M A S
D O C
Biased random walks on subcritical Galton-Watson
trees conditioned to survive
by
Adam Bowditch
Summer Research Project
Submitted for the degree of
Master of Science
Mathematics Institute
The University of Warwick
August 2014
Contents
Acknowledgments
ii
Declarations
iii
Abstract
iv
Chapter 1 Introduction
1
Chapter 2 Preliminaries
3
2.1
Galton-Watson trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Biased random walk on trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Electrical networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Directed trap model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 3 The infinite tree
12
3.1
The supercritical tree
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
The subcritical and critical trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1
The size conditioned tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.2
The height conditioned tree . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3
Proof that the height conditioned tree coincides with the construction . . . . . . 19
3.4
Regenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 4 The speed of the random walk
28
4.1
Speed on the supercritical tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2
Speed on the critical tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter 5 The random walk on the subcritical tree
33
5.1
Ballistic and sub-ballistic regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2
Speed in the ballistic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3
Escape in the sub-ballistic regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter 6 Conclusion
59
i
Acknowledgments
Firstly, I would like to thank my supervisor, Dr David Croydon for guidance, encouragement and advice
throughout my time on this project without whom this would have been impossible; my thanks also
go to my second supervisor, Dr Jon Warren, for helpful discussions and input. Secondly, I would like
to thank all the of lecturers at the University of Warwick who have taught me throughout the past
five years enabling me to obtain the breadth of knowledge required for this project. Most notably, Dr
Hendrik Weber for advanced probability and Dr Richard Sharp for ergodic theory. Thirdly, I would like
to thank the Engineering and Physical Sciences Research Council for funding and finally, the MASDOC
programme at the University of Warwick, in particular, Björn Stinner for organisation and guidance
throughout the year.
ii
Declarations
“I hereby certify that this material, which I now submit for assessment as a summer research project is
entirely my own work, that I have exercised reasonable care to ensure that the work is original, does not,
to the best of my knowledge, breach any copyright and has not been taken from the work of others save
such work that has been cited within the text of my work.”
iii
Abstract
In this thesis we survey known results concerning the limiting behaviour of biased random walks Xn on
supercritical and critical Galton-Watson trees conditioned to survive and extend them to the subcritical
case. We start by giving a proof that there exists a well-defined probability measure over such non-extinct
subcritical trees and that these exhibit a unique backbone with subcritical trees as leaves. We show that
the speed limn→∞ |Xn |/n exists a.s. for any bias β > 0 and is positive if and only if β belongs to some
determined region depending only on the mean and variance of the offspring distribution of the tree. We
then consider this as a directed trap model to determine the speed along the backbone in terms of the
bias of the walk and moments of the offspring distribution up to second order.
iv
Chapter 1
Introduction
The aim of this thesis is to survey known results concerning biased random walks on supercritical and critical Galton-Watson (GW) trees conditioned to survive and extend them to the
subcritical case. In Chapter 2 we state the necessary preliminaries for investigating random
walks in the random environments we consider including many notational aspects. This starts
with an introduction to GW trees as well as major results concerning the probability generating function and the random walks on these trees. We then give a brief overview of the links
between electrical networks and random walks including a characterisation of the expected time
spent by a random walk in fixed and random trees as well as the fundamental techniques used
to show recurrence and transience. The chapter finishes with an introduction to the directed
trap model and an explanation of how a random walk on certain trees can be considered as such
a model.
In Chapter 3 we consider how a GW tree can be conditioned to survive; in particular, we
show that it makes sense to condition critical and subcritical GW trees to survive using a proof
from Janson [12] and discuss the structure of these trees. One of the main results of the chapter
is that the subcritical GW tree conditioned to survive can be constructed by a fixing a single
infinite line of descent (ρ0 , ρ1 , ...), for each i allowing ρi to give birth onto ‘buds’ (according
to a size-biased distribution) which in turn behave as roots of independent GW trees with the
original offspring distribution. We conclude the chapter by proving some useful results for the
walk between regeneration times using the fact that the structure of the tree allows us to view
the random walk as a walk on (ρ0 , ρ1 , ...) taking excursions into finite trees.
We finish the literary review in Chapter 4 by stating the known results for the speed
of the biased random walk on non-extinct critical and supercritical GW trees. We start by
stating a result by Lyons, Pemantle & Peres [16] which identifies the conditions for positive
speed for the walk on the supercritical tree. We then include a result by Aı̈dékon [1] for a
characterisation of the speed in the ballistic regime on the supercritical tree before discussing
results of Ben Arous, Fribergh, Gantert & Hammond [2] which give an appropriate scaling in the
sub-ballistic regime under the extra condition that the offspring distribution has finite variance.
We then state results for the limiting behaviour of the first hitting time of backbone vertices on
the non-extinct critical GW tree from Croydon, Fribergh & Kumagai [6] before giving a brief
1
overview of the known results and expected behaviour for the walk on the subcritical tree.
In Chapter 5 we show similar results to those in Chapter 4 for the biased random walk on
subcritical GW trees conditioned to survive. We show that the speed exists almost surely and
is constant using an argument concerning the maximum height of a branch off the backbone.
We then determine that the walk is ballistic if and only if 1 < β < µ−1 < ∞ and σ 2 < ∞.
The intuitive reasoning for this region is that when β ≥ µ−1 the expected time spent in a trap
is infinite. This is also the case when σ 2 = ∞ since under this condition we have that the
size biased distribution has infinite mean and hence the expected number of offspring from a
backbone vertex is infinite. When β ≤ 1 the walk is recurrent and hence doesn’t ‘escape’. By
considering the walk on a two-sided tree at the transitions along the backbone and decomposing
the speed into a limit for the distance moved and a limit for the times between transitions we
show the speed of the β-biased random walk on the subcritical GW tree to be
rβ =
µ(β − 1)(1 − βµ)
µ(β + 1)(1 − βµ) + 2β(σ 2 − µ(1 − µ)
in the ballistic regime where µ, σ 2 are the mean and variance of the offspring distribution.
We end with a discussion as to whether there is an appropriate transformation of |Xn | which
converges to a non-trivial limit in the case that the walk on the subcritical GW tree conditioned
to survive is transient and sub-ballistic.
2
Chapter 2
Preliminaries
In this chapter we state the necessary preliminaries for investigating random walks on random
trees including many notational aspects. We start with an introduction to GW trees then move
on to biased random walks on trees and the links between these walks, electrical networks and
directed trap models.
2.1
Galton-Watson trees
In this section we describe GW trees and some of the major results concerning them. The
structure of the tree will be very important when considering the random walk since the size of
finite branches will determine trapping phenomena which slows the walk and a phase transition
exists for the structure of the backbone which determines transience.
Formally a rooted, labelled tree T is a non-empty collection of finite sequences of
positive integers where if hi1 , ..., in i ∈ T then hi1 , ..., ik i ∈ T for every k = 0, 1, ..., n and
hi1 , ..., in−1 , ji ∈ T for every j = 1, 1, ..., in . We denote the root ρ to be the empty sequence
th
φ, then say hi1 i is the ith
1 child of ρ and hi1 , ..., in i is the in child of hi1 , ..., in−1 i. We use
i0 = 1 to denote that there is a single vertex in level 0. For a given x = hi1 , ..., in i we write
←
− = hi , ..., i
x
1
n−1 i to be the parent of x, dx := |{j : hi1 , ..., in , ji ∈ T }| to be the number of
x
children of x, for j ≤ dx write xj = hi1 , ..., in , ji to be the j th child of x and c(x) := {xj }dj=1
to be the set of children of x. For x = hi1 , ..., in i ∈ T we define the descendent tree at x
to be Tx := {hj1 , ..., jk i : hi1 , ..., in , j1 , ..., jk i ∈ T } which is the sub-tree formed by fixing x as
the root and removing all individuals which are not descendants of x. The height H of a tree
T is the supremum length of sequences in the tree H(T ) := sup{n ≥ 0 : ∃ hi1 , ..., in i ∈ T }.
Notice that if the tree consists of just the root then it is considered to be of height 0. We
define the truncation T[n] of a tree T to be the restriction of the tree to the first n levels
T[n] := {hi1 , ..., ik i ∈ T : k ≤ n}. A tree T is called locally finite if for any n ∈ N we have
that T[n] is a finite tree. This holds whenever there isn’t a vertex with infinitely many offspring.
Apart from an important note concerning the construction of infinite trees in Section 3.2, we
will always consider locally finite trees. We write T[n] to be the set of locally finite trees of
height at most n, T to be the set of all locally finite trees and Tf to be the set of all finite trees.
3
We can define a metric d on T by setting
d(T, T 0 ) :=
1
0 }
1 + sup{n : T[n] = T[n]
which makes (T, d) a complete separable metric space (e.g. Lyons & Peres [14]). Figure 2.1
shows a graphical representation of such a rooted labelled tree.
Figure 2.1: A rooted labelled tree.
We always use |x| to denote the generation number of x on T . We will often use trees
−
that have been relabelled so that ←
ρ exists. In this case we have that |x| is not necessarily
←
−
−
positive. In particular for a fixed tree T we define T := T ∪ {←
ρ } to be the original tree with
the inclusion of an artificial parent of the root.
P∞
Z+ such that
Let (pk )∞
k=0 pk = 1 be a probability distribution and ξ a
k=0 ∈ [0, 1]
random variable such that P(ξ = k) = pk . We refer to the distribution of ξ as the offspring
distribution and denote its probability generating function
∞
X
f (s) =
pk sk
k=0
for 0 ≤ s ≤ 1. We write the mean of the offspring distribution as µ := E[ξ] = f 0 (1− ) =
lims→1− f 0 (s), the second moment µ2 := E[ξ 2 ] and the variance σ 2 := µ2 − µ2 .
A Galton-Watson process is a Markov chain Z = (Zn )n≥0 such that Z0 = 1 and
Zn+1 =
Zn
X
ξn,j ,
j=1
where {ξn,j : n ≥ 0, j ≥ 1} are i.i.d. copies of ξ. This is a branching process where Zn denotes
the number of individuals in the nth generation of the process. We will often write ZnT to be the
4
size of the nth generation when we need to specify a tree although we drop the superscript if it
is clear to which tree we refer. In this setting we start with a single (root) individual to which
we assign k children with probability pk . Each of these children independently of each other and
previous generations has k children with probability pk . This continues either forever or until
there is a generation containing no children. One immediate consequence is the branching
property which states that if {Z (j) }kj=1 , Z are independent GW processes then for l ≤ n
k
X
(j)
d
Zn−l = Zn |Zl = k
j=1
A Galton-Watson tree T is a tree constructed from a GW process where vertices
at level n are the individuals in generation n and edges connect parents to children. More
specifically ξn,j is the number of children of the j th individual in generation n.
We say that the process becomes extinct if there exists some generation n such that
S
Zn = 0 and we use q := P ( ∞
n=1 {Zn = 0}) to denote the extinction probability. If T is a tree
T
Tx
which does not become extinct then we define the backbone Y := x ∈ T : ∞
n=1 {Zn > 0}
to be the sub-tree consisting of all points which act as the root of a non-extinct tree. A well
known result is that, assuming p1 6= 1, we have that q = 1 if and only if µ ≤ 1 (e.g. Athreya &
Ney [4]). Due to this result we say that the process is subcritical if µ < 1, critical if µ = 1
and supercritical if µ > 1. In general we shall ignore the case that there is some k such that
pk = 1 since the behaviour is completely deterministic in this case. Another important fact is
that q is the smallest positive solution s to f (s) = s (e.g. [4]). We often denote fn to be the
nth convolution fn = f ◦ fn−1 with f1 = f and f (n) to be the nth derivative f (n) (s) =
dn f (s)
dsn .
We then have the useful result that by the Chapman-Kolmogorov equations
fn (0) = P(Zn = 0).
2.2
Biased random walk on trees
In this section we describe the biased random walk on a random tree and its speed which will
be the main areas of interest in this thesis. In simple terms a β-biased random walk on a fixed
tree T ∈ T is a random walk on the T which is β times more likely to make a transition onto a
given child of the current vertex than the parent (which are the only options).
We define a β-biased random walk on a fixed locally finite tree T ∈ T as the Markov
chain (Xn )n≥0 with state space T and transition probabilities
P T (Xn+1 = y|Xn = x) =
5

