Rare Events Statistics for Z d Map Lattices Coupled by Collision

Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $Z^d$ -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 $p^{\ast }\in Z^d$ 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 $p^{\ast }$ 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 $p^{\ast }$ .