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Gelation for Marcus-Lushnikov process

The Marcus-Lushnikov process is a simple mean field model of coagulating particles that converges to the homogeneous Smoluchowski equation in the large mass limit. If the coagulation rates grow sufficiently fast as the size of particles get large, giant particles emerge in finite time. This is known as gelation, and such particles are known as gels. Gelation comes in different flavors: simple, instantaneous and complete. In the case of an instantaneous gelation, giant particles are formed in a very short time. If all particles coagulate to form a single particle in a time interval that stays bounded as total mass gets large, then we have a complete gelation. In this article, we describe conditions which guarantee any of the three possible gelations with explicit bounds on the size of gels and the time of their creations.