1



 1+βdx
β
1+βdx



0
−
y=←
x
y ∈ c(x)
otherwise
for x 6= ρ and
T
P (Xn+1 = y|Xn = ρ) =


1
dρ
0
y ∈ c(x)
otherwise
so that the process is β times more likely to go down a given path away from the root as it is
to go towards the root.
It is important here to note that we always use P T to denote the quenched probability
measure over the random walk for a fixed tree T and P to denote the annealed probability
measure obtained by averaging P over some probability measure P on locally finite trees
Z
P(·) =
P T (·)P(dT ).
Recall that we use |Xn | to denote the generation number the position of Xn on T . Our
main object of interest is the speed of the random walk
|Xn |
.
n→∞ n
rβ := lim
Lyons, Pemantle & Peres [16] have shown this exists almost surely (and is constant) with respect
to P where P is a GW measure over supercritical trees conditioned to survive. We say that the
walk is ballistic if rβ > 0 and sub-ballistic otherwise. Interestingly, even when the bias is
directed towards the root it can be the case that the walk is ballistic. This is because there are
sufficiently many paths away from the root that the probability of moving away is large and
hence the drift positive. Another interesting phenomena is that the walk is sub-ballistic when
the bias is large assuming p0 > 0. The reason for this concerns the structure of the supercritical
GW tree on non-extinction. Such a tree, as will be explained in more detail in Chapter 3, can
be decomposed into an infinite backbone and finite branches. When the bias is large the time
spent in branches increases because, in order to escape, the walk needs to move towards the
root which happens with small probability.
For x ∈ T let τx := inf{n ≥ 0 : Xn = x} denote the first hitting time and τx+ :=
inf{n > 0 : Xn = x} to be the first return time. Define regeneration epochs θ0 = 0,
θk := inf{n > θk−1 : Xn ∈
/ {Xj }n−1
j=0 , Xm 6= Xn−1 ∀m ≥ n} for k = 1, 2, ...,. We refer to
the points Xθk as regeneration points then θk is the first time that the walk reaches the k
regeneration point which is the k th vertex from which it doesn’t backtrack. We write the slab
to be
Sn := TXθn \ TXθn+1 ∪ Xθn+1 ; Xθn +1 , ..., Xθn+1
(2.1)
which is the section of the tree between two regeneration epochs as well as the path through this
section; this is independent of the rest of the tree and the rest of the walk. In [16] it is shown
that for T supercritical, non-extinct GW trees we have that {Sn }n≥1 are mutually independent
random variables and using this, that the differences between successive regeneration epochs
{θn+1 − θn }n≥1 are independent, as are the differences {|Xθn+1 | − |Xθn |}n≥1 . We shall see in
Chapter 3 that this is also the case for subcritical GW trees conditioned to survive.
6
2.3
Electrical networks
Electrical networks are strongly related to random walks, especially with regards to transience,
recurrence and hitting times. The theory of electrical networks yields many useful results and
techniques concerning the biased random walk on trees. In this section we describe electrical
networks and some of the basic theory that allows random walks to be considered as such
networks. In particular, we determine the expected time for a β-biased random walk to return
to the root of a (subcritical GW) tree. Unless proven here or cited any result stated here can
be found in Lyons & Peres [14].
Let X be a Markov chain on some finite state space Ω with transition probabilities p.
We call X reversible if there is some positive function K : Ω → R such that K(x)p(x, y) =
K(y)p(y, x) for all x, y ∈ Ω. We then define a graph G = (Ω, E ) where e = (x, y) ∈ E if
p(x, y) > 0. To each edge we assign the weight κ(x, y) = K(x)p(x, y) = κ(y, x). The Markov
chain can then be described as a random walk on G where transitions occur with probabilities
proportional to the weights of the incident edges. Conversely, every finite connected graph with
positive weights gives rise to an irreducible, reversible Markov chain with transition probabilities
proportional to the weights. From such a graph we define K(x) to be the sum of the weights
incident to x
K(x) =
X
κ(x, y)
y∼x
and set
p(x, y) =
κ(x, y)
.
K(x)
We call a function f harmonic at x ∈ Ω if
f (x) =
1 X
κ(x, y)f (y)
K(x) x∼y
thus the value of f at a given point is the weighted average of neighbouring points. Harmonic
functions solve the discrete Laplace equation
∆f (x) :=
X
(f (y) − f (x))κ(x, y) = 0
y∼x
and therefore have many useful properties most of which won’t be discussed here (see [14] for
more details). The following are useful harmonic functions for the random walk:
1. For non-empty set A ⊂ Ω denote τA := inf{n ≥ 0 : Xn ∈ A} then for B ⊂ Ω disjoint
from A we have that the probability of reaching A before B : Fx (A, B) := Px (τA < τB ) is
harmonic.
2. For non-empty, disjoint sets A, B ⊂ Ω let R :=
P τB
n=0 χ{Xn ∈A}
then the expected time
spent in A before reaching B : Gx (A, B) := Ex [R] is harmonic. We call this the Green
function.
7
An electrical conductance network is a weighted graph in which we refer to the weights
κ(x, y) as conductances and their reciprocals ω(x, y) := κ(x, y)−1 as resistances. For disjoint
sets A, B ⊂ Ω, we call a function v a voltage if it is harmonic on Ω \ (A ∪ B) and constant
on A, B. For a given voltage function v we define a current ι(x, y) := κ(x, y)(v(x) − v(y)).
Intuitively this means that current flows towards regions of low voltage at a rate proportional
to the conductance.
Since ι(x, y) is the current flowing between x, y we have that
P
x∈Ω ι(a, x)
is the total
current flowing from a. We therefore have that the effective conductance of the electrical
circuit can be seen from a probabilistic standpoint
C(a, B) :=
1 X
ι(a, x) = K(a)Pa (τB < τa+ ).
v(a)
x∈Ω
Similarly we can then define effective resistance as R(a, B) = C(a, B)−1 . Effective resistance
and conductance are fairly easy to compute thus allowing us to determine return probabilities
and expected return times. Two important properties of the effective resistance and conductance
are the series and parallel laws. The series law states that resisters in series add; more
specifically, if w is a vertex of degree 2 connecting vertices x, y then by removing w and replacing
the two edges with a single edge with resistance ω(x, y) = ω(x, w) + ω(w, y) the effective
conductance between any two vertices (excluding w) is unchanged. The parallel law states that
conductances in parallel add; more specifically, if x, y are vertices connected by two edges with
conductances κ1 (x, y), κ2 (x, y) then by replacing these edges with a single edge of conductance
κ(x, y) = κ1 (x, y) + κ2 (x, y) the effective conductance between any two vertices is unchanged.
The parallel law is less important with trees because loops cannot form however, it remains a
useful tool for simplifying networks. Given x ∼ y, by cutting the edge (x, y) we mean we have
set κ(x, y) = 0 and by shorting the edge (x, y) we mean we have set κ(x, y) = ∞. Cutting
edges doesn’t increase conductance and shorting edges doesn’t decrease conductance.
For an infinite graph G fix a sequence of finite sub-graphs Gn such that Gn ⊂ Gn+1 and
G =
S
n Gn .
Write Bn := G \ Gn to be the set of vertices outside Gn . The conductance C(a, Bn )
can be defined for any n by removing the set of vertices which are not connected to Gn . The
sequence C(a, Bn ) is non-negative and decreasing so the limit C(a, ∞) := limn→∞ C(a, Bn ) exists
and is called the effective conductance from a to ∞. A random walk on an infinite connected
network is transient if and only if C(a, ∞) > 0 for any a ∈ Ω.
For a graph G and function f we define the energy to be
E(f ) :=
1X
(f (x) − f (y))2 κ(x, y)
2 x∼y
!1
2
.
It can be seen from the equivalency between transience and positive effective conductance to ∞
that a random walk is transient if and only if there exists function f of finite energy (from any
8
vertex to ∞) such that
P
x∈Ω f (x)
= 1 and in fact the effective conductance can be written as
C(ρ, ∞) = inf{E(f ) : f (ρ) = 1, f (∞) = 0}.
Let v be the voltage when a unit current flows from a to B then Ea [τB ] =
(2.2)
P
x∈Ω K(x)v(x)
is
the expected hitting time of B.
A β-biased random walk on a tree T can be described as an electrical conductance model
−|
− ) = β |←
x
with conductances κ(x, ←
x
.
Lemma 2.3.1. Let Zn denote the generation sizes of a finite tree T ∈ Tf
EρT [τρ+ ] = 2
X Zn β n−1
Z1
n≥1
.
Proof. Let π denote the stationary probability measure of Xn . The expected time that the
random walk takes to return to the root of a tree is equal to the reciprocal of the stationary
probability of the root. To see this we want to show EρT [τρ+ ] = π(ρ)−1 . Let S0 = 0, Sk :=
Pn−1
inf{n > Sk−1 : Xn = ρ} be the hitting times of the root and Rn := k=0
χ{Xn =ρ} be the time
spent at the root up to time n. By the strong law of large numbers Sn /n → EρT [τρ+ ] P -a.s.
Furthermore, by the ergodic theorem for Markov chains Rn /n → π(ρ) P -a.s. In particular
RSn /Sn → π(ρ) and since RSn = n we have that RSn /Sn = n/Sn → 1/EρT [τρ+ ].
We can consider the graph as an electrical network with conductances κ(x, y) = κ(y, x)
(since reversible). To find the stationary probability we want to solve the detailed balance
equations. The transition function is of the form
κ(x, y)
.
z∼x κ(x, z)
P (x, y) = P
It is then clear that to have π(x)P (x, y) = π(y)P (y, x) we must have
π(x) ∝
X
κ(x, z).
z∼x
In particular since it is a probability we have that
P
x∼ρ
π(ρ) = P
x,y
κ(ρ, x)
κ(x, y)
.
If x is the parent of y and x is in the nth generation (where we consider ρ to be the 0th generation)
then κ(x, y) = β n . Since there are Zn children in generation n each of which has a single parent
in generation n − 1 we have that
P
x∼ρ
π(ρ) = P
x,y
κ(ρ, x)
κ(x, y)
9
=
2
P
Z1
.
n−1
n≥1 Zn β
From this we have the expected time spent in a fixed tree. For our purposes, the expected
time spent in a subcritical GW tree will be important since these will form the traps in our
model which we explain in further detail in Section 2.4.
Corollary 2.3.2. Let E denote the annealed expectation over β-biased random walks on a
(sub)critical GW tree T then
Eρ [τρ+ ] = 2
X
β n µn
n≥0
Proof. By Lemma 2.3.1 we have that the expected time that the random walk takes to return
P
n−1
to the root of a tree is 2 n≥1 ZnZβ 1 . We then have that

