Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914688738000896 |
|---|---|
| author | Kabluchko, Zakhar Rosen, Daniel Thäle, Christoph |
| author_facet | Kabluchko, Zakhar Rosen, Daniel Thäle, Christoph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12820 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space Kabluchko, Zakhar Rosen, Daniel Thäle, Christoph Probability Metric Geometry 52A55, 60D05 (Primary), 60F05, 60G55 (Secondary) 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. |
| title | Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space |
| topic | Probability Metric Geometry 52A55, 60D05 (Primary), 60F05, 60G55 (Secondary) |
| url | https://arxiv.org/abs/2205.12820 |