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Journal ArticleDOI

Parking on a random tree

01 Jan 2019-Combinatorics, Probability & Computing (Cambridge University Press)-Vol. 28, Iss: 1, pp 23-45
TL;DR: In this article, a uniform random rooted labelled tree on n vertices is considered, where each node of the tree has space for a single car to park and a number of cars arrive one by one, each at a node chosen independently and uniformly at random.
Abstract: Consider a uniform random rooted labelled tree on n vertices. We imagine that each node of the tree has space for a single car to park. A number m ≤ n of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider m = ⌊α n⌋ and let An,α denote the event that all ⌊α n⌋ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if α ≤ 1/2, we have undergoes a discontinuous phase transition, which turns out to be a generic phenomenon for arbitrary offspring distributions of mean at least 1 for the tree and arbitrary arrival distributions.

Summary (2 min read)

1.1. The limiting model

  • Throughout this paper the authors write Po(α) for the Poisson distribution with mean α. Write PGW(α) for the law of the family tree of a Galton–Watson branching process with Po(α) offspring distribution (this is canonically thought of as an ordered tree rooted at the progenitor of the branching process, although they shall frequently ignore the ordering).
  • The authors begin by formally introducing their limiting model.
  • Then, for every n, add an independent PGW(1) tree rooted at n, with edges directed towards n .
  • This random tree has the same law as a PGW(1) tree conditioned on non-extinction, and the authors will write PGW∞(1) for its law.
  • There is only space for one of them, and any surplus cars drive towards the root, parking in the first available space.

1.2. A local weak limit

  • The model is the limit of the problem considered in [13] in the sense of local weak convergence, which the authors now introduce.
  • First, let G be the set of graphs G = (V (G), E(G)) with finite or countably infinite vertex set V (G) which are additionally locally finite i.e. all vertex degrees are finite,.

1.3. Main results

  • The authors summarise their results in the following theorem.
  • (We will discuss the definition and properties of the Lambert W-function in Section 2.)the authors.the authors.
  • Let X denote the number of cars that visit the root of a PGW(1) tree with, for some α ∈ (0, 1), an independent Po(α) number of cars initially picking every vertex.
  • In particular, the authors recover the phase transition and limiting probabilities of Theorem 1.1.

2. Parking on a critical Poisson Galton–Watson tree

  • The following simple proposition gives us a first piece of information about parking on critical Galton–Watson trees.
  • Proposition 2.1. Let α ∈ (0, 1) and let X denote the number of cars that arrive at the root of a critical Galton–Watson tree with Po(1) offspring distribution.
  • This turns out to be more complicated and the authors need to learn more about the exact form of G(s) in order to achieve it.
  • By the definition of the Lambert W-function, this implies that −s−1G(s) =.

3. Generalisations

  • Consider their parking process on a PGW(1) tree.
  • There are two aspects of this model which one might think of generalising: the distribution of the number of cars arriving at each vertex, and the offspring distribution of the Galton–Watson process, i.e. the laws of P and N respectively.
  • One specific such situation, which the authors shall summarise below, has been studied by Jones [10] in the context of a model for rainfall runoff down a hill.
  • The authors focus on the random variable X and potential analogues of the phase transition (1.2).

3.1. Binary branching, paired arrivals

  • (The authors parameterisation differs from the one used in [10] to provide an easier comparison with the results of Section 1.).
  • Note that the offspring distribution is critical for all values of β.
  • Jones observes completely analogous phenomena to those the authors have discussed above.

3.2. Subcritical branching

  • For completeness, the authors now show that a phase transition of the form (1.2) for E [X] cannot occur if the offspring distribution is subcritical.
  • Write Q for the total progeny of the branching process.

3.3. Critical branching

  • In both the Poisson case the authors study in this paper, and the situation studied by Jones, they take the smaller root, and this value is correct all the way up to the phase transition.
  • This argument leads us to make the following conjecture.

