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 inhomogeneities in the evolution of clusters, including variations in their shape or spatial distribution. We derive general, sufficient criteria for stochastic 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 ${\hat{I}}^3>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)\ge (m\wedge n)\log {(m\wedge n)}^{3+\hat{I}\mu }$ for $\hat{I}\mu >0$.

Shortened to 19 pages. The conditions for gelation (Theorem 2.2 and Corollary 2.3) have been significantly simplified incorporating referee suggestions. This version coincides with the one accepted for publication in Ann. Inst. H. Poincaré Probab. Statist