Parking on Cayley trees & Frozen Erd\"os-R\'enyi

TL;DR

This paper studies a parking process on Cayley trees coupled with a variant of Erd"os-Rényi graphs, revealing a phase transition in cluster sizes and connecting to growth-fragmentation trees related to stable processes.

Contribution

It introduces the frozen multiplicative coalescent to describe phase transitions in parking cluster sizes on Cayley trees coupled with Erd"os-Rényi processes.

Findings

Identifies a phase transition in the size of parked car components.

Describes the geometry of critical parked clusters.

Connects the structure to growth-fragmentation trees associated with 3/2-stable processes.

Abstract

Consider a uniform rooted Cayley tree $T_{n}$ 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 \approx \frac{n}{2}$. In this work, we couple this model with a variant of the classical Erd\"os-R\'enyi 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 Bienaym\'e-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.