Rare events statistics for $\mathbb Z^d$ map lattices coupled by collision

TL;DR

This paper studies collision statistics in $ extbf{Z}^d$ map lattices with simplified local dynamics, deriving approximations and distributional limits for collision times and counts, using transfer operators in an infinite-dimensional setting.

Contribution

It introduces a first order approximation for collision rates and proves distributional convergence results for collision times and counts in coupled map lattices.

Findings

First collision rate approximation at a lattice site

Distributional convergence of collision times to exponential

Collision counts converge to a compound Poisson distribution

Abstract

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $\mathbb Z^d$-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site $\textbf{p}^*\in \mathbb Z^d$ and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site $\textbf{p}^*$ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site $\textbf{p}^*$.