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

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 〈p^{N}〉 for integer p. Employing a replica conjecture-numerically validated by finite-M scaling-we extend this result to real p≥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.