Spouge's Conjecture on Complete and Instantaneous Gelation

Abstract

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij)^α, α>1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function ψ(i, j), increasing in both i and j such that ∑^∞ _j=1 1/(jψ(i, j))<∞ for all i, and K(i, j)≥ijψ(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)≥ij(log(i+1)log(j+1))^α, α>1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j→∞ ij/K(i, j)=0 and ∑^∞ _i, j=1 1/K(i, j)=∞, then gelation time T _g satisfies 0<T _g<∞. Moreover, the gelation is complete after T _g.