An answer to [another question](https://mathoverflow.net/questions/2084/what-is-the-volume-of-a-delta-ball-in-the-orthogonal-group-on-is-there-a-sim) 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 my purposes, an acceptance-rejection scheme is not viable. I am trying to study the behavior of an integral over $O(n)$ of a function that is very small outside of certain regions, so accepting samples that lie within the ball is no different (from a computational perspective) from sampling from $O(n)$ in its entirety.


Alternatively, any other non-uniform sampling scheme on $O(n)$ might be of interest, especially if it concentrates more samples around diagonal matrices $\text{diag}(\pm1,\dots,\pm1)$.



Sample from a delta-ball in the orthogonal group O(n)
In case anyone else comes across this: the model I ended up using is by León, Massé, and Rivest in the Journal of Multivariate Analysis (see [here](http://www.sciencedirect.com/science/article/pii/S0047259X05000382)). They give a distribution on the space of skew-symmetric matrices that gives an arbitrarily concentrated distribution on the orthogonal group: that is, for dispersion parameter $\kappa=0$ the resulting distribution is uniform, and as $\kappa\rightarrow\infty$ the distribution approaches a constant distribution.


EDIT: The limit is a Gaussian distribution (analogous to the degrees of freedom of a t distribution approaching $\infty$), not constant.


Thanks again to those who commented/answered as these suggestions put me on the path towards the method I ultimately found.