E[τρ+ ] = E 2
X Zn β n−1


Z1
n≥1
X
Zn
n−1
β
E
=2
.
Z1
n≥1
Notice, since Z1 = 0 =⇒ Zn = 0 for all n ≥ 1, using the branching property we have that
X
∞
Zn
P(Z1 = k)E[Zn |Z1 = k]
E
=
Z1
k
k=0
=
∞
X
P(Z1 = k)E[Zn−1 ]
k=0
= E [Zn−1 ]
= µn−1 .
2.4
Directed trap model
In this section we describe the directed trap model and explain how the biased random walk
can be associated with such a model. The model is very applicable because we can consider the
random walk on a tree as a random walk on the backbone of a tree taking excursions into finite
subtrees which we model as traps.
A directed trap model is a model of a continuous time random walk W = (Wt )t≥0 on
state space Z starting from W0 = 0 with transition rates
c(x, y) =


1
−1
1+β δx
 β δ −1
1+β x
10
y =x−1
y =x+1
where (δx )x∈Z is a family of i.i.d. random variables.
We can use such a model to describe a random walk on a tree. Let T be a tree formed
by starting with some locally finite tree T 0 in which every vertex has at least one child and
then attaching finite subtrees onto the vertices of T 0 . We shall observe later in Chapter 3 that
a GW tree conditioned to survive always takes such a form. More specifically let T 0 be the
deterministic tree in which each vertex gives birth onto precisely one vertex since this is the
structure of P-a.e. subcritical GW tree conditioned to survive as we shall show in Section 3.3.
Let X denote the random walk on T then define η0 = 0 and ηn := inf{k > ηn−1 : Xk , Xk−1 ∈ T 0 }
to be the time of the nth movement along T 0 . Then Yn := Xηn is a discrete time random walk
on T 0 moving away from the root ρ with probability β(1 + β)−1 and moving towards the root
with probability (1 + β)−1 . It is clear that a version of W on state space T 0 can be formed by
taking a continuous time version of Y using holding times of some distribution relative to the
size of the finite branches attached to T 0 . Equally by letting Π : T → T 0 denote the projection
of a vertex onto its closest ancestor on the backbone we have that the process Π(Xn ) is equal
in distribution to Y with holding times and hence a version of the directed trap model.
The directed trap model differs from Π(Xn ) because the time spent in subcritical trees
are not exponential holding times however, the random walk on a GW tree conditioned to
survive can be studied using these techniques which allow us to break the process between a
walk along the backbone and excursions into traps. Ben Arous, Cabezas, Černỳ & Royfman
[3] study such randomly trapped random walks. Once it has been established that rβ exists
P-a.s. and is constant we show that the distribution of time spent in a trap isn’t required to
determine the speed (which we show to be E[|Xη2 | − |Xη1 |]/E[η2 − η1 ] in Chapter 5) hence only
the expected time spent in a trap is important.
11
Chapter 3
The infinite tree
If the GW tree T is finite P-a.s. then |Xn | is bounded and hence |Xn |/n → 0 P-a.s. as n → ∞.
In particular, since any subcritical or critical GW tree is finite P-a.s. we have that rβ = 0. In
the supercritical case, since there is positive probability of non-extinction, it clearly makes sense
to condition on the event of non-extinction. In the critical and subcritical cases T is almost
surely finite and hence this conditioning is initially unclear. The behaviour of the random walk
depends greatly on the structure of the infinite tree therefore in this section we describe the
two different types of conditioning on T and properties concerning their structure.
We wish to consider the random walk on T conditioned on being infinite. This in itself
is an ill-defined statement since there are several ways in which we can condition T on being
infinite. Either we condition on the size of the tree being large (i.e. the number of vertices)
|T | ≥ n as n → ∞ or we condition on the height of the tree being large (i.e. non-extinction)
H(T ) ≥ n as n → ∞. As we shall shortly see both of these are well defined and can be
constructed using a graphical representation which demonstrates the structure of the infinite
tree. We shall also observe that, for GW trees, the two constructions coincide in the supercritical
and critical cases.
A result from Janson [12] is that if µ ≤ 1 then the distribution over {T : |T | ≥ n}
converges (as n → ∞) to a unique distribution over trees {T : |T | = ∞}. In fact, Janson shows
that the limiting distribution is equivalent to the distribution determining the construction of
T̂ which will be defined in Section 3.2.1. This is the first conditioning which we shall refer to
as size conditioning.
A modification of a result from Kesten [13] is that if µ ≤ 1 then for any rooted, labelled
tree T of k generations we have that
P∗ (T[k] = T ) := lim P(T[k] = T |Zn > 0)
n→∞
(3.1)
exists and has a unique extension onto a probability measure on rooted, labelled infinite
trees. This follows straightforwardly from Kolmogorov’s extension theorem and shows that there
is a unique probability measure over (sub)critical GW trees conditioned on non-extinction.
12
3.1
The supercritical tree
Almost every GW tree is locally finite and every non-extinct GW tree has infinitely many
vertices moreover, every locally finite GW tree with infinitely many vertices is non-extinct.
Since there is positive probability of non-extinction we trivially have that the two conditionings
are equivalent. We therefore devote this section to an explanation of the Harris decomposition
which demonstrates how a supercritical tree conditioned on non-extinction can be constructed
and reveals much about the structure of such trees which is important to the behaviour of the
random walk on such a tree. For this section when referring to a tree T we always (unless stated
otherwise) mean a supercritical GW tree with probability generating function f .
The Harris decomposition of a supercritical branching process involves splitting and
scaling the probability generating function into
g(s) =
f ((1 − q)s + q) − q
1−q
and
h(s) =
f (qs)
.
q
It is clear that g, h can be written as probability generating functions of GW trees.
h(s) =
=
=
f (qs)
q
∞
1X
q
pk (qs)k
k=0
∞
X
(pk q k−1 )sk
k=0
=
∞
X
phk sk
k=0
where
phk = pk q k−1
describes a subcritical tree with f 0 (q) = h0 (1− ) < 1 which is equivalent to the distribution of f
conditioned on extinction.
13
f ((1 − q)s + q) − q
1−q
!
∞
X
1
=
pk ((1 − q)s + q)k − q
1−q
g(s) =
k=0
=
∞
X
pgk sk
k=0
where

