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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 ...
LeftBrainDamaged's user avatar
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 ...
Samuel Johnston's user avatar
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^\...
thedude's user avatar
  • 1,549
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 ...
Peter's user avatar
  • 141
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^{...
Ben Deitmar's user avatar
  • 1,295
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 ...
John Jiang's user avatar
  • 4,466
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 ...
user97715's user avatar
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 ...
Marcel's user avatar
  • 2,552
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 ...
John Jiang's user avatar
  • 4,466
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 ...
bnaul's user avatar
  • 201
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 \...
john mangual's user avatar
  • 22.8k
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 ...
martin's user avatar
  • 300
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 ...
thedude's user avatar
  • 1,549
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} \...
user50394's user avatar
  • 123