Then

  • The authors conjecture that the jump from E [X] <∞ to E [X] =∞ coincides with the onset of long-range dependence in the model: above αc, the occupied cluster of the root appears to become macroscopic in the sense that it occupies a positive fraction of the tree.
  • Since the size of the tree has infinite expectation, this gives that X also has infinite expectation.
  • Consider now the tree conditioned to be infinite, work under the conditions of Conjecture 3.3 and suppose that the conjecture is true.

3.4. Supercritical branching

  • Observe that the assumption that E [X] is finite does not give us an explicit formula for pλ.
  • (On the other hand, if {|T | <∞} has positive probability then, conditionally on this event, |T | has finite mean.
  • Thus by the random walk interpretation of the parking process on a path, and by coupling the original parking process on T with the process the authors describe above, they see that the number X of cars that arrive at the root is infinite almost surely.
  • The authors are able to provide bounds on the critical value αc. Proposition 3.5.
  • Consider first only the vertices in the “even” generations of the tree (with the root being the 0th generation), with edges “inherited” from the original tree (so that every vertex is adjacent to its four grandchildren).

4. Acknowledgments

  • The authors are very grateful to Marie-Louise Lackner for introducing us to the problem, and to Owen Jones for telling us about his work and allowing us to see an early version of his manuscript [10].
  • During a large part of this project, the second author was affiliated with St Anne’s College and the Mathematical Institute of the University of Oxford.

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University of Birmingham
Parking on a random tree
Goldschmidt, Christina; Przykucki, Michal
DOI:
10.1017/S0963548318000457
License:
None: All rights reserved
Document Version
Peer reviewed version
Citation for published version (Harvard):
Goldschmidt, C & Przykucki, M 2018, 'Parking on a random tree', Combinatorics, Probability and Computing.
https://doi.org/10.1017/S0963548318000457
Link to publication on Research at Birmingham portal
Publisher Rights Statement:
Checked for eligibility 23/08/2018
First published in Combinatorics, Probability and Computing:
https://doi.org/10.1017/S0963548318000457
General rights
Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the
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Download date: 10. Aug. 2022

Parking on a random tree
Goldschmidt, Christina; Przykucki, Michal
Document Version
Peer reviewed version
Citation for published version (Harvard):
Goldschmidt, C & Przykucki, M 2018, 'Parking on a random tree' Combinatorics, Probability and Computing.
Link to publication on Research at Birmingham portal
General rights
Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the
copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes
permitted by law.
•Users may freely distribute the URL that is used to identify this publication.
•Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private
study or non-commercial research.
•User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?)
•Users may not further distribute the material nor use it for the purposes of commercial gain.
Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.
When citing, please reference the published version.
Take down policy
While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been
uploaded in error or has been deemed to be commercially or otherwise sensitive.
If you believe that this is the case for this document, please contact UBIRA@lists.bham.ac.uk providing details and we will remove access to
the work immediately and investigate.
Download date: 23. Aug. 2018

Combinatorics, Probability and Computing (xxxx) 00, 1–23.
c
xxxx Cambridge University Press
DOI: 10.1017/S0963548301004989 Printed in the United Kingdom
Parking on a random tree
C H R I S T I N A G O L D S C H M I D T
1
and M I C H A L P R Z Y K U C K I
2
1
Department of Statistics, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK
and Lady Margaret Hall, Norham Gardens, Oxford OX2 6QA, UK
(email: goldschm@stats.ox.ac.uk)
2
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
(email: m.j.przykucki@bham.ac.uk)
Received xxx
Consider a uniform random rooted labelled tree on n vertices. We imagine that each
node of the tree has space for a single car to park. A number m n of cars arrive
one by one, each at a node chosen independently and uniformly at random. If a
car arrives at a space which is already occupied, it follows the unique path towards
the root until it encounters an empty space, in which case it parks there; if there
is no empty space, it leaves the tree. Consider m = bαnc and let A
n,α
denote the
event that all bαnc cars find spaces in the tree. Lackner and Panholzer proved (via
analytic combinatorics methods) that there is a phase transition in this model. Then
if α 1/2, we have P (A
n,α
)
12α
1α
, whereas if α > 1/2 we have P (A
n,α
) 0.
We give a probabilistic explanation for this phenomenon, and an alternative proof
via the objective method. Along the way, we consider the following variant of the
problem: take the tree to be the family tree of a Galton–Watson branching process
with Poisson(1) offspring distribution, and let an independent Poisson(α) number of
cars arrive at each vertex. Let X be the number of cars which visit the root of the
tree. We show that E [X] undergoes a discontinuous phase transition, which turns
out to be a generic phenomenon for arbitrary offspring distributions of mean at least
1 for the tree and arbitrary arrival distributions.
2010 Mathematics subject classification: Primary 60C05
Secondary 60J80, 05C05, 82B26
1. Introduction
Let Π
n
be the directed path on [n] = {1, 2, . . . , n} with edges directed from i + 1 to i for
i = 1, 2, . . . , n 1. Let m n and assume that m cars arrive at the path in some order,
with the ith driver wishing to park in the spot s
i
[n]. If a driver finds their preferred
Research supported by EPSRC Fellowship EP/N004833/1.

