Shape theorem and surface fluctuation for Poisson cylinders

TL;DR

This paper establishes a shape theorem for Poisson cylinders and provides a power law bound on surface fluctuations, showing that points within a certain Euclidean distance are close in internal distance, conditioned on the origin being in the set.

Contribution

It introduces a shape theorem for Poisson cylinders and quantifies surface fluctuations with a power law bound, advancing understanding of geometric properties of these structures.

Findings

Proves a shape theorem for Poisson cylinders.

Establishes a power law bound on surface fluctuations.

Shows points within Euclidean radius R are close in internal distance, conditioned on the origin.

Abstract

In this work, we prove a shape theorem for Poisson cylinders and give a power law bound on surface fluctuations. We prove that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, every point in this set, whose Euclidean norm is less than $R$, lies at an internal distance less than $R+O(R^a)$ from the origin.