Exact Calculation of the Large Deviation Function for k-nary Coalescence

We study probabilities of rare events in the general coalescence process, \(kA\rightarrow \ell A\), where \(k>\ell \). For arbitrary \(k, \ell \), 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.