# Cross-Multiplicative Coalescent Processes and Applications

## TL;DR

This paper introduces the cross-multiplicative coalescent process, analyzes its mathematical properties, and applies it to determine the limiting mean length of minimal spanning trees on bipartite graphs.

## Contribution

It presents a new coalescent process model, analyzes its hydrodynamic limit, and applies it to bipartite graph spanning tree problems.

## Key findings

- The process is a gelling kernel with a specific gelation time.
- Derived the hydrodynamic limit equations for the process.
- Calculated the limiting mean length of minimal spanning trees on bipartite graphs.

## Abstract

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