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Statistics of spectral properties of matrix-valued random variables.

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43 views

Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
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41 views

Tail estimate of a random matrix [on hold]

How to estimate a tail of the random matrix if coordinates of each entry is known?
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36 views

Dominating powers of a random matrix

Let $A_n$ be a (sequence of) random matrix such that $ A_n = (a_{ij})_{1 \leq i,j \leq n}$ and the $a_{ij}$ are iid, $\mathbb E\left[ a_{ij}\right] = 0 \quad E\left[ \vert a_{ij}\vert^2\right] = 1$. ...
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0answers
21 views

Why do middle roots of the $\chi(p)$ graphs and percolation thresholds vary linearly with diagonal probability $q$ (in large random binary matrices)?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
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1answer
66 views

Generalization: (The “number” of) smaller sized clusters in large random binary matrices follow a descending order. Why?

This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix? In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
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1answer
117 views

Why is number of single cell clusters always greatest in a random matrix?

Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
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0answers
22 views

What is the distribution of engenvalues of covariance matrix when the covariance has some block diagonal structure

Let's say we have a matrix $X \in \mathbb R^{n\times p}$, where $X_{i,j}$ sampled from a Gaussian $N(\mu, \sigma^2)$, we use $\Phi$ to denote $\{\mu,\sigma\}$ for simplicity. Now, we sample $m$ ...
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1answer
207 views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
3
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3answers
131 views

Random complex eigenvalues and averages of traces

I have asked this in MSE here, but got no interesting answers. Suppose I have a random matrix $M$ of dimension $N$ which is real, but not symmetric. Suppose I know that, for large $N$, the marginal ...
10
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0answers
386 views

multi-dimensional integral of modified Vandermonde determinant

I'm looking for suggestions on how one might try to compute the following $(N-1)$-dimensional integral: $$I_N= \frac{1}{(2\pi)^{N-1}(N-1)!} \int\cdots\int \\ \begin{vmatrix} 1 ...
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1answer
84 views

Statistical independence of eigenvectors of real symmetric Gaussian random matrices

What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not ...
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0answers
82 views

norm and conorm of elliptic cocycle be different

Let $(M,\mathcal{B},\mu)$ be a probability space and $f:M \rightarrow M$ be a measure preserving map.Let $A:M \rightarrow SL(2,\mathcal{R})$be a measurable function with value invertiable $2\times2$...
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1answer
89 views

Are they the $N\times N$ random matrices with real or complex entries asymptotically free?

The $N\times N$ random matrices with real or complex entries are generalizations of non-hermitian gaussian ensembles, also known as Girko ensemble: the entries are independent and identically ...
4
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1answer
206 views

Confusion about Montgomery's Pair Correlation Conjecture

This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4 To set up the ...
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1answer
102 views

What are the disadvantage and advantages of the moment method and the resolvent method in Random Matrix Theory?

I am learning random matrix theory . I am aware that the most popular successful techniques for obtaining the limiting spectral measure of large Hermitian random matrices are the moment method and ...
27
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3answers
2k views

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 ...
2
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2answers
98 views

Spectrum of finite-band random matrices?

Let $X_n=(X_{ij})_{1 \leq i,j \leq n}$ such that : $$ \begin{cases} &X_{ij} = 0 \quad \text{if}\quad \vert i - j \vert > k\\ & X_{ij} \sim P_X \quad \text{otherwise} \end{cases}$$ And ...
4
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1answer
87 views

Haar unitaries with constraints

Given that one can sample unitaries from the Haar measure over $U(n)$ (as in F. Mezzadri, Notices of the AMS 54 (2007), 592-604), how can one sample from the uniform distribution over the following ...
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5answers
248 views

Reviews of Probability in High Dimension not by Van Handel

I'm completely in love with Ramon van Handel's lecture notes Probability in High Dimension and I would like to find more learning resources. Lecture notes or reviews would be ideal as anything in this ...
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1answer
67 views

How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the marginal distribution of each eigenvalue, using whatever you can. Background In my ...
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1answer
122 views

Pair of vectors multiplied by a random matrix and its inverse transpose are distributed randomly up to their dot product

Given arbitrary nonzero vectors $\vec{x}_1, \vec{y}_1, \vec{x}_2, \vec{y}_2 \in \mathbb{Z}^{n}_p$ ($p$ prime) with $\langle x_1, y_1 \rangle = \langle x_2, y_2 \rangle$, I am trying to show that: $(...
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0answers
46 views

How to bound the expectation of the power of sum of independent matrices?

Suppose I have $n$ independent zero mean random matrices $\{\Lambda_{i}\}_{1\leq i\leq n}$, where each $\Lambda_{i}\in \mathbb{R}^{n\times n}$ is symmetric. Also denote $E := \sum_{i=1}^{n} \Lambda_{i}...
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0answers
57 views

Conditonal convergence implies convergence?

