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Questions tagged [random-matrices]

For questions concerning random matrices.

In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems may be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

861 questions
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Random products of Jordan blocks — what distribution are they converging to?

Let $$ M_1=\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right), \qquad M_2=\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right),\qquad v = \left( \begin{array}{c} 1 \\ 1 \\ \end{array} \right) $$ and consider the $n$-fold…
Good Boy
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Random complex orthogonal matrices

How can I uniformly extract a random complex orthogonal matrix $\Omega\in O(3,\mathbb{C})$? It is easily found in the literature the uniform measure for unitary and real matrices, but I couldn't find anything about the complex case. Thanks for your…
mrf1g12
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How to generate random symmetric unitary matrices "close" to a given matrix?

I want to perform Monte Carlo simulation for the analysis of a circuit problem, where the generation of random symmetric unitary matrices "close" to \begin{equation}T=\left[ {\begin{array}{*{20}{c}} {}&{{I_n}} \\ {{I_n}}&{} \end{array}}…
Simon Wong
  • 101
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Expectation of scaled identity plus square Wishart matrix

I found that $\mathbf H^t\mathbf H$ known as an Wishart matrix, when each row of $\mathbf H$ is an realization of i.i.d. Gaussian random vector of zero mean and identity covariance matrix ($\mathbf H$ is square). Then what is the expectation of…
DYYANG
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Is there a way to quickly generate a random $n\times n$ matrix $M$ with uniformly distributed entries with $\mbox{Tr} (M^\dagger M) \leq 1$?

I am trying to reproduce the results of a paper where they generate $10^6$ $3\times 3$ matrices $M$ with complex elements both real and imaginary uniformly distributed from $-1$ to $1$. They impose another condition, namely that $\mbox{Tr}…
tefkah
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What does one-cut random matrix mean?

I am quite new to random matrix theory and recently I encountered the so-called "one-cut random matrix model" and even "two-cut" in physics. So what exactly does it mean?
M. Zeng
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Independence and equality of variance in random matrix multiplication and element-wise multiplication

Let $B\in\mathbb{R}^{m\times n}$ be a random matrix, sampled i.i.d. from a uniform distribution $\mathcal{U}$ with bounds $[-a, a]$ and $e\in\mathbb{R}^{m}$ be a vector. We define $e'\in\mathbb{R}^{n}$ with the multiplication $$ e'=eB $$ Under what…
Blade
  • 461
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Variance of the trace of a power of a gaussian random matrix

Let $H$ be a hermitian $N\times N$ random matrix from the gaussian unitary ensemble. Then it is a standard result that, in the large-$N$ limit, $${\mathbb E}\left[\frac1N{\rm Tr}\,H^{2k}\right]=c_k,$$ where $c_k$ is the the $k$th Catalan…
Mark Srednicki
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Eigenvalue distribution of real anti-symmetric Gaussian random matrices

Let $A_{ij} = - A_{ji}$ be a $n \times n$ matrix with real entries distributed according to a Gaussian distribution with zero mean and standard deviation $\sigma$. What is the eigenvalue distribution of such matrices? I am mostly interested in the…
Surgical Commander
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Concentration Result on Singular Values of iid Random Matrix with Nonzero Mean

For a random matrix $A\in\mathbb{R}^{N\times n}$ with iid entries, with mean zero, variance $\mathbb{E}[a_{11}^2]$, and bounded fourth moment, I know that when $N$ and $n$ are sufficiently large, the extreme eigenvalues of $A^TA$ concentrate around…
D. Ryan
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For a matrix model with $O(n)$ as a symmetry group is $\langle \det A \rangle = \det \langle A \rangle$

Suppose that I have a model of random $n \times n$ symmetric, positive definite matrices where the probability distribution is generated by the partition function: $Z = \int dA \; e^{-V(A)}$ where $V(A)$ is left invariant under orthogonal…
Kyle
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Random matrix theory

A random symmetric $2 \times 2$ matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22}\end{pmatrix}$ is a member of the gaussian orthogonal ensemble (GOE), if it satisfies three conditions: $(1)$ for any $2 \times 2$ orthogonal…
Idonknow
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Expressing functions other than trace in terms of limiting spectral density?

Suppose $h$ is vector of eigenvalues of a random matrix with unit trace and $e$ is a vector in $\mathbb{R}^d$ starting with $e_i=1$ and evolving according the following recurrence: $$e_i \leftarrow h_i \langle h, e\rangle - 2 h_i e_i $$ Is there a…
Yaroslav Bulatov
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On the density of states of GUE using SUSY method with saddle point method

I'm reading this paper Density of states for Gaussian unitary ensemble, Gaussian orthogonal ensemble, and interpolating ensembles through supersymmetric approach. In this work, the author developed a approach avoiding the Hubbard-Stratonovich…
Guoqing
  • 155
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Is it possible to generate a random complex $n \times p$ matrix $M$, that satisfies $M M^\dagger = \mathbb{1} $?

As stated in the title, I'm interested in generating isometric complex matrices of a given $n \times p$ size. Such a matrix $M$ fulfills the following condition $ M M^\dagger = \mathbb{1} $. Is there some way to generate such matrices efficiently?
brzepkowski
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