Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures

TL;DR

This paper proves Dettmann's geometric conjectures on the tail behavior of free path lengths in periodic Lorentz gases, extending results to more general convex body configurations and connecting to classical visibility problems.

Contribution

It establishes Dettmann's conjectures for general lattice-periodic convex bodies, broadening understanding of free path length distributions in Lorentz processes.

Findings

Probability tail behaves as C/t for large t

Results apply to intersecting convex bodies configurations

Provides asymptotic covariance in super-diffusive Lorentz process

Abstract

In the simplest case, consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann's first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than $t >>1$ is $\sim \frac{C}{t}$, where $C$ is explicitly given by the geometry of the model. In its simplest form, Dettmann's second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for $\mathcal{L}$-periodic configuration of - possibly intersecting - convex bodies with $\mathcal{L}$ being a non-degenerate lattice. These questions are related to P\'olya's visibility problem (1918), to theories of Bourgain-Golse-Wennberg (1998-) and of Marklof-Str\"{o}mbergsson (2010-). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if $d = 2$ and the horizon is infinite.