Fluctuations of $\lambda$-geodesic Poisson hyperplanes in hyperbolic space

Abstract

Poisson processes of so-called $\lambda$-geodesic hyperplanes in $d$-dimensional hyperbolic space are studied for $0\leq\lambda\leq 1$. The case $\lambda=0$ corresponds to genuine geodesic hyperplanes, the case $\lambda=1$ to horospheres and $\lambda\in(0,1)$ to $\lambda$-equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all $\lambda$-geodesic hyperplanes in the Poisson process within a hyperbolic ball of radius $R$ centred at some fixed point, as $R\to\infty$. It is shown that for $\lambda<1$ these random variables satisfy a quantitative central limit theorem precisely for $d=2$ and $d=3$. The exact form of the non-Gaussian, infinitely divisible limiting distribution is determined for all higher space dimensions $d\geq 4$. The special case $\lambda=1$ is in sharp contrast to this behaviour. In fact, for the total surface area of Poisson processes of horospheres, a non-standard central limit theorem with limiting variance $1/2$ is established for all space dimensions $d\geq 2$. We discuss the analogy between the problem studied here and the Random Energy Model whose partition function exhibits a similar structure of possible limit laws.