透过双曲空间可见：λ-测地超平面的可见性
Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes
AI总结：
该研究探讨了双曲空间中λ-测地超平面的可见性，证明了基本可见性性质对参数λ的不变性，并确定了临界强度γ_crit，区分了无界和有界可见区域。
摘要：
我们研究在双曲空间中固定点处存在λ-测地超平面的泊松过程时的可见性。λ-测地超平面家族介于完全测地超平面和水平面之间。我们的主要结果建立了该模型的普遍原则：我们证明了基本可见性性质对参数λ∈[0,1]是不变的。即存在一个临界强度γ_crit>0，使得γ<γ_crit时可见区域以正概率无界，而γ>γ_crit时可见区域几乎必然有界。对于d=2，我们也在临界点处证明了几乎必然的有界性。γ_crit的值是明确的，不依赖于λ。在有界阶段，我们显示可见体积的均值与已知的λ=0公式相同。关键的积分几何步骤是显式计算显示λ-测地超平面击中测地线段的测度是线性函数，与λ无关。
英文摘要：
We study visibility from a fixed point in the presence of a Poisson process of $λ$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $λ$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $λ\in[0,1]$. Namely, there is a critical intensity $γ_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $γ< γ_{\mathrm{crit}}$ and almost surely bounded for $γ> γ_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $γ_{\mathrm{crit}}$ is explicit and does not depend on $λ$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $λ=0$. The key integral-geometric step is an explicit computation showing that the measure of $λ$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$λ$.
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