 (1−q)k−1 P∞ p q j j!
j=k j k!(j−k)!
g
qk
pk =
0
k≥1
k=0
describes a GW tree with pg0 = 0 and mean
g 0 (1− ) = f 0 (1− ) = µ.
(3.2)
Figure 3.1 shows the probability generating function f for a supercritical branching
process. The generating function g can be formed by taking f on [q, 1] and rescaling it into the
form of Figure 3.2 and h can be formed by taking f on [0, q] and rescaling it into the form of
Figure 3.3.
Figure 3.1: Supercritical probability generating function.
14
Figure 3.2: Probability generating function without deaths g.
Figure 3.3: Subcritical probability generating function h.
It has been shown by Harris [11] that we can construct a GW tree T conditioned on
non-extinction by forming a g-GW tree T g (which will form the backbone) and then attach onto
each vertex x of T g a random number ξ˜x of h-GW trees where ξ˜x has a probability generating
function
h i f (dTx g ) (qs)
.
E sξ̃x = (dT g )
f x (q)
(3.3)
This generating function comes directly from (3.1) as follows. By summing over the
possible number of offspring at x onto finite branches we have that
∞
h i X
g
ξ̃x
E s
=
sk P(ξ˜x = k|dTx ).
k=0
15
(3.4)
This probability can be written in terms of the total offspring from x and using Bayes’
rule we have
g
g
g
P(ξ˜x = k|dTx ) = P(ξx = k + dTx |dTx )
g
g
P(ξx = k + dTx , dTx )
= P∞
.
Tg
Tg
j=0 P(ξx = j + dx , dx )
(3.5)
The probability in the numerator of (3.5) is of the event that a vertex gives birth onto
k+
g
dTx
g
offspring and precisely dTx are roots of non-extinct trees hence we have that
g
Tg
g
P(ξx = k + dTx , dTx ) = pk+dTx g q k (1 − q)dx
g
k + dTx
k
Tg
g
T
(1 − q)dx
k (k + dx )!
gq
=
.
·
p
T
g
k+dx
dTx !
k!
(3.6)
Multiplying by the factor sk from (3.4) we have
∞
X
Tg
k
s P(ξx = k +
g
dTx ,
g
dTx )
k=0
(1 − q)dx (dTx g )
f
(sq).
=
dTx g !
(3.7)
Substituting the form from (3.6) into the denominator of (3.5) we have that
∞
X
g
g
P(ξx = j + dTx , dxT ) =
j=0
∞
X
Tg
pj+dTx g q j (1 − q)dx
j=0
g
j + dTx
j
Tg
(1 − q)dx (dTx g )
f
(q)
=
dTx g !
(3.8)
and so substituting (3.7) and (3.8) back into (3.4) we have the result (3.3).
One useful result we can obtain about the distribution of Z1 is that it has finite mean
with no additional constraints on the distribution of ξ. We shall see in Section 3.2.2 that this
isn’t the case for the subcritical tree conditioned on non-extinction which has a similar structure.
P(Z1 = k)P(Zn > 0|Z1 = k)
n→∞
P(Zn > 0)
1 − P(Zn−1 = 0)k
= lim pk
n→∞
1 − P(Zn = 0)
k
1−q
= pk
1−q
µ − qf 0 (q)
E[Z1 ] =
1−q
µ
≤
1−q
lim P(Z1 = k|Zn > 0) = lim
n→∞
since P(Zn = 0) converges to the non-negative constant q as n → ∞.
16
(3.9)
3.2
The subcritical and critical trees
In this section we address the problem which occurs when conditioning T in the subcritical and
critical cases brought on by T being finite P-a.s. We consider two graphical constructions and
their implications for the random walk.
3.2.1
The size conditioned tree
In this section we describe the construction of a modified GW tree T̂ from Janson [12] which has
distribution coinciding with the limiting distribution of the (sub)critical GW tree T conditioned
on |T | ≥ n as n → ∞.
Consider two types of vertices: normal vertices having offspring according to independent
copies of ξ and special vertices having independent copies of ξˆ where

