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Kac Interaction Clusters: A Bilinear Coagulation Equation and Phase Transition

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One approach to a collisional particle system is to consider `particle histories’, which leads to the idea of interaction clusters for Kac’s process. If the collision kernel is of a particular quadratic form, the interaction clusters follow a bilinear Smoluchowski- style equation, for which a relatively complete analysis is possible using a connection to random graphs. Among other properties, we see that a macroscopic interaction cluster arises strictly before the mean free time.

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