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We consider the problem of gelation in the cluster coagulation modelintroduced 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 newparticle $z$ according to a transition kernel $K(x,y, \mathrm{d} z)$. Thismodel is general enough to incorporate various inhomogenieties in the evolutionof clusters, for example, their shape, or their location in space. We derivegeneral, sufficient criteria for stochastic gelation in this model, and fortrajectories associated with this process to concentrate among solutions of ageneralisation of the Flory equation; thus providing sufficient criteria forthe equation to have gelling solutions. As particular cases, we extend resultsrelated to the classical Marcus-Lushnikov coagulation process and Smoluchowskicoagulation equation, showing that reasonable `homogenous' coagulationprocesses with exponent $\gamma>1$ yield gelation; and also, coagulationprocesses 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 oflarge numbers for the trajectory of the empirical measure of the stochasticcluster coagulation process, by means of a uniqueness result for the solutionof the aforementioned generalised Flory equation. Finally, we use couplingarguments with inhomogeneous random graphs to deduce sufficient criterion forstrong gelation (the emergence of a particle of size $O(N)$).

Gelation, hydrodynamic limits and uniqueness in cluster coagulation processes