kp
k
P(ξˆ = k) :=
1 − µ
k ∈ N0
k=∞
in which normal vertices give rise onto normal vertices and special vertices give rise onto either
infinitely many normal vertices or finitely many vertices with one chosen uniformly at random
to be special. We choose the root to be special then this gives rise to two distinct cases:
1. µ = 1 critical case:
ξˆ < ∞ P-a.s. hence in every generation there is precisely one special vertex and the tree
has a single backbone. In this case we can consider an alternative construction starting
with a single infinite line of descent (ρ = ρ0 , ρ1 , ...) and attaching branches as ξˆ − 1 copies
of independent GW trees with offspring distribution ξ.
2. µ < 1 subcritical case:
With probability 1, the tree reaches a generation in which there are no special vertices but
there is one vertex with infinitely many offspring. We can construct this tree in a similar
way to the critical one by starting with a backbone of random length L given by a shifted
geometric distribution of parameter 1 − µ (i.e. P (L = l) = (1 − µ)µl−1 ). We then attach
infinitely many independent GW trees with offspring distribution ξ to the final vertex and
d
ˆ ξˆ < ∞ which gives the so
ξ ∗ − 1 independent copies to each other vertex where ξ ∗ = ξ|
called size-biased distribution P(ξ ∗ = k) = kpk /µ.
In either case we have the probability generating function
ϕ(s) = E[sξ̂ ] =
∞
X
k=0
for s ∈ [0, 1).
17
kpk sk = sf 0 (s)
The random walk X on T̂ for µ = 1 then acts as a directed trapping model with traps
formed by ξ ∗ − 1 independent copies of subcritical f -GW trees. The holding times should then
be modelled as a geometric number of return times to the root of a tree with first generation
distributed according to the size biased distribution and subsequent independent critical GW
trees. In the case µ < 1 the random walk eventually reaches the vertex x with infinite degree.
− but explores infinitely many independent
At this point the walk almost surely never reaches ←
x
subcritical GW trees and is therefore sub-ballistic.
3.2.2
The height conditioned tree
In this section we construct the tree whose distribution coincides with the limiting distribution
P∗ of the (sub)critical GW tree conditioned to survive. This will be the main object of interest
in Chapter 5 where we consider biased random walks on this random graph.
Consider T ∗ to be a tree in which we start with a single special vertex. At each generation
every normal vertex gives birth onto normal vertices according to ξ and every special vertex
gives birth onto vertices according to the size-biased distribution ξ ∗ (P(ξ ∗ = k) = kpk µ−1 ),
one of which is chosen uniformly at random to be special. In this construction the special
vertices form the infinite backbone which is a single infinite line of descent Y := (ρ0 = ρ, ρ1 , ...)
−
where ρk = ←
ρ k+1 for k = 0, 1, ..., and each vertex on the backbone to gives birth onto buds
ξρ∗ −1
i
{ρi,ij }ij =1
, each of which is then the root of a ξ-GW tree Tρi,ij . We call each Tρi,ij a trap and
ξρ∗ −1
i
the collection {Tρi,ij }ij =1
from a single backbone vertex a shrubbery. From this it is clear
that this tree can be constructed similarly to the Harris decomposition in which we start with
a backbone and attach finite trees. It will be important to refer to the finite subtrees from
vertices on Y which we call branches and denote Tρ∗−
:= Tρ∗i \ Tρ∗i+1 .
i
An important property of the size biased distribution is its relation to the offspring
distribution
E[f (ξ ∗ )] =
∞
X
k=1
f (k)
kpk
= E[f (ξ)ξ]µ−1
µ
(3.10)
in particular
E[log(ξ ∗ )] = E[log(ξ)ξ]µ−1
and
E[ξ ∗ ] =
µ2
.
µ
(3.11)
(3.12)
(3.11) will be useful for showing properties of the maximum height of a branch in Chapter 5
and (3.12) shows that we only have finite mean of the size-biased distribution if the variance of
the offspring distribution is finite which leads to very different behaviour for the random walk
which isn’t experienced when µ > 1 since we obtain the bound (3.9).
Figure 3.4 shows subcritical and supercritical trees conditioned to survive. The dotted
lines represent finite branches and the solid lines represent the backbone structure. In both
cases the branches have a very similar structure as traps hence the main difference between
the random walks on these structures concerns the probability of entering a trap and moving
18
towards the root or away from the root from the backbone. For this reason we would expect that
the region for which the random walk is ballistic in the subcritical case to follow a very similar
structure as in the supercritical case. This is indeed true under certain additional conditions of
which we shall show the exact form in Chapter 5.
Figure 3.4: Tree diagrams in supercritical and subcritical cases.
3.3
Proof that the height conditioned tree coincides with the
construction
In this section we give a proof from Janson [12] that the tree constructed in Section 3.2.2 is
distributed according to the limiting measure P∗ from Kesten’s result (3.1). Before showing
this we state an important result for subcritical branching processes. This gives the condition
required for showing convergence of P(Zn > 0)/µn to a positive constant which will be important
for determining P∗ . This clearly fails when µ = 1 since limn→∞ P(Zn > 0) = 0 however we still
clearly have that P(Zn > 0)/µn is decreasing which will be important for showing the slowly
varying form of P(Zn > 0).
Theorem 3.3.1 (Lyons, Pemantle & Peres [15]). The sequence {P(Zn > 0)/µn } is decreasing.
If µ < 1 then TFAE
1. limn→∞ P(Zn > 0)/µn > 0;
2. supn E[Zn |Zn > 0] < ∞;
3. E[ξ log(ξ)] < ∞.
The result that limn→∞ P(Zn > 0)/µn > 0 is helpful for proving the form of the distribution of T ∗ however we require a corresponding result when µ = 1. This result is actually
much stronger than required and the result in Lemma 3.3.2 is sufficient which doesn’t require
that E[ξ log(ξ)] < ∞.
19
Lemma 3.3.2. If 0 < µ ≤ 1 is the mean of the GW process Z then for any l ∈ N
P(Zn > 0)
= µl .
n→∞ P(Zn−l > 0)
lim
Proof. Since P(Zn > 0) > 0 for any n we have that, by the branching property, for k = 1, 2, ...
P(Zn > 0) =
∞
X
P(Z1 = k)P(Zn > 0|Z1 = k)
k=1
=
∞
X
pk (1 − P(Zn−1 = 0)k ).
k=1
In particular,
∞
X pk (1 − P(Zn−1 = 0)k )
P(Zn > 0)
=
P(Zn−1 > 0)
P(Zn−1 > 0)
=
k=1
∞
X
pk
k−1
X
P(Zn−1 = 0)j .
j=0
k=1
For any j we have that P(Zn−1 = 0)j increases to 1 hence by monotone convergence we have
that
∞
X
P(Zn > 0)
kpk
=
n→∞ P(Zn−1 > 0)
lim
k=1
= µ.
The result then follows from
l−1
Y P(Zn−k > 0)
P(Zn > 0)
= lim
= µl .
n→∞ P(Zn−l > 0)
n→∞
P(Zn−k−1 > 0)
lim
k=0
From this we can conclude Corollary 3.3.3 which shows a very useful form of P(Zn > 0)
which we will use later to obtain results concerning the height of a branch.
Corollary 3.3.3. If 0 < µ ≤ 1 is the mean of the GW process Z then P(Zn > 0) = µn L(µn )
where L(µn ) is slowly varying as n → ∞ and is decreasing in n.
Proof. Define L(µn ) := P(Zn > 0)/µn then by Theorem 3.3.1 this is decreasing in n so it suffices
20
to consider a = µm for m ∈ Z then
L(µn )
L(µn )
P(Zn > 0) m
=
=
µ
n
n+m
L(aµ )
L(µ
)
P(Zn+m > 0)
which converges to 1 as n → ∞.
We are now ready to state the main result of Chapter 3 which confirms that the subcritical GW tree conditioned to survive has the form defined in Section 3.2.2.
Theorem 3.3.4. The distribution P∗ over (sub)critical GW trees conditioned on non-extinction
coincides with the distribution of T ∗ .
Proof. Let T ∈ T[l] for l ≥ 0 then we will show that
P∗ (T[l] = T ) =
ZlT P(T[l] = T )
= P(T[l]∗ = T ).
µl
By the branching property of the GW process we have that
P∗ (T[l] = T ) = lim P(T[l] = T |Zn > 0)
n→∞
P(T[l] = T )P(Zn > 0|T[l] = T )
n→∞
P(Zn > 0)
P(Zn−l > 0|Z0 = ZlT )
= P(T[l] = T ) lim
n→∞
P(Zn > 0)
= lim
T
1 − P(Zn−l = 0)Zl
= P(T[l] = T ) lim
n→∞
P(Zn > 0)
T
Zl −1
1 − P(Zn−l = 0) X
= P(T[l] = T ) lim
P(Zn−l = 0)j
n→∞
P(Zn > 0)
j=0
ZlT −1
=
P(T[l] = T )
P(Zn−l > 0)µl X
lim
P(Zn−l = 0)j
n→∞
P(Zn > 0)
µl
=
ZlT P(T[l]
µl
j=0
= T)
.
Where we have used Lemma 3.3.2 to show
(sub)critical to ensure P(Zn−l =
0)j
P(Zn−l >0)µl
P(Zn >0)
→ 1 and the fact that the tree T is
→ 1 for any j.
We now show that the constructed tree T[l]∗ has the same distribution. Clearly for fixed
u (possible offspring of a given special vertex) we have
P(ξ ∗ = k, u special) =
21
pk
1
P(ξ ∗ = k) = .
k
µ
We want to show that
P(T[l]∗ = T, u special) = µ−l P(T[l] = T ).
(3.13)
We will show this inductively hence suppose l = 1 then
∗
= T, u special) =
P(T[1]
P(ξ ∗ = Z1T , u special)
Z1T
P(ξ = Z1T )
µ
P(T[1] = T )
=
µ
=
so (3.13) holds for all T ∈ T[1] . Assume (3.13) holds for all T ∈ T[l] and consider T ∈ T[l+1] .


− special)P {ξ ∗ = dT←
∗
u
= T, u special) = P(T[l]∗ = T[l] , ←
P(T[l+1]
−} ∩
u
=
=
P(T[l] = T[l] ) pdT←
−
u
l
µ
µ
Y
−)
v∈T[l] \(T[l−1] ∪←
u
\
{ξv = dTv }
−)
v∈T[l] \(T[l−1] ∪←
u
pdTv
P(T[l+1] = T )
µl+1
therefore by induction we indeed have that (3.13) holds for all T ∈ Tf .
If we sum over u in the final generation of T in (3.13) we have that
P(T[l]∗ = T ) = µ−l ZlT P(T[l] = T ) = P∗ (T[l] = T ).
So indeed, for any l ∈ N we have that T[l]∗ has the same distribution as T[l] under P∗ therefore
since P∗ is a probability measure over rooted, labelled, non-extinct trees we have that P∗ is the
distribution over the construction T ∗ .
We will, therefore, henceforth denote T ∗ to denote the f -GW tree conditioned to survive
and T to denote the unconditioned f -GW tree unless specified otherwise.
3.4
Regenerations
Recall that the slabs Sn (2.1) are the sections of the tree together with the walk between
regeneration points. These have been shown by Lyons, Pemantle & Peres [16] to be mutually
independent random variables in the case of the transient random walk on the supercritical
GW tree conditioned on non-extinction. Now that we have determined the distribution over
22
subcritical GW trees conditioned on non-extinction we will extend their results to the transient
random walk on the T ∗ . More specifically, let θ̃k satisfy θk = ηθ̃k be the regeneration times for
the walk Yn on Y so that Yθ̃k = Xθk then we show that E[θ̃k − θ̃k−1 ] < ∞ for k = 1, 2, ... when
the random walk X is transient.
The following lemma shows the critical value of β for recurrence and transience of the
random walk on the T ∗ . The idea is that the random walk X is recurrent if and only if the
random walk Y is recurrent since the branch Tρ∗−
is a finite tree P-a.s. for any i and so starting
i
from X0 ∈ Tρ∗−
\ {ρi } we have that τρ+i < ∞ P-a.s. Y is a simple random walk on the backbone
i
Y = (ρ0 , ρ1 , ...) with transition probabilities
P (Yn+1 = ρi ) =

