Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space

Abstract

Poisson processes of so-called $λ$-geodesic hyperplanes in $d$-dimensional hyperbolic space are studied for $0\leqλ\leq 1$. The case $λ=0$ corresponds to genuine geodesic hyperplanes, the case $λ=1$ to horospheres and $λ\in(0,1)$ to $λ$-equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all $λ$-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 $λ<1$ 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 $λ=1$ 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.

Figures (3)

• Figure 1: Left panel: Simulation of a hyperbolic Poisson hyperplane process in the hyperbolic plane (corresponding to the choice $\lambda=0$). Right panel: Simulation of a Poisson process of horospheres in the hyperbolic plane (corresponding to the choice $\lambda=1$).
• Figure 2: $\lambda$-geodesic lines in the upper half-plane model for a hyperbolic plane: (i) and (ii) correspond to $\lambda=0$ (geodesic lines), (iii) and (iv) to $0<\lambda<1$ (equidistants), (v) and (vi) to $\lambda=1$ (horocycles).
• Figure 3: Left panel: Horospheric section of the ball. Right panel: A linear equidistant.

Theorems & Definitions (28)

• Example 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Example 2.7
• Theorem 2.8
• Remark 2.9
• Remark 2.10