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 $p\ge 0$. 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 large-deviation function for small $N/M$.

Close

PrevNext

View figure in article

• Aggregation
• Growth processes
• Irreversible processes
• Nonequilibrium statistical mechanics
• Phase transitions
• Aggregates
• Large deviation & rare event statistics
• Path-integral methods