Parking on a Random Tree

TL;DR

This paper analyzes a parking problem on an infinite random tree, deriving an explicit formula for the parking constant based on the degree distribution's generating function.

Contribution

It introduces a novel variation of Renyi's parking problem on random trees and provides an explicit expression for the parking constant.

Findings

Derived an explicit formula for the parking constant.

Extended parking problem analysis to trees with random degrees.

Connected the parking constant to the generating function of degree distribution.

Abstract

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.