Probabilities of rare events in product kernel aggregation: An exact formula and phase diagram

TL;DR

This paper derives an exact formula for the probability of rare fluctuations in product-kernel aggregation, revealing a phase diagram with a tricritical point and singular behaviors in the large deviation function.

Contribution

It introduces an exact integral representation for the probability distribution and large deviation function in product-kernel aggregation, including a phase diagram with a tricritical point.

Findings

Exact integral representation for $P(M,N,t)$

Identification of a tricritical point in the phase diagram

Singular behavior in the large deviation function

Abstract

We present an exact method for calculating the large deviation function describing rare fluctuations in the number of particles for product-kernel aggregation. Starting from the master equation, we derive an exact integral representation for the probability $P(M,N,t)$ of observing $N$ particles at time $t$ starting from $M$ monomers for any finite $M, N, t$. From this, we obtain an exact expression for the exponential moment $\langle p^N\rangle$ for integer $p$. Employing a replica conjecture -- numerically validated by finite-$M$ scaling -- we extend this result to real $p \geq 0$. The convex envelope of the large deviation function, obtained via a Legendre-Fenchel transform of the exponential moment, shows singular behavior. The singular structure allows us to construct the full phase diagram of product-kernel aggregation, which contains a tricritical point, separating continuous and discontinuous transitions. We also compute the asymptotic form of the LDF for small $N/M$.