All Questions
Tagged with random-matrices dg.differential-geometry
9 questions
15
votes
3
answers
875
views
Laplacian on manifolds and random matrix theory
Let $M$ be a compact Riemannian manifold with a metric $g$, and consider the spectrum of the Laplacian operator $\Delta$.
What is known about the relationship between this spectrum and random matrix ...
- 34.7k
2
votes
0
answers
100
views
Stiefel C-manifold: volume and uniform distribution
I'm trying to define the uniform distribution on the Stiefel $C$-manifold (Downs 1972), given by $\mathcal{V}_{p,n}(C) = \{ X \in \mathbb{R}^{n \times p} : X' X = C \}$ for $n \geq p$ and $C > 0$. ...
- 21
2
votes
0
answers
83
views
Biased ensemble in the unitary group
I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows
$$
\mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2}...
- 121
4
votes
2
answers
240
views
A 'projective' property of the Haar U(n) measure
Let $U(n)$ be the compact manifold of unitary $(n \times n)$-matrices and let $\mu_n$ denote the Haar-probability measure on $U(n)$. For $m < n$ does there exists a measurable (maybe even ...
CommunityBot
- 1
6
votes
1
answer
1k
views
An integral with respect to the Haar measure on a unitary group
Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...
- 101
0
votes
0
answers
98
views
Genes mirror geography on a torus?
Disclaimer: this is an open-ended, imprecise question, asking for speculation in a topic that I know relatively little about (random matrix theory and principal component analysis). I originally asked ...
- 1,225
13
votes
2
answers
655
views
Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 4,361
7
votes
2
answers
599
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 ...
- 188k
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123