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Parking on a random tree

Goldschmidt, Christina; Przykucki, Michal

DOI:

10.1017/S0963548318000457

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Goldschmidt, C & Przykucki, M 2018, 'Parking on a random tree', Combinatorics, Probability and Computing.

https://doi.org/10.1017/S0963548318000457

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Checked for eligibility 23/08/2018

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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

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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

) →

√

1−2α

1−α

, 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+1−m)(n+

1)

m−1

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)

m−1

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

) =

(

√

1−2α

1−α

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,