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

by Goutham R. (The Institute of Mathematical Sciences, Chennai, India)

Tuesday 22 Sept 2026, 10:00 → 11:00 Europe/Rome

P1A (Dipartimento di Fisica e Astronomia - Edificio Ricci-Curbastro (ex-Paolotti))

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 $N/M$.