Sharpness of the phase transition for parking on random trees

TL;DR

This paper extends the understanding of phase transitions in parking processes on random trees, demonstrating sharpness of the transition and revealing geometric similarities with Erdős-Rényi graphs.

Contribution

It generalizes previous results to Galton-Watson trees with variable car arrival distributions and proves the sharpness of the phase transition via large deviations.

Findings

Phase transition in parking on random trees is sharp.

Cluster geometry resembles Erdős-Rényi graph structure.

Results apply to general Galton-Watson trees with variable arrivals.

Abstract

Recently, a phase transition phenomenon has been established for parking on random trees. We extend the results of Curien and H\'enard on general Galton--Watson trees and allow different car arrival distributions depending on the vertex outdegrees. We then prove that this phase transition is sharp by establishing a large deviations result for the flux of exiting cars. This has consequences on the offcritical geometry of clusters of parked spots which displays similarities with the classical Erd\H{o}s-Renyi random graph model.