有谱，使得Wiegmann和Zabrodin发现的随机正规矩阵与共形映射之间的关系变得严格.证明了对于适当的位势类，特征值的渐近密度在简单平滑曲线的内部区域内是一致的。
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。
On the spectral norm of Rademacher matrices
本文研究非齐次加权 Rademacher 矩阵平均谱范数的非渐近界，证明相关猜想对任意 n×n 矩阵在 log log log n 因子内成立，并指出 {0,1} 系数矩阵可消除该三重对数因子。
Simple Norm Bounds for Polynomial Random Matrices via Decoupling
该研究提出一种基于解耦与线性化思想的新方法，可简单推导元素为独立实值随机变量低次多项式的随机矩阵的范数界，迭代后能以初等证明复现此前需复杂技术得到的诸多结果。
Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms
本文从两个方向改进并扩展了由随机向量诱导的复矩阵范数理论，给出了埃尔米特矩阵上弱酉不变范数分类的简单证明，并在无需Lewis框架的情况下将文献[7]主定理的指数范围从 $d\geq 2$ 扩展至 $d \geq 1$。
In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\{ \pm1 \}$ with probability $1/2$) and prove $$2^{-\left(0.5+o(1)\right)n^2} \le P\left(M_n \text{ is normal}\right) \le 2^{-(0.302+o(1))n^{2}}. $$ We conjecture that the lower bound is sharp.
We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius.
Concentration of norms and eigenvalues of random matrices
证明了矩形随机矩阵的$\ell_p^n$算子范数和自伴随机矩阵的特征值的集中结果。随机矩阵，我们认为有界的项目是独立的，到一个可能的自伴约束。我们的结果是基于Talagrand关于积空间的一个等周不等式。
On Spectral Norm of Large Band Random Matrices
研究了在主对角线宽度为b的带外元素为零的N维厄米特随机矩阵的谱范数。在这个频带内，元素由高斯中心联合独立随机变量给出，方差为1/b阶。我们表明，如果b趋于无穷快于三次幂的log N，那么谱范数是有界的概率为1。
Growth in products of matrices: fastest, average, and generic
本文研究2x2矩阵乘积中最大条目随长度增长的最快、平均与典型行为，部分回答最快增长，完整回答平均增长，并提出非负矩阵情形下Lyapunov指数上界的简单方法。
We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erdős-Rényi and $d$-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the Matrix Dyson Equation as developed in [Aj…
We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter's positivity theorem for the complete homogeneous symmetric polynomials.
In contemporary applied and computational mathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent random matrices. This quantity is controlled by the norm of the expected square of the random matrix and the expectation of the maximum squared norm achieved by one of the summands; there is also a weak dependence on the dimension of the random matrix. The purpose of this paper is to give a complete, elementary proof of this important, but underappreciated, inequality.
In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution functions and the Airy_2 process. The link is however sometimes fragile. For example, the connection between the eigenvalues in the Gaussian Orthogonal Ensembles (GOE) and growth on a flat substrate is restr…
Random matrices and determinantal processes
我们调查最近的结果行列式过程，随机增长，随机平铺和它们的关系，随机矩阵理论。
We study random normal matrix models whose eigenvalues tend to be distributed within a narrow "band" around the unit circle of width proportional to $\frac1n$, where $n$ is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.
On the distance to low-rank matrices in the maximum norm
本文利用 Hanson--Wright 不等式改进最大范数下具有不相干列空间和行空间的矩阵到低秩矩阵的距离估计，并通过数值实验验证理论界对交替投影方法近似误差的描述效果。
We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.
In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does…
Random Matrices, the Ulam Problem, Directed Polymers & Growth Models, and Sequence Matching
在这些课堂讲稿中，我将对4个不同问题的一些常见方面进行教学介绍：（i）随机矩阵（ii）最长递增子序列问题（也称为Ulam问题）（iii）随机介质中的定向聚合物和（1+1）维中的生长模型以及（iv）一对随机序列的对齐问题。每一个问题本身几乎都是一个子领域，在这里我将只讨论其中的一些具体方面。在过去的几十年里，这四个问题几乎都是独立研究的，但直到最近几年，才发现一条共同的线索将它们联系起来。特别是，所有这些都有一个共同的极限概率分布，称为Tracy-Widom分布，它描述了随机矩阵最大特征值的渐近概率分布。我将在这里提到，没有数学推导，在过去几年中发现的一些美丽的结果。然后，我将考虑两个特定的模型（a）弹道沉积生长模型和（b）称为伯努利匹配模型的序列比对模型，并详细讨论如何在这些模型中精确推导Tracy-Widom定律。这些讲座的重点是如何将一个模型映射到另一个模型。最后讨论了一些尚待解决的问题。
The link between a particular class of growth processes and random matrices was established in the now famous 1999 article of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. During the past ten years, this connection has been worked out in detail and led to an improved understanding of the large scale properties of one-dimensional growth models. The reader will find a commented list of references at the end. Our objective is to provide an introduction highlighting random matrices. From the outset it should be emphasized that this connecti…
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