Shape theorem and surface fluctuation for Poisson cylinders

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.

19 pages, 3 figures