• Corpus ID: 119612054

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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