Statistics of spectral properties of matrix-valued random variables.

**6**

votes

**0**answers

45 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**

votes

**0**answers

53 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**

votes

**1**answer

120 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 ...

**6**

votes

**4**answers

342 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 ...

**-1**

votes

**1**answer

93 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 ...

**2**

votes

**1**answer

77 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} ...

**5**

votes

**1**answer

143 views

### rigidity of eigenvalues of circular ensemble

Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...

**3**

votes

**2**answers

236 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 ...

**2**

votes

**0**answers

68 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 ...

**4**

votes

**1**answer

103 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.
...

**5**

votes

**2**answers

324 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 ...

**2**

votes

**1**answer

170 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 ...

**1**

vote

**1**answer

183 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
...

**4**

votes

**0**answers

220 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 ...

**1**

vote

**0**answers

66 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
...

**0**

votes

**0**answers

29 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**

votes

**0**answers

58 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 ...

**1**

vote

**0**answers

35 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 ...

**2**

votes

**1**answer

251 views

### Distribution of sum of freely independent Marchenko-Pastur measures

Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables ...

**1**

vote

**1**answer

78 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 ...

**3**

votes

**1**answer

131 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 ...

**5**

votes

**1**answer

173 views

### concentration of random matrices involving normal random variables

Define the random variable
\begin{align*}
A=|a_1|^2\mathbf{a}\mathbf{a}^*
\end{align*}
where $\mathbf{a}\in\mathbb{c}^n$ is a random vector distributed as ...

**1**

vote

**0**answers

29 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

**1**answer

115 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 ...

**2**

votes

**1**answer

107 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
...

**0**

votes

**0**answers

81 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**

votes

**0**answers

82 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 ...

**6**

votes

**1**answer

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 ...

**-3**

votes

**3**answers

304 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**

votes

**3**answers

379 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 ...

**14**

votes

**4**answers

823 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 ...

**1**

vote

**1**answer

106 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 ...

**6**

votes

**1**answer

148 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 ...

**8**

votes

**1**answer

671 views

### Matrix integral identity

1) How to prove that $N\times N$ matrix integral over complex matrices $Z$
$$
\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det ...

**28**

votes

**2**answers

1k views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...

**2**

votes

**0**answers

55 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 ...

**1**

vote

**1**answer

73 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 ...

**21**

votes

**1**answer

2k views

### When should we expect Tracy-Widom ?

The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the ...

**2**

votes

**0**answers

78 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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

363 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 ...

**3**

votes

**1**answer

133 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 $m<n$, ...

**0**

votes

**0**answers

34 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 ...

**7**

votes

**3**answers

713 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 $v$ ...

**0**

votes

**0**answers

67 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 ...

**0**

votes

**1**answer

368 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...

**1**

vote

**1**answer

113 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) := ...

**0**

votes

**0**answers

46 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 ...

**2**

votes

**0**answers

867 views

### Distribution of Inverse of a Random Matrix

Recently i got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose i have a fat random matrix (i,e $R$ has dimensions $k\times d$ where $k<d$) ...

**0**

votes

**0**answers

273 views

### expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$.
Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...