2021/07/05 by Alice Contat, Nicolas Curien, Contat, Alice +1
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2107.02116
openalex publication_date 2021/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a uniform rooted 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 parks as soon as possible. Lackner & Panholzer (arXiv:1504.04972) established a phase transition for this process when m ≈ (n)/(2). In this work, we couple this model with a variant of the classical Erdös-Rényi random graph process. This enables us to describe the phase transition for the size of the components of parked cars using a modification of the multiplicative coalescent which we name the frozen multiplicative coalescent. The geometry of critical parked clusters is also studied. Those trees are very different from Bienaymé-Galton-Watson trees and should converge towards the growth-fragmentation trees canonically associated to the 3/2-stable process that already appeared in the study of random planar maps.