论文标题
$λ$ -GEODESIC POISSON超平面在双曲空间中的波动
Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space
论文作者
论文摘要
以$ 0 \ 0 \leqλ\ leq 1 $研究，研究了所谓的$λ$ - 地理超纯平面的泊松过程。 $λ= 0 $的情况对应于真正的大地测量平面，holosphers的$λ= 1 $，$λ\ in（0,1）$到$λ$ - 等式的人。重点是在泊松过程中，在某个固定点的半径$ r $ r $ radius $ r $中，所有$λ$ - 地理性超平面的集中和归一化的总表面积的波动，以$ r \至\ infty $。结果表明，对于$λ<1 $，这些随机变量满足定量中心限制定理，恰恰是$ d = 2 $和$ d = 3 $。非高斯，无限划分的限制分布的确切形式是针对所有较高空间尺寸$ d \ geq 4 $的。特殊情况$λ= 1 $与这种行为形成鲜明对比。实际上，对于霍斯莱斯泊松过程的总表面积，为所有空间尺寸$ d \ geq 2 $建立了限制差异$ 1/2 $的非标准中心极限定理。我们讨论了此处研究的问题与随机能量模型之间的类比，其分区函数表现出可能的极限定律的相似结构。
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.
