Statistics of spectral properties of matrix-valued random variables.

**4**

votes

**2**answers

571 views

### Distribution of eigenvalue spacings

I have been doing some experiments on classes of random matrices, and it seems (visually) that the distribution of eigenvalue spacings is consistent with GOE or GUE or GSE. Unfortunately, to test ...

**8**

votes

**0**answers

343 views

### Has the technique of “sprinkling” been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...

**11**

votes

**4**answers

860 views

### Why only three classical matrix ensembles in RMT? (Newbie question)

I am just starting out on understanding random matrix theory from a background in applied mathematics. I have a very basic question about the Gaussian ensembles: why are there only three classical ...

**4**

votes

**2**answers

487 views

### Induced p-norm of a Random matrix

This question is related to my earlier question
here .
Given an $n\times n$ random matrix $A$, is determining the properties (mean, variance,moments,etc.) of its induced $p$-norm ($p\neq ...

**2**

votes

**1**answer

692 views

### Expectation of product of Gaussian random vectors

Say we have two multivariate Gaussian random vectors $p(x_1) = N(0,\Sigma_1), p(x_2) = N(0,\Sigma_2)$, is there a well known result for the expectation of their product $E[x_1x_2^T]$ (matrix result) ...

**2**

votes

**1**answer

495 views

### What are the origin and applications of this result?

In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose
$$ M = \begin{bmatrix} A & B \\\\ B^T & C ...

**8**

votes

**1**answer

618 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 ...

**0**

votes

**1**answer

510 views

### randomized SVD singular values

randomized SVD decomposes a matrix by extracting the first k singular values/vectors using k+p random projections.
my question concerns the singular values that are output from the algorithm. why ...

**15**

votes

**1**answer

1k views

### Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...

**30**

votes

**1**answer

3k views

### Anti-concentration bound for permanents of Gaussian matrices?

In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of ...

**5**

votes

**1**answer

253 views

### Expected inverse determinant with independent rows

Let $a_1,a_2,\dots,a_n$ be independent identically distributed random vectors in $\mathbb R^n$. I need a bound for $E[|\det A|^{-1}]$, where $A$ is the matrix composed out of these vectors.
More ...

**4**

votes

**0**answers

196 views

### q-deformation of the unitary group integral

There is a well-known orthogonality property of $U(N)$ group characters
$$
\int d U \chi_{\mu}(U)\chi_\lambda(U^\dagger V)=\delta_{\mu\lambda}\frac{\chi_\mu(V)}{\dim_\mu}
$$
where the integral is ...

**0**

votes

**2**answers

251 views

### Question about “wide” random matrices

Let $A \in \mathbb{R}^{m \times n}$ be a random matrix with i.i.d. entries (the distribution is not important), where $m < n$ (i.e. $A$ is a "wide" matrix). I would like a lower bound on
$$
...

**5**

votes

**2**answers

499 views

### Dependence of trace norm on matrix size for smooth vs. random matrices.

Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...

**2**

votes

**4**answers

409 views

### any known universality results of random matrices with non-independent entries?

GUE, GOE, hermitian/symmetric wigner matrices, ... already known with some mild assumptions.
but is there any this kind of results without independent entries condition. thanks a lot.

**11**

votes

**2**answers

916 views

### Evaluation of a combinatorial sum (that comes from random matrices)

I'm looking for an elementary combinatorial/generating function/etc proof of the following result:
For nonnegative integers $r$,
$$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} ...

**12**

votes

**4**answers

956 views

### An experiment on random matrices

A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me ...

**9**

votes

**1**answer

652 views

### U(3) Sato-Tate measure.

An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix ...

**10**

votes

**3**answers

972 views

### Random Walks and Lyapunov exponents

Given a sequence $Y_1, Y_2, \dots$ of i.i.d. matrices in $GL_n(\mathbb R)$, there is a theorem of Furstenberg and Kesten which says that if $\mathbb E(\log||Y_1||)$ is finite, there exists a constant ...

**6**

votes

**1**answer

454 views

### distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field

This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...

**0**

votes

**1**answer

211 views

### Probability on the distance

Let $A$ be an $n\times n$ gaussian matrix whose entries are i.i.d.
copies of a gaussian variable, and $\left\{ a_{j}\right\} _{j=1}^{n}$
be the column vectors of $A$. How to show that the probability
...

**12**

votes

**7**answers

4k views

### Expected determinant of a random NxN matrix.

What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?

**8**

votes

**0**answers

283 views

### A formula for moments of the limit distribution of singular values in the proof of the circular law

One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - ...

**6**

votes

**5**answers

778 views

### L-functions and random matrices

I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one ...

**2**

votes

**4**answers

2k views

### Symmetrical Presentation of 4-Dimensional Rotation Matrix

This question is not urgent; just a matter of curiosity...
It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...

**8**

votes

**3**answers

815 views

### Singularity of sparse random matrices

The following topic came up in conversation with my office-mate Lionel: Let $p$ be a fixed prime, $c$ a fixed positive real parameter and $n$ a large number. Consider a random $(0,1)$ matrix with ...