Spouge's conjecture on complete and instantaneous gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation e quation, namely the Marcus-Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluste r in an arbitrarily small lime, which is known as a complete and instantane ous gelation phenomenon. Indeed, Spouge (also Domilovskii et nl. for a spec ial case) conjectured that K(i, j) = (ij)(alpha), alpha > 1, are such kerne ls. In this paper, we extend the above conjecture and prove rigorously that if there is a function psi(i, j), increasing in both i and j such that Sig ma(j)(= 1)(infinity), 1/(j psi(i, j)) < infinity for all i, and K(i, j) gre ater than or equal to ij psi(i, j) for all i, j, then complete and instanta neous gelation occurs. Evidently, this implies that any kernels K( i, j) gr eater than or equal to ij(log(i + 1) log(j + 1))(alpha), alpha > 1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a c ritical (or metastable) sol state: if lim(i + j --> infinity) ij/K(i, j) = 0 and Sigma(i,)(j = 1)(infinity) 1/K(i, j) = infinity, then gelation time T -g satisfies 0 < T-g < infinity. Moreover, the gelation is complete after T -g.