# Tagged Questions

Statistics of spectral properties of matrix-valued random variables.

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### Is there a stochastic process with zero mean but nonzero value in each time instant? [on hold]

Background: In a signal processing application, we want to randomized a signal by 'whiten' it, that is to say we want the 'whitened' signal with zero mean. However, there is a constraint on the signal ...
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### Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
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### On the least singular value of a random matrix and its minors

Let $A$ be a $n \times n$ random matrix with entries i.i.d from the continuous uniform distribution U([-1,1]). Is it true that the least singular value of $A$ and all its minors is greater than ...
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### Smallest Singular Value of a Random Matrix with Dependent Entries

Overview I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
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### Upper bound involving random orthogonal projection

Let $R$ be an $n\times N$ random matrix with i.i.d. standard Gaussian entries, $n<N$, and let $M:=(RR^T)^{-1/2}R$. Let $u,v\in \mathbb{R}^N$ non-random and s.t. $u^Tv=0$ and $\|u\|>\|v\|$. I ...
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### Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...
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### Normalizing Entries In Defining Random Matrices (Wigner Matrix)

In the definition of Wigner Matrix (a certain type of random Matrices) we take to independent family of i.i.d zero mean distributions $\{Z_{i,j}\}_{1<i<j}$ and $\{Y_{i}\}_{1\leq i}$ and then the ...
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### Estimating a Selberg-type integral (or a Fredholm determinant)

I am concerned with the asymptotical behavior of integrals like this for large $n$ $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$ ...
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### What is the reason the eigenvalues of GUE and CUE matrices tend locally to the same distribution?

It's well known in random matrix theory that locally the eigenvalues of a random matrix from the Gaussian unitary ensemble tend to a sine-kernel determinantal point process. Likewise, locally the ...
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### Can the GUE be thought of as a uniform point in a high-dimensional polytope

I have thought about this question for a long time and could only find partial answers. The Gaussian Unitary Ensemble (or GUE) is the eigenvalues of a random Hermitian matrix with complex Gaussian ...
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### Symmetry type of non-cohomological automorphic forms

By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...
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### When does a d.r.v. take a value very close to the mean? [closed]

Suppose that $X$ is a discrete random variable with values $x_{1},x_{2},\ldots,x_{n}$ (not known precisely, but there is some information available about the mean and variance). Is there a result ...
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### Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows: Pick $k$ distinct numbers out of numbers ...
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### Exact growth rate of Longest Increasing Subsequence expectation

Let $S_n$ be the symmetric group, $\pi\in S_n$ a uniformly random permutation and $L_n:=L_n(\pi)$ denoting the length of the longest increasing subsequence (LIS). We know that ...
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### Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
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### Statistics of strongly connected components in random directed graphs

I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for. ...
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### An expectation of the product of random unitaries

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$ Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
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### Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
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### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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### Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$. We would like to bound the following expectation on the trace norm ...
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### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$. Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s. Does ...
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### Restricted singular values of Wishart matrices

This is an extended question of Restricted singular values of random matrix. It is well-known that the smallest singular value of a $p \times \frac{p}{2}$ matrix consisting of i.i.d. ...
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### Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
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### Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
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### Moments of random special unitary matrices

This should be both well-known and probably easy, but I was wondering if the following is known (and, if so, how to easily calculate the thing or where to read about how to calculate it): what is ...
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### Inverse of the covariance of the estimate of a covariance

I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
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### Quaternion Wishart matrices of half-integer dimension?

