Statistics of spectral properties of matrix-valued random variables.

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17 views

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 ...
3
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0answers
44 views

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 ...
0
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0answers
21 views

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 ...
2
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2answers
156 views

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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1answer
142 views

Upper bound for a Selberg-type integral over a rectangular region

(Cross-posted from math-SE). I am trying to estimate the values of the following integral for large $n$, $$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq ...
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1answer
77 views

Norm of matrix with randomly deleted entries

Let $A$ be an $n \times n$ matrix with real entries and let $B$ be the random matrix whose $(i,j)$ entry is $$B_{i,j}=v_{i,j}A_{i,j}$$ where the $v_{i,j}$ are i.i.d Bernoulli random variables with ...
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2answers
76 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
7
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0answers
73 views

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 ...
4
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0answers
59 views

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 ...
5
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1answer
173 views

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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1answer
97 views

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 ...
6
votes
4answers
356 views

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 ...
2
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0answers
72 views

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 ...
3
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2answers
254 views

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 ...
4
votes
2answers
143 views

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. ...
2
votes
1answer
176 views

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 ...
5
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2answers
346 views

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 ...
4
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229 views

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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0answers
83 views

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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1answer
184 views

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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0answers
32 views

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. ...
5
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0answers
65 views

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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0answers
38 views

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 ...
3
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1answer
132 views

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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0answers
30 views

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 ...
2
votes
1answer
109 views

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 ...
2
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1answer
78 views

Distribution of the $\alpha$-parameter of a $2\times 2$ Haar-distributed, unitary matrix

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} ...
0
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0answers
86 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 ...
2
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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 ...
2
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1answer
118 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 ...
6
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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 ...
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3answers
308 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 ...
6
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3answers
389 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 ...
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1answer
131 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 ...
2
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0answers
65 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 ...
6
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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 ...
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1answer
74 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 ...
2
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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 ...
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1answer
79 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 ...
0
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0answers
76 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*} ...
9
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1answer
411 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 ...
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0answers
37 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 ...
3
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1answer
136 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 $m<n$, ...
14
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4answers
846 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 doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
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0answers
69 views

Bounding multiplications of PSD random matrices

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 ...
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1answer
116 views

Invertibility of random Vandermonde matrix

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) := ...
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0answers
47 views

Tail Bounds for the minimum value of a function

Consider y to be the minimum value of an objective function over some subspace. More specifically $y= \min_x \|e+Bx\|_\infty \quad s.t. \quad x\in \mathcal{S}$ where $e$ is a known vector, $B$ is a ...
7
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3answers
718 views

A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...
2
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1answer
93 views

limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...
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0answers
26 views

the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...