November 2023 Parking on Cayley trees and frozen Erdős–Rényi
Alice Contat, Nicolas Curien
Author Affiliations +
Ann. Probab. 51(6): 1993-2055 (November 2023). DOI: 10.1214/23-AOP1632


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 and Panholzer (J. Combin. Theory Ser. A 142 (2016) 1–28) established a phase transition for this process when mn2. In this work, we couple this model with a variant of the classical Erdős–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.

Funding Statement

We acknowledge support from ERC 740943 GeoBrown.


We thank Linxiao Chen, Armand Riera and especially Olivier Hénard for several motivating discussions during the elaboration of this work. We thank Svante Janson and Cyril Banderier for comments about the first version of this work. Last, but not least, we warmly thank the anonymous referee for a thorough reading of our paper and precious comments which were greatly appreciated.


Download Citation

Alice Contat. Nicolas Curien. "Parking on Cayley trees and frozen Erdős–Rényi." Ann. Probab. 51 (6) 1993 - 2055, November 2023.


Received: 1 March 2022; Revised: 1 April 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1632

Primary: 60C05
Secondary: 05C05 , 05C80 , 05C85

Keywords: Erdős–Rényi random graph , multiplicative coalescent , Parking process , planar maps , Random trees

Rights: Copyright © 2023 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.51 • No. 6 • November 2023
Back to Top