1



 1+β
β
1+β



0
Yn = ρi+1
Yn = ρi−1
otherwise
for i 6= 0 and P (Yn+1 = ρ1 |Yn = ρ0 ) = 1 hence by coupling with a simple random walk on Z+
with a reflecting barrier at 0 we have that Y is recurrent if and only if β ≤ 1.
Lemma 3.4.1. In the case µ < 1 the random walk on T ∗ is transient if and only if β > 1.
Proof. From (2.2) the random walk is transient if and only if the effective conductance
C(ρ, ∞) = inf{E(f ) : f (ρ) = 1, f (∞) = 0}
if positive. Cutting bonds between vertices x, y (i.e. setting κ(x, y) = 0) doesn’t increase the
conductance hence cutting all bonds between the backbone and buds shows that in order to show
that the walk is transient when β > 1 it suffices to show that the graph N with conductances
κ(x, x + 1) = β |x| has finite effective resistance. Resisters in series add hence
R(ρ, ∞) ≤
∞
X
β −n
n=0
which is finite when β > 1 so indeed the β-biased random walk on T ∗ is transient when β > 1.
For β ≤ 1 we want to show that
(
inf
)
X
(f (x) − f (y))2 κ(x, y) : f (ρ) = 1, f (∞) = 0
= 0.
x∼y
Recall Π : T ∗ → Y is the projection of a vertex onto its closest ancestor on the backbone then
defining f (x) = f (Π(x)) for x ∈ T ∗ \ Y it suffices to find a sequence of functions fn : N → [0, 1]
such that fn (0) = 1, limx→∞ fn (x) = 0 and
lim
n→∞
X
(fn (m) − fn (m + 1))2 β m = 0.
m≥0
23
Choosing

