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Parking on supercritical geometric Bienaymé--Galton--Watson trees

Linxiao Chen, Alice Contat

Abstract

Consider a supercritical Bienaymé--Galton--Watson tree $ \mathcal{T}$ with geometric offspring distribution. Each vertex of this tree represents a parking spot which can accommodate at most one car. On the top of this tree, we add $(A_u : u \in \mathcal{T})$ i.i.d.\ non negative integers sampled according to a given law $ μ$, which are the car arrivals on $ \mathcal{T}$. Each car tries to park on its arriving vertex and if the spot is already occupied, it drives towards the root and takes the first available spot. If no spot is found, then it exits the tree without parking. In this paper, we provide a criterion to determine the phase of the parking process (subcritical, critical, or supercritical) depending on the generating function of $ μ$.

Parking on supercritical geometric Bienaymé--Galton--Watson trees

Abstract

Consider a supercritical Bienaymé--Galton--Watson tree with geometric offspring distribution. Each vertex of this tree represents a parking spot which can accommodate at most one car. On the top of this tree, we add i.i.d.\ non negative integers sampled according to a given law , which are the car arrivals on . Each car tries to park on its arriving vertex and if the spot is already occupied, it drives towards the root and takes the first available spot. If no spot is found, then it exits the tree without parking. In this paper, we provide a criterion to determine the phase of the parking process (subcritical, critical, or supercritical) depending on the generating function of .
Paper Structure (13 sections, 6 theorems, 52 equations, 3 figures)

This paper contains 13 sections, 6 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Parking on trees.
  3. Examples.
  4. Outline of the paper.
  5. Acknowledgments.
  6. Phase transition
  7. Two regimes
  8. Fully parked trees
  9. Decomposition into fully-parked components
  10. Probabilistic consequences
  11. Location of the thereshold: Theorem \ref{['thm:locthereshold']}
  12. Examples.
  13. Perspectives

Key Result

Theorem 1

Suppose that there exists $t_c$ such that Then the parking process is subcritical if and only if where $\varphi(y) = (y+1) G(y) - y(y-1)G'(y)$.

Figures (3)

  • Figure 1: Illustration of the "a la Tutte" recursive decomposition. On the left, a fully parked tree of size $n$ and $p$ outgoing cars. On the right, the different possibility depending on the number $j$ of the children of the root vertex.
  • Figure 2: On the left, a tree together with its car arrivals configuration. On the right, the (black) cluster of root of this tree together with all empty spots attached to it in the initial tree. All this empty spots are the root of independent Bienaymé--Galton--Watson trees where in the final configuration, the root is empty. This cluster is a fully parked tree with $7$ vertices and there are $8$ empty spots attached to it. The probability to observe this cluster is $( \mu_0^2 \mu_1^2 \mu_2^2 \mu_3) q^6 (1-q)^7 (q p_ \circ)^8$
  • Figure 3: From left to right, the critical value for $q$ as a function of $\alpha$ for car arrivals with respectively binary, geometric and Poisson distribution. Note that for the geometric and Poisson distribution, this function has a finite derivative as $\alpha$ goes to $0$, whereas it is of the form $1- \mathrm{cst} \sqrt{a}$ for some $\mathrm{cst}>0$ in the binary case

Theorems & Definitions (7)

  • Theorem 1
  • lemma 1: Dichotomy subcritical/supercritical regime
  • lemma 2
  • proposition 1
  • lemma 3
  • proof
  • proposition 2: F-characterization of the subcritical regime