2022/05/31 by David Aldous, Alice Contat, Aldous, David +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2205.15932
openalex publication_date 2022/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (Au : u ∈ \mathbbB) be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree \mathbbB. 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 \mathbbB 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".