Spouge's Conjecture on Complete and Instantaneous Gelation

摘要

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)~(#alpha#), #alpha#>1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function #PHI#(i, j), increasing in both i and j such that sum_(j=1)~(infinity) 1/(j#PHI#(i, j))< infinity for all i, and K(i, j)>=ij#PHI#(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))~(#alpha#), #alpha#>1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim_(i+j->infinity)ij/K(i, j) = 0 and sum_(i, j=1)~(infinity) 1/K(i, j)=infinity, then gelation time T_g satisfies 0