# Kac Interaction Clusters: A Bilinear Coagulation Equation and Phase Transition

@article{Heydecker2019KacIC, title={Kac Interaction Clusters: A Bilinear Coagulation Equation and Phase Transition}, author={Daniel Heydecker and Robert I. A. Patterson}, journal={arXiv: Probability}, year={2019} }

We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing well-posedness, and a closed form for the time of the gelation phase transition $t_\mathrm{g}$ when a…

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