Large deviations for sparse systems of moving particles

We study large deviations for rare clusters in sparse systems of moving particles. In the regime \(nr_n^d\to0\) and \(ρ_{k,n}=n^kr_n^{d(k-1)}\to\infty\), we prove a large deviation principle for the empirical measure of isolated \(k\)-particle trajectory clusters. The speed is \(ρ_{k,n}\), and the r