Parking on supercritical Galton-Watson trees

TL;DR

This paper studies parking dynamics on supercritical Galton-Watson trees, revealing phase transitions, growth rates of expected cars at the root, and effects of initial distribution concentration, with specific bounds for d-ary trees.

Contribution

It establishes finiteness of the expected number of cars at the critical point, characterizes growth above criticality, and analyzes how initial distribution affects outcomes, including bounds for d-ary trees.

Findings

$EX$ is finite at the critical threshold.

Expected cars grow at a specific rate above criticality.

Probability of zero cars at the root is discontinuous in certain cases.

Abstract

At each site of a supercritical Galton-Watson tree place a parking spot which can accommodate one car. Initially, an independent and identically distributed number of cars arrive at each vertex. Cars proceed towards the root in discrete time and park in the first available spot they come to. Let $X$ be the total number of cars that arrive to the root. Goldschmidt and Przykucki proved that $X$ undergoes a phase transition from being finite to infinite almost surely as the mean number of cars arriving to each vertex increases. We show that $EX$ is finite at the critical threshold, describe its growth rate above criticality, and prove that it increases as the initial car arrival distribution becomes less concentrated. For the canonical case that either 0 or 2 cars arrive at each vertex of a $d$-ary tree, we give improved bounds on the critical threshold and show that $P(X = 0)$ is discontinuous.