1 −
fn (m) =
0
m
n
m>n
we have that
X
m≤n
(fn (m) − fn (m + 1))2 β m ≤
m≥0
1
→0
n
as n → ∞.
Lemma 3.4.2 gives the exact probability that the walk on the backbone never returns to
its current location. This is both positive and a lower bound for the probability that a given
point on the backbone is a regeneration point for X when β > 1.
Lemma 3.4.2. When µ < 1, β > 1 we have that
Pρ1 (|Yj | > 1 ∀j > 0) =
β−1
.
β+1
Proof. Pρ1 (|Yj | > 1 ∀j > 0) = Pρ1 (Y1 = ρ2 )Pρ2 (|Yj | ≥ 2 ∀j ≥ 0) so write pk := Pρk (|Yj | ≥
1 ∀j ≥ 0) then p0 = 0 and for k = 1, 2, ...
pk =
1
β
pk+1 +
pk−1 .
β+1
β+1
Rearranging gives
1
(pk − pk−1 )
β
pk+1 − pk =
= β −k p1 .
By transience we have that
1=
∞
X
pk+1 − pk
k=0
= p1
=
1
∞
X
β −k
k=0
p1
.
− β −1
So indeed we have that
Pρ2 (|Yj | ≥ 2 ∀j ≥ 0) = p1 =
and
Pρ1 (|Yj | > 1 ∀j > 0) =
β−1
β
β
β−1
β−1
·
=
.
β+1
β
β+1
24
We are now able to prove that when the walk is transient we have that there are infinitely
many regeneration epochs. This proof follows a similar structure to the corresponding proof in
[16] for the transient biased random walk on the supercritical tree conditioned to survive.
Lemma 3.4.3. There exist infinitely many regeneration epochs P-a.s. for the walk Xn on T ∗
for µ < 1, β > 1.
Proof. Since regeneration points occur on the backbone it suffices to show that there are infinitely many regenerations epochs for Yn on Y.
Denote Θ = {θ̃k }k≥0 to be the set of all regeneration epochs of Yn and Θn := {θ̃ ∈ Θ :
θ̃ ≤ n} to be the set of all regeneration epochs before time n. We want to show that Θ \ ΘN 6= φ
for any N .
By Lemma 3.4.1 the random walk Yn is transient hence each ρi has a first hitting time
∇i := inf{n ≥ 0 : Yn = ρi }. {∇i }∞
i=0 are stopping times and P (∇i ∈ Θ) = P (∇1 ∈ Θ) for
all i ≥ 1. It follows from Lemma 3.4.2 that the probability that the first hitting time is a
regeneration time is positive since
P (∇1 ∈ Θ) = Pρ1 (|Yj | > 0 ∀j > 0) ≥ Pρ1 (|Yj | > 1 ∀j > 0) =
β−1
> 0.
β+1
Clearly ∇i ≥ i hence without loss of generality let ρi be the closest point to ρ on Y that
hasn’t been reached by time N and let FN denote the σ-field generated by the walk up to time
N.
P (Θ \ ΘN 6= φ|FN ) ≥ P (∇i ∈ Θ|FN )
= P(∇1 ∈ Θ).
By martingale convergence we have that
P (Θ \ ΘN 6= φ|F∞ ) = lim P (Θ \ ΘN 6= φ|FN +k )
k→∞
≥ lim P (Θ \ ΘN +k 6= φ|FN +k )
k→∞
≥ P(∇1 ∈ Θ).
Therefore, by Kolmogorov’s 0 − 1 law we have that there are P-a.s. infinitely many regeneration
epochs.
Since regenerations of X only occur on Y (P-a.s. ) and only depend on the walk Yn along
d
the backbone we have that since the backbone is deterministic and Tρ∗i = Tρ∗j for all i, j ≥ 0 we
25
have that {Sn }∞
n=1 are independent and identically distributed. Lemma 3.4.4 shows that the
expected time between regeneration points of Yn is finite; this will be fundamental in proving
that the speed exists P-a.s. in Chapter 5.
Lemma 3.4.4. For µ < 1, β > 1 we have that
E[θ̃2 − θ̃1 ] < ∞.
Proof. Since the sequence {θ̃k }k≥1 only depends on the process Yn on the backbone it suffices
to restrict to this process. The expected number of regeneration epochs by time n is
E [|Θn |] =
=
n
X
k=1
n
X
P (k ∈ Θ)
P (Yj 6= Yk ∀j < k)P (Yj 6= Yk−1 ∀j ≥ k)
k=1
= P (∇1 ∈ Θ)E [|Ψn |]
where Ψn is the set of distinct vertices reached by time n. |Ψn | is the number of distinct vertices
reached by time n which is also the number of k ≤ n for which the process doesn’t revisit Yk
for times k + 1, ..., n hence we have that
E [|Ψn |] = E
" n
X
#
χ{Yj 6=Yk ∀j=k+1,...,n}
k=1
=
n
X
P (Yj 6= Yk ∀j = k + 1, ..., n)
k=1
≥ nP (Yj 6= Y0 ∀j > 0).
This follows because the events that the process doesn’t return to the parent are identically
distributed (except when the starting vertex is the root in which case this probability is 1 hence
the bound still holds). Since the time between regenerations is identically distributed we have
that the frequency of regeneration epochs is
E [|Θn |]
n→∞
n
E [θ̃2 − θ̃1 ]−1 = lim
E [|Ψn |]
n→∞
n
≥ P (∇1 ∈ Θ)Pρ1 (Yj 6= ρ ∀j ≥ 0)
β−1 2
≥
β+1
= P (∇1 ∈ Θ) lim
by Lemma 3.4.2, which is positive since β > 1.
26
A straightforward but important conclusion of Lemma 3.4.4 is that the expected distance
between regeneration points is finite which we shall use for proving results for the speed. Much
stronger results can be shown for the times between regeneration times; in particular Ben Arous,
Fribergh, Gantert & Hammond [2] use that the time between regenerations for the walk on the
supercritical tree has exponential moments.
Corollary 3.4.5. For µ < 1, β > 1 we have that
E[|Xθ2 | − |Xθ1 |] < ∞.
Proof. |Xθ2 | − |Xθ1 | ≤ θ̃2 − θ̃1 hence then result follows trivially from Lemma 3.4.4.
27
Chapter 4
The speed of the random walk
In this chapter we survey the known results for the speed of the β-biased random walk on
supercritical and critical GW tree conditioned to survive. We then prove corresponding results
in Chapter 5. Recall that we say the walk is ballistic if rβ > 0 and sub-ballistic if rβ = 0. In
Chapter 3 we saw that the subcritical trees have the backbone structure of the critical tree and
traps similar to the supercritical trees. Typically, the backbone structure determines recurrence
and transience and hence the critical lower bound for β in the ballistic regime. Similarly, the
traps create a slowing down effect which determines the corresponding critical upper bound for
β in the ballistic regime. It is, therefore, clear that understanding the behaviour of the walk on
the supercritical and critical trees is instrumental to understanding the walk on the subcritical
tree however we shall observe in Chapter 5 that the subcritical tree has several additional
complications.
4.1
Speed on the supercritical tree
In this section we state results concerning the speed in the case of the supercritical tree, most
notably the region under which the speed is positive and its value. Recall that the speed rβ of
the random walk is the almost sure limit of |Xn |/n as n → ∞ for which we have the following
theorem distinguishing the ballistic and sub-ballistic regimes.
Theorem 4.1.1 (Lyons, Pemantle & Peres [16]). Let T ∗ be a supercritical GW tree conditioned
on non-extinction. The speed rβ exists P-a.s. is constant and depends only on the offspring
distribution ξ and bias parameter β. Furthermore, rβ is positive if and only if µ−1 < β <
f 0 (q)−1 .
As previously remarked, this shows several interesting phenomena. Firstly, when β ∈
(µ−1 , 1)
the bias acts towards the root. The reason that the speed can still be positive is that
by (3.2) the expected number of offspring on the backbone from a single individual on the
backbone vertex is µ hence the walk still has positive drift. Secondly, for β > f 0 (q) the speed
is zero because the walk spends too much time in traps. In the subcritical case q = 1 hence
f 0 (q) = f 0 (1− ) = µ suggesting that the walk is sub-ballistic when β ≥ µ−1 . This is in fact
28
true as we shall show in Chapter 5, however in the supercritical case µ−1 is also the lower
bound which would suggest that there is no ballistic regime for the walk on the subcritical tree.
This is false (assuming certain regularity conditions) because the lower bound is determined by
the mean number of offspring on the backbone and by conditioning on survival we change the
expected number of offspring from backbone vertices.
∗
+
For a fixed supercritical, non-extinct tree T ∗ denote λ(x) := PxT (τ←
− = ∞) to be the
x
quenched probability that the walk never returns to the parent of the starting vertex x. Recall
←
−
−
that T ∗ is the tree T ∗ with the addition of an artificial parent ←
ρ of the root. Then by λ(ρ) we
←
−
−
refer to the quenched probability that the walk never returns to ←
ρ on T ∗ . For β > µ−1 this
probability is positive if and only if the sub-tree Tx∗ is infinite (by transience of X), i.e. if x
i.i.d.
belongs to the backbone. Letting {λi }i≥0 ∼ λ(ρ) be P∗ random variables independent of T ∗
we have the following result:
Theorem 4.1.2 (Aı̈dékon [1]). For 1 < µ < ∞, µ−1 < β < f 0 (q)−1 we have
E
rβ =
E
(βξ−1)λ0
P
1−β+β ξi=0 λi
(βξ+1)λ0
P
1−β+β ξi=0 λi
.
This characterises the speed with respect to the annealed expectation. Notice that β
is constant hence the only truly random variables are λi , ξ which are independent. We would
expect the speed to be continuous with respect to β. With this in mind we should expect that
rµ−1 = 0 = rf 0 (q)−1 which isn’t immediately obvious from this equation.
In the sub-ballistic regime we have that rβ = 0 because either the bias is too large or
too small. If the bias is too large then the walk loses time in traps meaning that time spent
out of deep traps becomes negligible. If the bias is too small then the walk drifts towards the
root. In the latter case the walk is recurrent whereas in the former case the walk is transient
and hence will ‘escape’ (i.e. |Xn | → ∞ P-a.s. as n → ∞).
Suppose that E[ξ 2 ] < ∞ and β > f 0 (q)−1 so that the walk is transient and strongly
biased away from the root. We denote the exponent
γ := −
log(f 0 (q))
<1
log(β)
and define the hitting time of the nth level as
∆n := inf{i ≥ 0 : |Xi | = n}.
Theorem 4.1.3 (Ben Arous, Fribergh, Gantert & Hammond [2]). The laws of ∆n n−1/γ and
|Xn |n−γ are tight with respect to P. Furthermore, we have that P-a.s.
log |Xn |
=γ
n→∞ log n
lim
29
however ∆n n−1/γ doesn’t converge in distribution (at least for sufficiently large β).
The reason that the sequence ∆n n−1/γ doesn’t converge in distribution is because of a
lattice effect which means that |Xn | cannot be rescaled properly. It can be shown that most
of the time spent by the random walk before reaching level n is spent in ‘deep’ traps. The
deeps traps are roughly independent and the trapping times cluster around powers of β. This
creates a discrete inhomogeneity in the displacement of the walker which is persistent in all
time steps and prevents the distribution belonging to the domain of attraction of a stable law.
This is discussed by Hammond [10] who considers a model with random biases according to
some distribution ν satisfying the non-lattice assumption that log(supp(ν)) is dense in R which
is sufficient to remove the discrete inhomogeneity. We return to this discussion in Section 5.3
where we consider the difficulties in extending the proof of Theorem 4.1.3 to the sub-ballistic
walk on the subcritical tree.
4.2
Speed on the critical tree
In this section we give an overview of the results by Croydon, Fribergh & Kumagai [6] for the
walk on the critical tree. Details concerning the Skorohod topology will not be needed for the
results we prove for the subcritical tree hence I will not discuss this further; for additional detail
see Whitt [18].
We have seen that the critical and subcritical GW trees conditioned to survive have a
very similar structure. Both are formed by a single infinite line of descent with branches as
finite trees composed of a collection of critical or subcritical GW trees respectively. For this
reason we may expect that the random walk has very similar behaviour on the two trees however
this will turn out to be false. Although the walk Yn on the backbone will have very similar
behaviour it is the trapping in shrubbery that slows the walk down. In particular, critical trees
are typically much larger than subcritical trees hence they form larger traps and slow the walk
down significantly more. We see here that when µ = 1 (assuming p1 6= 1) we always have that
the walk on T ∗ is sub-ballistic.
It can be shown that when the offspring distribution is non-trivial, critical and belongs
to the domain of attraction of a stable law with index α ∈ (1, 2] then the escape follows a
logarithmic regime. Denote m(t) to be the maximum process of the Poisson point process with
intensity measure x−2 dxdt.
Theorem 4.2.1 (Croydon, Fribergh & Kumagai [6]). Let ξ be a critical offspring distribution
in the domain of attraction of an α-stable law where α ∈ (1, 2]. As n → ∞ the laws of the
processes
(α − 1) log+ ∆nt
n log(β)
t≥0
converge weakly with respect to the Skorohod J1 topology on D([0, ∞), R) to the law of (m(t))t≥0 .
This shows that the random walk follows a logarithmic escape along the backbone dependent on α, β in the sense that if α is large then the leaves are smaller and the walk escapes
30
faster and if β is large then the time spent in the traps increases and the walk escapes more
slowly.
An interesting extension to this result is the following theorem which shows that the
sequence of quenched expectations grows more quickly than the hitting times because it explores
every trap in every branch whereas the hitting time only explores those which are visited deeply.
Theorem 4.2.2 (Croydon, Fribergh & Kumagai [6]). Let α ∈ (1, 2]. As n → ∞, the laws of
the processes
∗
(α − 1) log+ (EρT [∆nt ])
nα log(β)
!
t≥0
under P converge weakly with respect to the Skorohod J1 topology on D([0, ∞), R) to the law of
(m(t))t≥0 .
4.3
Overview
The known results can largely be summarised by Figure 4.1 which shows the phase transition
for the speed in the case that the offspring distribution is geometric. The regions will be similar
for any offspring distribution with finite variance however one of the curves will have a slightly
different parametrisation. The reason for this is that the upper bound for the ballistic regime
when E[ξ] > 1 depends on the extinction probability which doesn’t have a direct correspondence
with µ. For µ > 1 there is a ballistic region µ−1 < β < 1/f 0 (q), a sub-ballistic region in which
polynomial scaling yields a tight sequence β > 1/f 0 (q) and a sub-ballistic region in which the
random walk is recurrent β < µ−1 . For µ = 1 the walk is always sub-ballistic; when β ≤ 1
the walk is recurrent and when β > 1 the walk follows a logarithmic escape regime. It is
straightforward to show that the recurrent region extends into the case µ ≤ 1 such that β ≤ 1
as we shall see in Chapter 5. The unknown region remaining is µ < 1, β > 1; intuitively
one expects that there will be some separation between a ballistic and sub-ballistic polynomial
regime as indicated in Figure 4.2 because the regime largely depends on the time spent in traps
which will be very similar to the supercritical traps because both have branches consisting of
some collection of independent subcritical GW trees. This is precisely what we show in Section
5.1 in the case that σ 2 < ∞ whereas when σ 2 = ∞ there is no ballistic regime. The reason
for this being that the size-biased distribution has infinite mean and hence the time spent in
a shrubbery in infinite. Furthermore, when σ 2 = ∞ the tightness result for the sub-ballistic
regime µ > 1, β > f 0 (q)−1 is unknown and so in this case the known results are indicated by
Figure 4.3.
31
Figure 4.1: Rate of escape for geometric offspring distribution.
Figure 4.2: Expected rate of escape for geometric offspring distribution.
Figure 4.3: Rate of escape for offspring distribution with infinite variance.
32
Chapter 5
The random walk on the subcritical
tree
In this chapter we prove the main results for the β-biased random walk on the subcritical GW
tree conditioned to survive. Section 5.1 is used to distinguish between the ballistic and subballistic regimes. The first main result of the section is Proposition 5.1.4 which shows that the
recurrent random walk on the subcritical tree is sub-ballistic as expected. The second main
result is Corollary 5.1.6 which shows that the walk is also sub-ballistic when β ≥ µ−1 . Recall
that the walk on the supercritical tree was sub-ballistic for β ≥ f 0 (q)−1 and that since q = 1
in the subcritical case f 0 (q)−1 = µ−1 so this is precisely what we expected. Corollary 5.1.7
then shows that the walk is sub-ballistic when σ 2 = ∞ which differs from the walk on the
supercritical tree due to the nature of the size biased distribution (3.12). Section 5.1 finishes
with Proposition 5.1.8 which shows that the remaining case 1 < β < µ−1 < ∞ with σ 2 < ∞ is
the ballistic regime for the walk on the subcritical tree and hence that the previous bounds are
tight. In Section 5.2 we prove the main result of this report which is the characterisation of the
speed for the ballistic random walk on the subcritical tree. We conclude the chapter in Section
5.3 with an outline of the proof of Theorem 4.1.3 and a discussion concerning the extension of
this to the walk on the subcritical tree.
5.1
Ballistic and sub-ballistic regimes
Before delving further, there is a trivial case that should be addressed. If pk = 0 for all k ≥ 2
then the subcritical tree conditioned to survive is identical in distribution the the GW tree with
offspring distribution p1 = 1 and pk = 0 for k 6= 1 hence the speed is the same as the speed
along the backbone which is (β − 1)(β + 1). For this reason, for the remainder of the chapter
we assume that there is some k ≥ 2 such that pk > 0.
Recall that we refer to branches as the finite trees Tρ∗−
:= Tρ∗i \Tρ∗i+1 which are comprised
i
of a backbone vertex and a shrubbery which itself consists of a collection of traps. We introduce
the following lemma for the maximum height of the trap in a single shrubbery which will be
used to prove the almost sure existence of the speed on the subcritical tree even if the offspring
33
distribution has infinite variance.
Lemma 5.1.1. If µ < 1 and E[ξ log(ξ)] < ∞ then for any i ∈ N0
E[H(Tρ∗−
)] < ∞.
i
ξρ∗ −1
i
Proof. H(Tρ∗−
) is the maximum height of the subcritical GW trees {Tρ∗i,i }ij =1
plus 1.
i
j
Fix 0 < c < − log(µ) so that ec µ < 1 then we have that
!
E[H(Tρ∗−
)] = 1 +
i
X
n≥1
≤1+
X
P
max H(Tρ∗i,i ) > n
ij ≤ξρ∗i −1
j
P(ξ ∗ > ecn ) +
X
P(ξ ∗ ≤ ecn , max
H(Tρ∗i,i )) > n).
∗
ij ≤ξ −1
n≥1
n≥1
j
The first sum describes the tails of the size biased distribution which we can bound using the
condition E[ξ log(ξ)] < ∞ as
X
P(ξ ∗ > ecn ) =
n≥1
X
P(log(ξ ∗ )c−1 > n)
n≥1
= E[log(ξ ∗ )]c−1
= E[log(ξ)ξ]c−1 µ−1
< ∞.
Here the third equality comes from properties of the size biased distribution (3.11).
By Corollary 3.3.3 P(H(T ) > n) = PT (Zn > 0) = µn L(µn ) for a function L(µn ) which
is slowly varying as n → ∞ and decreasing in n, hence bounded above with L(1) = 1. Using a
union bound we therefore have that

