Questions tagged [random-matrices]
Statistics of spectral properties of matrix-valued random variables.
871 questions
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Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
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Maximum probability of a set of vectors from $ \mathbb{F}_2^n $ being linearly independent
Suppose $ m $ vectors from the vector space $ \mathbb{F}_2^n $ are selected independently according to a distribution $ P $ over $ \mathbb{F}_2^n $. Here $ \mathbb{F}_2 $ denotes the field with two ...
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Concentration inequality for minimal eigenvalue of sample covariance
I was reading an article of matrix completion and met the following lemma
The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\...
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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 ...
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
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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 - zI)^\...
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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 ...
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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. ...
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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}).$$
The fact that this ...
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On the spectrum of random regular graph
For a random $d$-regular graph, where $d$ can be fixed or can grow slowly with the size of the graph $n$, what can we say about its spectrum - Do you believe it has simple spectrum?
Thank you,
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distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field
This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...
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Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\DeclareMathOperator\GL{GL}$A few months ago, the following discussion took place on AoPS, concerning matrices that have nonzero determinant under any permutation of their entries: https://...
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Generating function of $SO(N)$ random matrix
I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute
$$
Z_N[J]=\int dM e^{{\rm Tr} (J^T M)},
$$
where $dM$ is the $SO(N)$ Haar measure, and $J$ is an ...
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Analysis of $AB^{-1}$, where $A,B$ are random matrices
I am looking for help pointing me in the direction of any literature or other known work that analyze the probability distribution or other important properties of random variables of the form $AB^{-1}...
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Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
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Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$,
and $np\to c=2,3$.
The plots above are already ...
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What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?
Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$...
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Distribution of dot product of two unit random vectors
Consider $\mathbf{u}, \mathbf{v}\in \mathcal{C}^M$ to be two independent unit norm random vectors on the $M-1$ dimensional complex sphere $\mathcal{S}^{M-1}$. In addition, $\mathbf{u}$ follows an ...
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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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Extension of Wigner's semicircle law?
It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries.
My question is: is there any study about the more general ...
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Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding ...
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Smallest eigenvalue of a random matrix
Let $A \in \mathbb R^{n\times n}$ be a positive semi-definite matrix,
and let $b \in \mathbb R^n$. For a random vector $x \sim \mathcal N(0, I_{n\times n})$, consider the random matrices
$$
B_1 = A + ...
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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 ...
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Iterating projections to random halfspaces
Consider the following process:
Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \...
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Minimizing the largest eigenvalue of random matrices
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric matrix with entries $A_{ij} \sim \mathcal{N} (0,1)$, all independent except for the symmetry condition.
Consider the following minimization problem:...
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Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$
We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
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Product of two random Gaussian matrices - orthant probability
Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal ...
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Can one "smooth over" k-wise independence to get actual independence?
I came across the following toy problem and was curious if there was a simple solution or counterexample. Suppose you have a distribution $p$ on $m$ random variables $X_1, \ldots, X_m$, each with ...
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irregular LDPC code construction algorithm
I want to construct a sparse random binary matrix ${{\bf{H}}_{m \times n}}$ that has the following properties
1- Faction of columns of weight $i$ is ${v_i}$ .
2- Fraction of rows of weight $i$...
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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 ...
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Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements
Consider a random matrix $A \in \mathbb{R}^{N \times N}$ where the elements are random gaussian variables. The mean and variance of the elements are different on the diagonal and the off-diagonal:
$\...
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Is there a way to simplify the following trace expression?
I'd like to simplify the following expression:
$$\text{tr}\{\mathbf{A}^HE(\mathbf{C}^H \begin{bmatrix} \mathbf{0}_{M\times M} & \mathbf{0}_{M\times N} \\ \mathbf{0}_{N\times M} & \mathbf{I}_{...
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Phase transition in matrix
Playing around with Matlab I noticed something very peculiar:
Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by
$$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$
...
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2
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Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices
Let $A_n$ be the matrix product of $n$ i.i.d. N-by-N random complex matrices. The matrix distribution is not fixed and can be tuned to suit specific solution if needed, as long as it's not too "...
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Upper bound for a Selberg-type integral over a rectangular region
(Cross-posted from math-SE).
I am trying to estimate the values of the following integral for large $n$,
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-...
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Central limit theorem for 3d rotations
Let $X_i$ (where $i=1, \dots , n$) be independent and identically distributed 3d rotations. What is the distribution of $X_1X_2\dotsb X_n$ in the limit of large $n$?
I'm especially interested in the ...
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Poisson kernel for the orthogonal groups
For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
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A second-order recursion (functional equation)
In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion,
$$L(s+1)=L(s)+2s(2s+1)L(s-1).$$
Is it familiar to someone ? Is ...
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Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
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Probability that sum of each row and sum of each column is greater than 0 for a random matrix
Given a matrix $W_{n,m}$ whose each entry $w_{ij}$ is 1 or -1 or 0 with probability $p$, $p$ and $1-2p$ respectively, $0<p<0.45$
Let $R_i$ be the sum of $i^{th}$ row and $C_{j}$ is the sum of $...
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Probability of a large random integer Matrix to have zero determinant
Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...
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An integral with respect to the Haar measure on a unitary group
Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...
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Distribution of the permanent modulo $p$
We know that the order of $SL_n({\mathbb F}_p)$ is
$$p^{n(n-1)/2}(p^n-1)(p^{n-1}-1)\cdots(p^2-1).$$
Dividing by $p^{n^2}$, we deduce the probability that $\det$ takes the value $1$ over $M_n({\mathbb ...
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Lower bound of integral involving Laguerre polynomials
I want to lower bound the expected value of the square root of a randomly chosen eigenvalue of a Wishart matrix.
To get the bound I want I need a lower bound on
$$T_n = \int_0^\infty\sqrt{x}e^{-x}...
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Largest eigenvalues of a (random) correlation matrix?
I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
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Average minimum number of random k-sparse vectors in GF(2) to span the whole space?
What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
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200
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Modification of matching
Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...
- 903
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633
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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,n]...
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Smallest eigenvalue gap of a non-symmetric random matrix
The question: Let $A$ be the matrix whose each element is an independently generated random variable which is uniform on $[0,1]$. One can see that the eigenvalues of $A$ will be distinct almost surely....
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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 ...
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