Gelation in cluster coagulation processes

TL;DR

This paper investigates gelation phenomena in a general cluster coagulation model, providing criteria for gelation and extending classical results to more complex, inhomogeneous systems with diverse kernels.

Contribution

It derives general criteria for stochastic gelation in a broad coagulation model, extending classical results to inhomogeneous and complex kernels.

Findings

Homogeneous coagulation processes with exponent > 1 exhibit gelation.

Processes with kernels ≥ (m ∧ n) log(m ∧ n)^{3+ε} also show gelation.

General criteria applicable to inhomogeneous cluster evolution models.

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 $\gamma>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +\epsilon}$ for $\epsilon>0$.