A quantitative central limit theorem for Poisson horospheres in high dimensions (Q6628939)

The study of random geometric systems in non-Euclidean geometries is a recent and fast growing branch of stochastic geometry, see [\textit{F. Besau} et al., Math. Ann. 381, No. 3--4, 1345--1372 (2021; Zbl 1486.52007)]. The study of Euclidean Poisson hyperplanes is by now classical and was extended in [\textit{I. Benjamini} et al., Ann. Probab. 46, No. 4, 1917--1956 (2018; Zbl 1430.60019)] to hyperbolic space, where Poisson processes of totally geodesic hypersurfaces are studied. A horosphere in a \(d\)-dimensional hyperbolic space \(\mathbf{H}^d\) is, intuitively speaking, a sphere of infinite radius. More formally, it is a complete totally umbilic hypersurface of constant normal curvature 1. For concreteness, in the Poincaré ball model of hyperbolic space, horospheres are realized as Euclidean spheres tangent to the boundary. Let \(\mathcal H\) be the space of all horospheres in \(\mathbf{H}^d\). This space admits a transitive action by the group of hyperbolic isometries and an invariant measure for this action, which is unique up to a multiplicative constant and will be denoted by \(\Lambda\). Now, let \(\eta_d\) be a Poisson process on \(\mathcal H\) with intensity measure \(\Lambda\). For \(R >0\), consider the total surface area \(S_{R, d} := \sum_{H\in \eta_d} {\mathcal H}^{d-1}(H\cap B^d_R)\) of \(\eta_d\) within a hyperbolic ball \(B_r^d\) around an arbitrary but fixed point in \(\mathbf{H}^d\) and having hyperbolic radius \(R > 0\). Here, \(\mathcal{H}^{d-1}\) stands for the \((d -1)\)-dimensional Hausdorff measure with respect to the hyperbolic metric. To measure the distance between two random variables \(X\) and \(Y\) the authors use the Kolmogorov \(d_{\mathrm{Kol}}(X, Y)\) and Wasserstein \(d_{\mathrm{Wass}}(X, Y)\) metrics. The main result of the paper is the following quantitative non-standard central limit theorem. \N\NTheorem 1. Let \(N_{1/2}\) be a centred Gaussian random variable of variance \(1/2\). Consider the surface functional \(S_{R, d}\), for \(d\ge 2\) and \(R\ge 1\). Then there exists a universal constant \(C >0\) such that for any choice \(t\in \{\mathrm{Kol, Wass}\}\) one has \N\[\Nd_t\left(\frac{S_{R, d} - ES_{R, d}}{\sqrt{\mathrm{Var} S_{R, d}}}, N_{1/2}\right)\le \N\begin{cases}\NC \cdot e^{-R/2},&\text{ if } R-\log d\le 1\\\NC \cdot\left(\frac{1}{\sqrt{d}(R -\log d)} + \frac{1}{d \sqrt{R -\log d}}\right)&\text{ if }R- \log d > 1.\N\end{cases}\N\]