2 C. Goldschmidt and M. Przykucki
parking spot empty, they stop there. If not, they drive along the path towards 1, taking
the first available place. If no such place is found, they leave the path without parking.
If all drivers find a place to park then we call (s
1
, s
2
, . . . , s
m
) a parking function for Π
n
.
Konheim and Weiss [12] introduced parking functions in the context of collisions of
hashing functions. Imagine that we have a hash table consisting of a linear array of n cells,
where we want to store m items. We use a hashing function h : [m] [n] to determine
where each item is stored. Item i is stored in cell h(i), unless some item j < i has already
occupied it, in which case we have a collision. We can resolve a collision by allocating
item i to the smallest cell k > h(i) such that k is empty at time i, if such a cell can be
found. If not, our scheme fails, and we cannot allocate our items to the hashing table.
This collision resolving scheme is clearly modelled by the parking functions described in
the first paragraph.
Konheim and Weiss showed that for 1 m n cars there exist exactly (n+1m)(n+
1)
m1
parking functions for Π
n
. Hence, taking α (0, 1) and m = bαnc, if the ith driver
independently picks a uniformly random preferred parking spot S
i
then the probability
that (S
1
, S
2
, . . . , S
m
) is a parking function for Π
n
is
(n + 1 m)(n + 1)
m1
n
m
(1 α)e
α
,
as n . In particular, this limiting probability is strictly positive for every α (0, 1).
Some generalisations of parking functions and their connections to other combinatorial
objects have been studied by, for example, Stanley [16, 17, 18, 19]. In a recent paper,
Lackner and Panholzer [13] studied parking functions on other directed graphs, in partic-
ular on uniform random rooted labelled trees (uniform random rooted Cayley trees). Let
T
n
denote such a tree on n vertices. Each of the m cars independently picks a uniform
vertex and tries to park at it. If it is already occupied, the car moves towards the root
and parks at the first empty vertex it encounters. If it finds no empty vertex, it leaves
the tree. Lackner and Panholzer (see Theorem 4.10 and Corollary 4.11 in [13]) prove that
in this setting there is a phase transition.
Theorem 1.1. Let T
n
denote a uniform random rooted labelled tree on n vertices. Let
A
n,α
be the event that all bαnc cars, with uniform and independent random preferred
parking spots, can park on T
n
. Then
lim
n→∞
P(A
n,α
) =
(
12α
1α
if 0 α 1/2,
0 if α > 1/2.
In fact, the result proved in [13] is much sharper: it not only demonstrates that there is
a phase transition, but it also gives an asymptotic formula for P(A
n,α
) which specifies its
behaviour in n, including at the critical point α = 1/2. However, the analytic methods
used in [13] offer no explanation for why the phase transition occurs. The purpose of the
present paper is to find a probabilistic explanation for this phenomenon. We employ the
objective method, pioneered by Aldous and Steele [3], to reprove Theorem 1.1. Much
of our analysis is performed in the context of a limiting version of the above model

