Cross-multiplicative coalescent processes and applications

We introduce and analyze a novel type of coalescent processes called cross-multiplicative that models system with two types particles, $A$ $B$. The bonds are formed only between the pairs particles opposite same rate for each bond, producing connected components made both types. solve Smoluchowski coagulation equations obtained as hydrodynamic limit corresponding Marcus-Lushnikov process. establish kernel is gelling kernel, find gelation time. As an application, we derive limiting mean length minimal spanning tree on complete bipartite graph $K_{\alpha[n], \beta[n]}$ partitions sizes $\alpha[n]=\alpha n +o(\sqrt{n})$ $\beta[n]=\beta independent edge weights, distributed uniformly over $[0, 1]$.