Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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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 i....
Scott Aaronson's user avatar
42 votes
3 answers
5k 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 $p_n=...
Denis Serre's user avatar
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39 votes
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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 ...
Adrien Hardy's user avatar
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36 votes
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Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
Carlo Beenakker's user avatar
31 votes
3 answers
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What is the Katz-Sarnak philosophy?

It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
Henry.L's user avatar
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26 votes
3 answers
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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 ...
Andrew's user avatar
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24 votes
1 answer
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What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
Qiaochu Yuan's user avatar
23 votes
7 answers
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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?
Jason Knight's user avatar
23 votes
2 answers
831 views

Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
Abdelmalek Abdesselam's user avatar
22 votes
2 answers
1k views

Laws of Iterated Logarithm for Random Matrices and Random Permutation

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then $$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, $...
Gil Kalai's user avatar
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Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
ght's user avatar
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21 votes
1 answer
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Unexpected $\sqrt{3}$

A somewhat lengthy calculation, involving integrals, reveals that the probability an $n\times n$ Hermitian matrix, drawn from the Gaussian unitary ensemble, is positive definite, decays as $\left(\...
Veit Elser's user avatar
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21 votes
0 answers
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The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
Gil Kalai's user avatar
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20 votes
6 answers
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Intuition for Haar measure of random matrix

What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices? My understanding for what Haar measure means for $U(1)$ is that it ...
Jiahao Chen's user avatar
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20 votes
3 answers
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Jensen Polynomials for the Riemann Zeta Function

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) the abstract includes In the case of the Riemann zeta function, this proves the ...
Stopple's user avatar
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19 votes
2 answers
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Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating a random orthogonal matrix $M$, uniformly distributed among all such, and a random displacement $v$, whose coordinates are drawn from a ...
Joseph O'Rourke's user avatar
19 votes
1 answer
2k 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 ...
Suvrit's user avatar
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19 votes
0 answers
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What does a product of many Gaussian matrices converge to?

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is ...
Yaroslav Bulatov's user avatar
18 votes
5 answers
3k 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 ...
J. E. Pascoe's user avatar
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18 votes
1 answer
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How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
sbahmani's user avatar
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17 votes
4 answers
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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 ...
Piero D'Ancona's user avatar
17 votes
0 answers
457 views

Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
Leonid Petrov's user avatar
16 votes
5 answers
2k views

Expected value of determinant of simple infinite random matrix

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$ I would like to ...
Hipstpaka's user avatar
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16 votes
1 answer
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Intuitive understanding of the Stieltjes transform

I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does. The gist of my work is that I have an $N\times N$ true covariance ...
user avatar
16 votes
4 answers
580 views

The lattice spanned by $m$ random 0-1 vectors of length $n$

Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
Gil Kalai's user avatar
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16 votes
2 answers
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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} \frac{1}{(p_0!)^2(p_1!)^2\...
Brad Rodgers's user avatar
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15 votes
3 answers
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Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
Guido Li's user avatar
15 votes
0 answers
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PT Symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
Stopple's user avatar
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14 votes
2 answers
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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 ...
Adrien Hardy's user avatar
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14 votes
1 answer
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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 ...
ght's user avatar
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14 votes
1 answer
408 views

References for reasoning about the spectrum of a convex body?

By "spectrum of a convex body", I mean: start with a convex body $B$ in $\mathbb{R}^d$, then consider the corresponding $d \times d$ covariance matrix resulting from a uniform distribution over $B$ -- ...
Barbot's user avatar
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14 votes
0 answers
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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 ...
Brad Rodgers's user avatar
  • 2,141
13 votes
3 answers
573 views

GOE/GSE duality and Bott periodicity

Many papers in random matrix theory make passing references to duality between eigenvalue statistics of the GOE and GSE, for which the most concrete reference I can find is https://arxiv.org/pdf/math-...
Roger Van Peski's user avatar
13 votes
2 answers
843 views

The expected square of the determinant of a random row stochastic matrix

In this question Anthony Quas asks about the expected absolute value of the determinant of an $n\times n$ row stochastic matrix $A$, where the rows are independently selected from the uniform ...
Richard Stanley's user avatar
13 votes
1 answer
865 views

Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular? For example, for $n=3$ and $k=2$, the first ...
Joseph O'Rourke's user avatar
13 votes
1 answer
988 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 ...
Louigi Addario-Berry's user avatar
13 votes
1 answer
640 views

$\ell^1$-norm of eigenvectors of Erdős-Renyi Graphs

Setting. Let $G(n,p)$ denote the usual Erdős-Renyi (random) graphs. For each such graph there is an associated Laplacian matrix $L = D - A$ where $D$ collects the degrees on the diagonal and $A$ is ...
Stefan Steinerberger's user avatar
13 votes
2 answers
628 views

Random matrix with given singular values

Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let $$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
neverevernever's user avatar
13 votes
0 answers
696 views

Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it

In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
Itay's user avatar
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13 votes
0 answers
565 views

What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
Christopher A. Wong's user avatar
13 votes
2 answers
5k views

Distribution of inverse of a random matrix

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 $k<d$) whose ...
Salman's user avatar
  • 131
12 votes
4 answers
1k views

Why only three classical matrix ensembles in random matrix theory?

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 ...
Jiahao Chen's user avatar
  • 1,870
12 votes
1 answer
250 views

GOE Version of Longest Increasing Subsequence

Let $S_n$ be the symmetric group equipped with uniform measure. For any $\pi\in S_n$, let $L_n=L_n(\pi)$ denote the longest increasing subsequence. A celebrated result of Baik, Deift and Johansson ...
Alex R.'s user avatar
  • 4,902
11 votes
8 answers
2k views

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
Alex R.'s user avatar
  • 4,902
11 votes
3 answers
1k views

Maximum singular value of a random $\pm 1$ matrix

Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...
Kostas's user avatar
  • 199
11 votes
1 answer
3k views

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 $x_1,\...
Suvrit's user avatar
  • 28.4k
11 votes
1 answer
590 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
Kurisuto Asutora's user avatar
11 votes
1 answer
615 views

A simple proof for a theorem of Szekeres and Turán

Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...
Gil Kalai's user avatar
  • 24.2k
11 votes
2 answers
2k views

Expected values of traces of products of random matrices

Suppose I want to compute a quantity of the type: $\mathbb{E}\mathrm{tr}(AUBU^{\ast})$ where averaging is over Haar measure on the unitary group $\mathcal{U}(n)$ (one can of course consider higher ...
Marcin Kotowski's user avatar
11 votes
1 answer
502 views

A function with unexpectedly simple Legendre transformation

Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and \begin{equation} I(x)= \begin{cases} \frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\ \...
Pluviophile's user avatar
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