Parking on a random tree 3
(its so-called local weak limit). Instead of T
n
, we consider a critical Galton–Watson tree
with Poisson mean 1 offspring distribution, conditioned on non-extinction. We replace
the multinomial counts of cars wishing to park at each vertex by independent Poisson
mean α numbers of cars at each vertex. Once we have analysed this limiting model, it is
relatively straightforward to then show that the probability all cars can park really gives
the limit of P (A
n,α
) as n .
1.1. The limiting model
Throughout this paper we write Po(α) for the Poisson distribution with mean α. Write
PGW(α) for the law of the family tree of a Galton–Watson branching process with Po(α)
offspring distribution (this is canonically thought of as an ordered tree rooted at the
progenitor of the branching process, although we shall frequently ignore the ordering).
We begin by formally introducing our limiting model.
Let T be an infinite random tree defined as follows. Start with an infinite directed path
Π
on N = {1, 2, . . .}, with edges directed from n + 1 to n for all n 1. Then, for every
n, add an independent PGW(1) tree rooted at n, with edges directed towards n (see
Figure 1). Finally, root the resulting (infinite) tree at 1. This random tree has the same
law as a PGW(1) tree conditioned on non-extinction, and we will write PGW
(1) for
its law. (Since extinction occurs with probability 1, the conditioning must be obtained
by a limiting procedure such as conditioning the tree to survive to generation k and then
letting k ; see Kesten [11]. We will discuss a more general case of this result in
Theorem 3.1 below.) At every vertex of the resulting tree, place an independent Po(α)
number of cars. There is only space for one of them, and any surplus cars drive towards
the root, parking in the first available space.
1 2 3 4 5
· · ·
Figure 1.The tree T , a critical Poisson–Galton–Watson tree conditioned on
non-extinction. The trees attached to the path on N are almost surely finite.
1.2. A local weak limit
Our model is the limit of the problem considered in [13] in the sense of local weak
convergence, which we now introduce.
First, let G be the set of graphs G = (V (G), E(G)) with finite or countably infinite
vertex set V (G) which are additionally locally finite i.e. all vertex degrees are finite,

Citations
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Posted Content
TL;DR: In this paper, the authors established a phase transition for the parking process on critical Galton-Watson trees, and showed that the cars go down the tree and try to park on empty vertices as soon as possible.
Abstract: We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $\sigma^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $\Sigma^{2}$ conditioned to be large. The cars go down the tree and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ \Theta:= (1-m)^2- \Sigma^2 (\sigma^2+m^2-m).$$ Specifically, if $ \Theta>0,$ then most cars will manage to park, whereas if $\Theta<0$ then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki.

14 citations

Book ChapterDOI
TL;DR: In this article, the authors considered a simple max-type recursive model which was introduced in the study of depinning transition in presence of strong disorder, by Derrida and Retaux.
Abstract: We consider a simple max-type recursive model which was introduced in the study of depinning transition in presence of strong disorder, by Derrida and Retaux [5]. Our interest is focused on the critical regime, for which we study the extinction probability, the first moment and the moment generating function. Several stronger assertions are stated as conjectures.

10 citations

Posted Content
TL;DR: In this paper, the authors show that the root is almost surely visited infinitely many times when $p \geq 1/2$, and only finitely many times otherwise. And they show that if a car encounters an available parking spot it parks there.
Abstract: Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p \geq 1/2$, and only finitely many times otherwise.

10 citations

Journal ArticleDOI
TL;DR: In this paper, a continuous-time version of the Derrida and Retaux model, built on a Yule tree, is presented, which yields an exactly solvable model belonging to this universality class.
Abstract: To study the depinning transition in the limit of strong disorder, Derrida and Retaux (2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal.

9 citations

Journal ArticleDOI
TL;DR: In this paper, a continuous-time version of the Derrida and Retaux model, built on a Yule tree, is presented, which yields an exactly solvable model belonging to this universality class.
Abstract: To study the depinning transition in the limit of strong disorder, Derrida and Retaux (J Stat Phys 156(2):26–290, 2014) introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux’ model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal.

9 citations

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Q1. What contributions have the authors mentioned in the paper "Parking on a random tree" ?

In this paper, the authors consider a directed path with edges directed from i+ 1 to i for i = 1, 2, ..., n− 1 and assume that m cars arrive at the path in some order, with the ith driver wishing to park in the spot si ∈ [ n ].