Statistics of spectral properties of matrix-valued random variables.

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8
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1answer
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Matrix inversion lemma with pseudoinverses

The utility of the Matrix Inversion Lemma has been well-exploited for several questions on MO. Thus, with some positive hope, I'd like to field a question of my own. Suppose we pick $n$ values ...
24
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0answers
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 ...
12
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2answers
1k views

What do we actually know about logarithmic energy ?

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
9
votes
1answer
580 views

Average over Random Permutations

Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the ...
6
votes
1answer
611 views

Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces?

Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. ...
8
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0answers
423 views

Framework for primes vs random matrices

This is inspired by What results would follow from or imply "randomness" of the primes? , but I think it is sufficiently different to ask separately. We can formalise probability in ...
0
votes
1answer
669 views

Uniform correlation matrix sampling and not so uniform laws

Hi everyone, I am looking for a way of simulating correlation matrices of fixed dimension in (at least) two ways. First, I would like to determine the "uniform" distribution over the "correlation ...
4
votes
3answers
917 views

Marginal distribution of the diagonal of an inverse Wishart distributed matrix

This is a cross-posting of a question I asked at CrossValidated. It hasn't generated much activity so I'm trying here: Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the ...
11
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0answers
322 views

Lower Bound on the Volume of Certain Polytopes

Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks $$ \rho=\{B_1,B_2,\ldots,B_{k}\} $$ we can define the set of equations $$ E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in ...
12
votes
1answer
816 views

A Question on Random Matrices

Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by $$ V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q}) $$ where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
4
votes
4answers
573 views

efficient way to compute the inversion of the following matrix

Hi, there I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
4
votes
2answers
278 views

analogue of GUE and Ginibre in higher dimensions

This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N ...
2
votes
1answer
247 views

Why doesn't the argument of circular law convergence of Ginibre spectrum give the same result for GUE?

It appears I am profoundly confused in the following nice argument of Ginibre and Mehta and beautifully presented in Djalil Chafai's blog ...
3
votes
3answers
2k views

Distribution of trace of inverse-Wishart matrix $W_n(I,n)$

Hello, I'm interested in the distribution of the trace of an inverse-Wishart matrix $W_n^{-1}(I,n)$, where $I$ is $n\times n$ identity matrix. More precisely, I seek for an asymptotic estimate (when ...
4
votes
0answers
249 views

Stable distributions for Lindeberg exchange strategy?

Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
4
votes
2answers
649 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
0answers
365 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
4answers
879 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
2answers
510 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
1answer
808 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
1answer
507 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
1answer
691 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
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1answer
603 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 ...
17
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1answer
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
1answer
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
1answer
259 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
0answers
209 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 ...
2
votes
4answers
857 views

On the spectrum of random regular graph

For a random $d$-regular graph, where $d$ can be fixed or can grow slowly with the size of the graph $n$, what can we say about its spectrum - Do you believe it has simple spectrum? Thank you,
0
votes
2answers
257 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
2answers
516 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
4answers
424 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
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2answers
933 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} ...
13
votes
4answers
1k 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
1answer
669 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
3answers
1k 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
1answer
469 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
1answer
212 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 ...
13
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7answers
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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
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0answers
292 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 - ...
7
votes
5answers
815 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
4answers
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
3answers
862 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 ...