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

TL;DR

This paper derives an exact large deviation function for the probability of rare events in a general coalescence process, providing insights into the likelihood of specific particle configurations over time.

Contribution

It introduces a method to compute the large deviation function for arbitrary coalescence parameters, including the most probable trajectories for rare events.

Findings

Derived the large deviation function for general $k, \, \\ell$ coalescence processes.

Provided explicit formulas for the probability of having $N$ particles at time $t$.

Identified the most probable trajectories associated with rare particle configurations.

Abstract

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.