Gelation in cluster coagulation processes

Abstract: We consider the problem of gelation in the cluster coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)], where pairs of clusters of types $(x,y)$ taking values in a measure space $E$, merge to form a new particle of type $z\in E$ according to a transition kernel $K(x,y, \mathrm{d} z)$.This model possesses enough generality to accommodate inhomogenieties in the evolution of clusters, including variations in their shape or spatial distribution. We derive general, sufficient criteria for stochastic gelation and strong gelation in this model. As particular cases, we extend results related to the classical Marcus--Lushnikov coagulation process, showing that reasonable `homogenous' coagulation processes with exponent $\gamma>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +\epsilon}$ for $\epsilon>0$. Finally, we use coupling arguments with inhomogeneous random graphs to deduce sufficient criterion for strong gelation (the emergence of a particle of size $O(N)$).