# Rare Events Statistics for ℤd Map Lattices Coupled by Collision

## TL;DR

This paper studies collision statistics in a lattice model of gas particles to better understand rare collision events.

## Contribution

The paper introduces a novel approach using transfer operators to analyze collision rates and times in infinite-dimensional systems.

## Key findings

- A first-order approximation for the collision rate at a lattice site is derived.
- The first collision time converges to an exponential distribution with a sharp error term.
- The number of collisions converges 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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {Z}^d$$\end{document}Zd-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 \documentclass[12pt]{minimal}
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\begin{document}$${\textbf{p}}^*\in \mathbb {Z}^d$$\end{document}p∗∈Zd 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 \documentclass[12pt]{minimal}
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\begin{document}$${\textbf{p}}^*$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$${\textbf{p}}^*$$\end{document}p∗.