A quantitative central limit theorem for Poisson horospheres in high dimensions

TL;DR

This paper establishes a quantitative central limit theorem for the total surface area of Poisson horospheres in high-dimensional hyperbolic spaces as both the radius and dimension grow, revealing asymptotic normality.

Contribution

It introduces a novel non-standard CLT for the surface area of Poisson horospheres in high dimensions, considering simultaneous growth of radius and dimension.

Findings

Asymptotic normality of surface area in high dimensions

Quantitative bounds for convergence to normal distribution

Behavior of horospheres in large-scale hyperbolic spaces

Abstract

Consider a stationary Poisson process of horospheres in a $d$-dimensional hyperbolic space. In the focus of this note is the total surface area these random horospheres induce in a sequence of balls of growing radius $R$. The main result is a quantitative, non-standard central limit theorem for these random variables as the radius $R$ of the balls and the space dimension $d$ tend to infinity simultaneously.