Exact calculation of the large deviation function for $k$-nary coalescence

We study probabilities of rare events in the general coalescence process, $kA\to \mathrm{\ell }A$, where $k>\mathrm{\ell }$. For arbitrary $k,\mathrm{\ell }$, by rewriting these probabilities in terms of an effective action, we derive the large deviation function describing the probability of finding $N$ particles at time $t$, when starting with $M$ particles initially. Additionally, the most probable trajectory corresponding to a fixed rare event is derived.