Note : All measures below are probability measures. Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$. Actually,...
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0answers
85 views

A general question on random matrices

To give some context, suppose you have a sequence of random variables $X_n$ that you know weakly converges to a random variable $X$. Suppose you also have a sequence of random variables $Y_n$. Then, ...
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0answers
23 views

Confused about the generality of statements regarding rank one deformations of Wigner matrices

I'm interested in the behavior of the largest eigenvalue of random $n \times n$ Hermitian matrices of the form $A^{n} = \frac{1}{2}(Q^{n}+(Q^{n})^{\dagger})$ where the real and imaginary components of ...
3
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0answers
106 views

A guide to reading Alice Guionnet's Random Matrix book

I was reading Alice Guionnet's book "Large Random Matrices: Lectures on Macroscopic Analysis". I would need some help in understanding the author intends to do in Part III, "Matrix Models". In the ...
7
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2answers
365 views

Generating function of $SO(N)$ random matrix

I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute $$ Z_N[J]=\int dM e^{{\rm Tr} (J^T M)}, $$ where $dM$ is the $SO(N)$ Haar measure, and $J$ is an ...
6
votes
2answers
289 views

Probability of a large random integer Matrix to have zero determinant

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$ I would like to ...
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3answers
274 views

Non combinatorial random matrix theory

I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, ...
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1answer
101 views

Distribution of eigenvalues of a Wishart matrix

Is there a known expression for the eigenvalue distribution of a matrix of the form $$\sum\limits_{i=1}^n k_ia_ia_i^T$$ where $a_i \in \mathcal{R}^m$, with $n > m$, $a_i \sim \mathcal{N}(0,\Sigma)...
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1answer
64 views

Moment generating function of spectral norm of iid N(0,1) data matrix

Let $W^{p\times p}$ be a normal data matrix with $W_{ij}$ i.i.d. $N(0,1)$. Are there any results on the evaluation, or upper bound for the Moment Generating Function of the spectral norm of W, that is,...
3
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2answers
145 views

Does free probability have anything to say about the eigenvalue correlations of random matrices?

Free probability provides a compact route to compute the average eigenvalue density for various families of random matrices in the large $N$ limit. Does it provide any route to eigenvalue correlations,...
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0answers
55 views

Probability of collision of sums of vectors

Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function. Does there exist a random matrix $R \...
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1answer
171 views

Can this particular random matrix model be converted/related to any existing graph theory model?

Context: This a sequel to the question: Is the Erdős–Rényi giant component result applicable here? Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ ...
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0answers
77 views

Concentration of a Gaussian function around its mean

I am interested in showing a certain function of a Gaussian random vector is concentrated around its mean. Let $\mathbf{g}\in\mathbb{R}^n$ be distributed as $\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. I ...
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1answer
115 views

Random rotation of a set of distinct points in $\mathbb{R}^n$ [closed]

Consider a set $\{\mathbf{X}_1,\cdots , \mathbf{X}_M\}$ of distinct points in $\mathbb{R}^n$ with $M$ finite. The $M$ values of the $i$-th coordinate do not all have to be dinstinct. For example, in $\...
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0answers
161 views

Why do larger random matrices maximize their number of clusters with lower densities?

Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$. Define a cluster of cells as a maximal connected component in the ...
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0answers
43 views

A question on linear separability after random projection

Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, and let the data set $S = \{($ $\textbf{x}_i$ $\in \mathbb{R}^{d}, y_i \in ...
3
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1answer
113 views

Proving tail bound of a random projection

Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\...
2
votes
1answer
69 views

Independent decomposition of coordinate distribution

Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\...
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1answer
95 views

Two theorems about incoherence

These are two theorems I have heard being referred to in "folklore" but I cant find the proofs for these in any compressed sensing or high-dimensional probability reviews (like, https://www.math.uci....
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1answer
76 views

Eigenvalue density expansion in random matrices

First, some background and context. Curently I am studying the basics of "random matrices" and their large N expansion. At large $N$ I know that the eigenvalues condense in what is called the Wigner ...
16
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5answers
1k 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 ...
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vote
0answers
126 views

A “conjectured” concentration inequality for operators, probably related with random matrix theory

I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as ...
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1answer
69 views

Probability of collision of sums of vectors multiplied by random matrix

Let $S$ and $T$ be sets of vectors from $\mathbb{R}^d$ such that $S$ and $T$ are at least different in one element. Does there exist a random matrix $M \in \mathbb{R}^{d \times k}$, e.g., a gaussian ...
3
votes
1answer
217 views

Help with understanding a proof on angle preservation

Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\...
0
votes
0answers
68 views

Zero-mean unit variance complex value independent Gaussian Random Variables in Matrix

Let $G$ be $m\times n (m\ge n)$ random matrix of zero-mean unit variance complex value independent Gaussian Random Variables. Then what is distribution of eigenvalues of $G^HG$ and how to obtain the ...
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0answers
76 views

Dixon-Anderson-Selberg integral variant

I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$ $$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\...
1
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1answer
100 views

Random matrix and spherical spin-glass

The Hamiltonian of the p-spherical spin glass model is $$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$ where $\sigma \in ...
8
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1answer
272 views

Gaussian integrals over the space of symmetric matrices

Let $S\in\mathcal S_N$ be a $N\times N$ symmetric matrix over the reals, and introduce the (normalised) gaussian measure $$ \mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\...