# Distribution of the hyperbolic distance between random points in the Poincaré disc

Let two points at polar coordinates $(r_i, \theta_i)$ and $(r_j, \theta_j)$ be hyperbolic points in $\mathbb{H}^2_\zeta$ with curvature $K=-\zeta^2$. The radial coordinates of these points are exponentially distributed with exponent $\alpha$: $p(r) = \alpha e^{\alpha(r - R)}$ with $R \approx \ln(N)$ (the radius of the Poincaré disc), $N$ being the number of points in this disc, exponent $\alpha > 0$ and $r \in [0, R]$. The angular coordinates are uniformly distributed: $p(\theta) = \frac{1}{2\pi}$ with $\theta \in [0, 2\pi]$.

For simplicity, let $\zeta = 1$. As a result, the hyperbolic distance between points in $\mathbb{H}^2_1$ is defined as follows:

$d_{i, j} = \text{acosh}[\cosh(r_i)\cosh(r_j) - \sinh(r_i)\sinh(r_j)\cos(\theta_{i, j})]$

with $\theta_{i, j} = \pi - |\pi - |\theta_i - \theta_j||$. What's the distribution of distances between random points with the above characteristics in the hyperbolic disc of radius $R$?

Edit

I've been playing with the generation of random points using $p(r)$ and $p(\theta)$ for different values of $\alpha, N=1000$ and $\zeta=1$. These are the resulting point densities and hyperbolic distance distributions:

By eye, the hyperbolic distances seem to be $Beta$ distributed. Of course a proper analysis has to be carried out to confirm this hypothesis...

• I've been playing with the generation of random points using $p(r)$ and $p(\theta)$ for different values of $\alpha$, $N = 1000$ and $\zeta = 1$. These are the resulting point densities and hyperbolic distance distributions: link – gal Aug 21 '13 at 18:00
• Because $r$ can lie only in the interval $[0, R]$, $p$ is not a density function. When renormalized, it would be the density of a truncated exponential distribution. This makes it highly unlikely the resulting distances will have a (rescaled) Beta distribution. What statistical problem is this distribution attempting to model? – whuber May 11 '15 at 14:03