Parking on Cayley trees and Frozen Erdös-Rényi
Consider a uniform Cayley tree Tn with n vertices and let m cars arrive
sequentially, independently, and uniformly on its vertices. Each car tries
to park on its arrival node, and if the spot is already occupied, it drives
towards the root of the tree and park as soon as possible. Using
combinatorial enumeration, Lackner & Panholzer established a phase
transition for this process when m is approximately n/2. We couple
this model with a variation of the classical Erdös–Rényi random graph
process. This enables us to completely describe the phase transition for
the size of the components of parked cars using a modification of the
standard multiplicative coalescent which we named the frozen multiplicative
coalescent. The geometry of critical parked clusters in the parking process
is also studied. Those trees are very different from usual random trees and
should converge in the scaling limit towards Jean's growth-fragmentation
trees canonically associated to 3/2-stable process that already appeared in
the study of random planar maps.
based on joint work with Alice Contat.