On the occurrence of a gelation transition in Smoluchowski's coagulation equation

It has been conjectured by Lushnikov and Ziff that Smoluchowski's coagulation equation describes a gelation transition, i.e., the mean cluster size diverges within a finite timetc (gelpoint) if the coagulation rate constantsK(i,j) have the propertyK(ai,aj)=aλK(i,j), with λ>1. The existing evidence was based on self-consistency arguments. Here we prove this conjecture for an appropriate class of physically acceptable rate constants by constructing a finite upper bound fortc and a nonvanishing lower bound. Apart from the exactly solved caseK(i,j)=ij this result provides the first solid proof of the occurrence of a gelation transition in a description based on Smoluchowski's coagulation equation.