Parking on the infinite binary tree

TL;DR

This paper analyzes a parking process on an infinite binary tree where cars arrive randomly and attempt to park, revealing a phase transition between partial and complete parking regimes based on the distribution of arrivals.

Contribution

It explicitly characterizes the phase transition regimes for parking on the binary tree in terms of the distribution of car arrivals and studies the critical and discontinuous nature of the transition.

Findings

Explicit characterization of subcritical and supercritical regimes

Analysis of the critical regime behavior

Identification of a discontinuous phase transition

Abstract

Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $ \mathbb{B}$. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known that the parking process on $ \mathbb{B}$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of $A$ in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be "discontinuous".