Renormalization group for Gibbs point processes
Proving the existence of phase transitions and condensation for particles in continuous space is a challenging open problem in mathematical statistical physics. We propose a program to make the droplet picture of condensation rigorous. The key tool to be developed is a renormalization group theory for Gibbs point processes that is applied to mixtures of droplets, for example mixtures of hard spheres of different sizes. The relevant systems are multiscale; different scales correspond to different droplet sizes. In the new renormalization theory, the superposition principle of Poisson point processes plays a role similar to the additivity of covariances of independent Gaussian fields. We apply the theory to compute free energies and density functionals and to prove first-order phase transitions. The program is of direct relevance to Gibbs hard-core models in stochastic geometry and connects to diagrammatic techniques for point processes. The droplet picture is also highly relevant for effective coagulation-fragmentation models in dynamic studies of condensation and metastability.