For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution ...
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It is well known that any $2\times 2$ unitary matrix $\mathbf{U}$ can be parametrized as $$\mathbf{U}=\begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{\mathrm{j}\beta_1}\end{pmatrix} \begin{pmatrix} ... 0answers 89 views ### Is there an improvement for the Schur-Horn inequalities for positive semi-definite matrices? By the Schur-Horn inequality I am thinking of the statement that for any Hermitian matrix H its diagonal n-tuple (H_{11},H_{22},..,H_{nn}) for any choice of basis lies in the convex hull of the ... 0answers 84 views ### “Semiclassical approximation” in random matrix theory I am reading Planar Diagrams by Brezin, Parisi, Itzykson and Zuber. If you specialize the discussion in section 5 there we seem to derive the eigenvalue distribution of N \times N random Hermitian ... 1answer 120 views ### Restricted singular values of random matrix Let X \in \mathbb{R}^{p\times p} be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of X follow the Marchenko-Pastur law. Now let's ... 1answer 186 views ### Distribution of the permanent modulo p We know that the order of SL_n({\mathbb F}_p) is$$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$Dividing by p^{n^2}, we deduce the probability that \det takes the value 1 over M_n({\mathbb ... 3answers 309 views ### Determinant of matrix from set {-1, 1} [closed] Let A \in \mathbb{R}^{11 \times 11} and it's elements are form set \{ -1,1 \}. \mathbb{P}(-1) = \mathbb{P}(1) = 0.5. What is a probability to get such a matrix, that \det A > 4000? I have ... 3answers 391 views ### An infinite product associated with random matrices Motivation Let {\mathbb F}_q be the field with q (a power of some prime number) elements. Then the order of GL_n({\mathbb F}_q) is$$(q^n-1)(q^n-q)\cdots(q^n-q^{n-1}).The fact that this ... 1answer 137 views ### Expected value of the inverse of a random, truncated Haar matrix Let Q be a (say 4x4) unitary matrix, distributed according to the Haar distribution. Denote the upper left 2x2 submatrix of Q as Q_{1:2,1:2}. I am interested in the following expectation: E(I ... 0answers 69 views ### Random matrices whose limit gives exact Wigner surmise Let M come from an ensemble of N\times N matrices. The Wigner surmise is density function p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}. From a random matrix point of view, we can write ... 1answer 149 views ### Closure of random rotations Are matrix Fisher random variables closed under multiplication? For those unfamiliar with the jargon, let me unpack the terms above and repose my question. This is a question about probability ... 1answer 75 views ### Asymptotic eigenvalue analysis for a sparse random matrix We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix: H=\{h_{ij}\}_{N_r\times N_t}, where each entry h_{ij} is with a probability p to obey the ... 0answers 79 views ### Stricter Notion of Crossing in a Partition Let k be an integer. Traditionally a partition \pi=V_1\cup \dots \cup V_n of the set [k]:=\{1,\dots, k\} is called crossing when there exist a,c\in V_i and b,d\in V_j\not= V_i such that ... 1answer 81 views ### What is the relation between the eigenvectors of a sample covariance matrix and those of the true covariance matrix? As is known, the covariance matrix of a set of random vectors \{\mathbf{x}_i\}_{i=1}^N can be estimated by their sample covariance matrix: \mathbf{\hat ... 0answers 78 views ### Bounding Random Quadratic Gauss sums I'm interested in seeing whether the following is true. Assume u is uniform on [0,1] and |\epsilon_k|=1 for all k=1,2,\ldots,n. We have \begin{align*} ... 1answer 450 views ### What is known about the distribution of eigenvectors of random matrices? Let A be a real asymmetric n \times n matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of A? In particular: How are individual eigenvectors ... 0answers 38 views ### Distribution of Wishart Sample Eigenvalues for Multiple Roots I am interested in finding an asymptotic approximation to the latent roots l_1>\dots>l_p of a white noise Wishart matrix nS\sim W_p(n,I) as n\rightarrow\infty (where p is fixed). In ... 1answer 137 views ### Strictly positive solutions of a random linear system Suppose B\in\mathbb{R}^{m\times n} is a random binary matrix with i.i.d entries and c\in \mathbb{R}^m is a strictly positive vector, that is c_i>0 for i=1,2,\cdots m. Also assume m<n, ... 4answers 849 views ### Moments of the trace of orthogonal matrices Let O_n be the (real) orthogonal group of n by n matrices. I am interested in the following sequence which showed up in a calculation I was doinga_k = \int_{O_n} (\text{Tr } X)^k dX where ...
Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let \begin{equation*} p = p\left(\kappa,d\right) := ...