Scholarlink

<p>Poisson processes of so-called <i>λ</i>-geodesic hyperplanes in <i>d</i>-dimensional hyperbolic space are studied for 0 ≤ <i>λ</i> ≤ 1. The case <i>λ</i> = 0 corresponds to genuine geodesic hyperplanes, the case <i>λ</i> = 1 to horospheres and <i>λ</i> ∈ (0, 1) to <i>λ</i>-equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all <i>λ</i>-geodesic hyperplanes in the Poisson process within a hyperbolic ball of radius <i>R</i> centred at some fixed point, as <i>R</i> → ∞. It is shown that for <i>λ</i> &lt; 1 these random variables satisfy a quantitative central limit theorem precisely for <i>d</i> = 2 and <i>d</i> = 3. The exact form of the non-Gaussian, infinitely divisible limiting distribution is determined for all higher space dimensions <i>d</i> ≥ 4. The special case <i>λ</i> = 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 <i>d</i> ≥ 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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Fluctuations of λ-geodesic Poisson hyperplanes in hyperbolic space

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年6月29日

查看全文
  • Zakhar Kabluchko,
  • Daniel Rosen,
  • Christoph Thäle

摘要

Poisson processes of so-called λ-geodesic hyperplanes in d-dimensional hyperbolic space are studied for 0 ≤ λ ≤ 1. The case λ = 0 corresponds to genuine geodesic hyperplanes, the case λ = 1 to horospheres and λ ∈ (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 → ∞. 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 ≥ 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 ≥ 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.