Rare events statistics for $\mathbb Z^d$ map lattices coupled by collision
Wael Bahsoun, Maxence Phalempin
TL;DR
This work analyzes rare-event statistics for a $\,\mathbb{Z}^d$-map lattice where neighboring sites interact by collision. The authors build a robust transfer-operator framework around a decoupled site dynamics and prove a spectral gap for the relevant operators, yielding a first-collision rate $-\ln\lambda_{\varepsilon,\delta}$ and a sharp exponential law for the first hitting time. They then derive an extremal index $\theta$ governing clustering of collisions in the small-collision limit $\delta\to0$, and establish a compound Poisson limit for the counting process $Z_\delta(t)$ with intensity $\theta\,\mathrm{Leb}$ and a structured jump distribution determined by a twisted transfer operator. An explicit example in one dimension demonstrates $\theta=1-5^{-4}$ and, under the general theory, the counting process converges to a Poisson process when clustering is absent. A key methodological contribution is the use of spectral analysis of rare-event transfer operators, coupled with a perturbation approach in the Keller–Liverani framework, adapted to the infinite-dimensional setting of a $\mathbb{Z}^d$ lattice.
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 $\textbf{p}^*$.