All Questions
Tagged with random-matrices lie-groups
14 questions
6
votes
1
answer
252
views
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^\...
- 188k
8
votes
3
answers
508
views
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
- 484
5
votes
2
answers
324
views
Is there a 'natural' projection from $O(n)$ into $S_n$?
Is there an easily definable projection $F$ from the orthogonal group $O(n)$ into the permutation group $S_n$, which has the following properties?
$F(P_\sigma) = \sigma$ for all $\sigma \in S_n$
$F^{...
0
votes
0
answers
96
views
Integral of elements of random unitaries
It is known how to calculate the integral of elements of $N\times N$ Haar random unitaries using the Weingarten function:
$$\int \prod_{k=1}^n U_{i_kj_k} U_{m_kr_k}^* \mathrm d U = \sum_{\sigma,\tau} \...
- 123
1
vote
2
answers
922
views
Eigenvectors of random unitary matrices
Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$,
$$U=VDV^\dagger,$$
where $D={\rm diag}(z_1,z_2,...,z_N)$ is diagonal.
If $U$ is taken at random uniformly with respect to Haar ...
- 63.9k
6
votes
1
answer
1k
views
An integral with respect to the Haar measure on a unitary group
Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...
- 101
4
votes
1
answer
321
views
Average of product of matrix elements in the special orthogonal group
Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see ...
8
votes
3
answers
3k
views
How to correctly generate uniformly distibuted random elements from SO(n)?
I already found some way to produce such matrices from SO(n) with a method called subgroup algorithm but I would like some advice on the method I used. Nowhere I could really find any paper relating ...
- 63.9k
4
votes
1
answer
172
views
Haar unitaries with constraints
Given that one can sample unitaries from the Haar measure over $U(n)$ (as in F. Mezzadri, Notices of the AMS 54 (2007), 592-604), how can one sample from the uniform distribution over the following ...
- 43
2
votes
0
answers
95
views
what kind of Gaussian matrix models are these?
In a physics paper I found a very complicated Gaussian matrix model:
$$ Z = \int \frac{d\mu}{(2\pi)^n} \frac{d\nu}{(2\pi)^n}
\frac{
\prod_{i < j}\left[2 \sinh \frac{\mu_i - \mu_j}{2} \right]^2 \...
- 22.8k
5
votes
1
answer
287
views
rigidity of eigenvalues of circular ensemble
Given a circular unitary ensemble, with the following joint density:
$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,
is the following statement true? With ...
CommunityBot
- 1
2
votes
0
answers
284
views
Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices
I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...
- 300
3
votes
2
answers
513
views
Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
CommunityBot
- 1
4
votes
1
answer
560
views
Decomposition of Haar measure other than Hurwitz's
Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
- 188k