Gelation, hydrodynamic limits and uniqueness in cluster coagulation processes

Luisa Andreis, Tejas Iyer, Elena Magnanini

Published 2023-08-20Version 1

We consider the problem of gelation in the cluster coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)], where clusters take values in a measure space $E$, and merge to form a new particle $z$ according to a transition kernel $K(x,y, \mathrm{d} z)$. This model is general enough to incorporate various inhomogenieties in the evolution of clusters, for example, their shape, or their location in space. We derive general, sufficient criteria for stochastic gelation in this model, and for trajectories associated with this process to concentrate among solutions of a generalisation of the Flory equation; thus providing sufficient criteria for the equation to have gelling solutions. As particular cases, we extend results related to the classical Marcus-Lushnikov coagulation process and Smoluchowski coagulation equation, showing that reasonable `homogenous' coagulation processes with exponent $\gamma>1$ yield gelation; and also, coagulation processes with kernel $\alpha(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +\varepsilon}$ for $\varepsilon>0$. In another special case, we prove a law of large numbers for the trajectory of the empirical measure of the stochastic cluster coagulation process, by means of a uniqueness result for the solution of the aforementioned generalised Flory equation. Finally, we use coupling arguments with inhomogeneous random graphs to deduce sufficient criterion for strong gelation (the emergence of a particle of size $O(N)$).