X
n≥1
H(Tρ∗i,i ) > n) ≤
P(ξ ∗ ≤ ecn , max
∗
ij ≤ξ −1
X
P
j
n≥1
≤
X
[
H(Tρ∗i,i
j≤ecn
ecn P(H(T ) > n)
n≥1
=
X
ecn µn L(µn )
n≥1
≤
X
(ec µ)n
n≥1
< ∞.
34
j

) > nξ ∗ = ecn 
Recall that {H(Tρ∗−
)}i≥0 are identically distributed under P so we actually have a unii
form bound over this expectation which will be used in Corollary 5.1.3. Lemma 5.1.2 shows the
reverse implication of Lemma 5.1.1 and hence that the criterion E[ξ log(ξ)] < ∞ is precisely
what is required so that the expected height of a trap is finite. We use this to show that the
walk doesn’t deviate too far from the backbone and hence the speed exists and coincides with
the speed of the process projected onto the backbone Π(Xn ).
Lemma 5.1.2. If µ < 1 and E[ξ log(ξ)] = ∞ then for any i ∈ N0
E[H(Tρ∗−
)] = ∞.
i
Proof. Recall that P(Zn > 0) = µn L(µn ) for a function L(s) slowly varying as s → 0+ .
E[H(Tρ∗−
)]
i
≥
≥
=
∞
X
n=1
∞
X
n=1
∞
X
P(ξ ∗ > µ−n L(µn )−1 , max∗ H(Tρ∗i,j ) > n)
j≤ξ
P(ξ ∗ > µ−n L(µn )−1 )(1 − P(H(T ) < n)µ
−n L(µn )−1
)
−n L(µn )−1
P(ξ ∗ > µ−n L(µn )−1 )(1 − (1 − µn L(µn ))µ
).
(5.1)
n=1
We have that
−n L(µn )−1
lim (1 − µn L(µn ))µ
n→∞
= e−1
−n L(µn )−1
hence we can find N1 ∈ N such that (1 − (1 − µn L(µn ))µ
) ≥ 1/2 for all n ≥ N1 . By
properties of slowly varying functions (see Proposition 1.3.6 [5]) we have that ∃N2 ∈ N such
that L(µn ) ≥ µn for all n ≥ N2 . Let N := max{N1 , N2 } and c = −2 log(µ). By dropping the
first N − 1 terms in (5.1) we have
E[H(Tρ∗−
)]
i
∞
1 X
≥
P(ξ ∗ > µ−2n )
2
=
1
2
n=N
∞
X
P(log(ξ ∗ )c−1 > n)
n=N
1
=
2c
∗
E[log(ξ )] −
N
−1
X
!
∗
−1
P(log(ξ )c
> n)
n=1
1
E[ξ log(ξ)]µ−1 − N
2c
= ∞.
≥
We will use Lemma 5.1.1 to show that when E[ξ log(ξ)] < ∞ the walk cannot deviate too
far from the backbone. In particular we will use Corollary 5.1.3 which gives a useful convergence
result.
35
Corollary 5.1.3. For µ < 1, E[ξ log(ξ)] < ∞ and any 0 < β < ∞ we have that P-a.s.
H(TX∗−
)
η
k
lim
k→∞
ηk
= 0.
Proof. Since |Xηk | ≤ k ≤ ηk we have that
H(TX∗−
)
η
k
ηk
≤
maxm≤k H(Tρ∗−
)
m
.
k
{H(Tρ∗−
)}m≥1 are P-i.i.d. with finite mean hence we have that P-a.s.
m
H(Tρ∗−
)
m
=0
m→∞
m
lim
and hence P-a.s.
maxm≤k H(Tρ∗−
)
m
= 0.
m→∞
k
lim
In the case that Xn is a recurrent random walk we have that |Xn |/n = 0 along certain
infinite subsequences (such as the return times to the root) which suggests that rβ = 0. This
is indeed true as we shall see however it is not as straightforward as one would expect since if
the walk is null recurrent then the walk could take excursions of arbitrary length making the
almost sure convergence non-trivial.
The proof that rβ exists P-a.s. and is constant splits into the two distinct cases that Xn
is recurrent or transient since the proof in the transient case requires the existence of infinitely
many regeneration epochs which doesn’t occur in the recurrent case but does in the transient
case by Lemma 3.4.3. In the case of the supercritical tree there is a phase transition between
the walk being ballistic and sub-ballistic where there is a transition between the walk being
transient and recurrent. We will have similar behaviour for the walk on the subcritical tree
hence it is important to distinguish the cases. Recall that in Lemma 3.4.1 we saw that the
β-biased random walk on the subcritical tree T ∗ was recurrent if and only if β ≤ 1. Proposition
5.1.4 shows that the random walk is sub-ballistic when it is recurrent by coupling with a random
walk on Z and using Corollary 5.1.3.
Proposition 5.1.4. If µ < 1, β ≤ 1 and E[ξ log(ξ)] < ∞ then rβ = 0 P-a.s.
d
Proof. Let Zn be a simple symmetric random walk on Z then for β = 1 we have that |Yn | = |Zn |.
P
We can write Zn := ni=1 Ui where Ui are i.i.d with zero mean and unit variance so by the strong
law of large numbers Zn /n converges a.s. to 0 and so |Zn |/n converges a.s. to 0.
d
Since |Yn | = |Zn | for β = 1 we therefore have that the unbiased random walk Yn on the
backbone Y satisfies |Yn |/n → 0 P-a.s. as n → ∞.
(β)
Write Yn to be the unbiased walk on the backbone and Yn
to be the β-biased random
(β)
(β)
walk on the backbone coupled to Yn starting from the root with transitions P (Yn+1 = ρ1 |Yn
36
=
ρ0 ) = 1 and for i ≥ 1
(β)
2β
β+1
1−β
= ρi , |Yn+1 | > |Yn |) =
β+1
P (Yn+1 = ρi+1 |Yn(β) = ρi , |Yn+1 | > |Yn |) =
(β)
P (Yn+1 = ρi−1 |Yn(β)
(β)
P (Yn+1 = ρi−1 |Yn(β) = ρi , |Yn+1 | < |Yn |) = 1.
(β)
(β)
In this setting we have that |Yn | ≥ |Yn | since the only situation in which |Yn | can increase
(β)
and |Yn | decrease is the case in which Yn
2) which results in
(β)
|Yn+1 |
= ρ0 and |Yn | ≥ 2 (since the processes have period
= 1 ≤ |Yn+1 |.
(β)
(β)
Since |Yn | stochastically dominates |Yn | for β ∈ (0, 1) we have that |Yn |/n converges
to 0 P-a.s. as n → ∞ whenever β ∈ (0, 1].
Recall that ηk denote the times of movements along the backbone Yk = Xηk . So in
particular
|Xηk |
|Xηk |
|Yk |
≤
=
→0
ηk
k
k
P − a.s.
For ηk ≤ n < ηk+1 we have that
|Xn | ≤ |Xηk | + H(TX∗−
).
η
k
In particular, since by Corollary 5.1.3 we have that P-a.s.
lim
k→∞
)
H(TX∗−
η
k
ηk
=0
and limk→∞ ηk = ∞ it follows that P-a.s.
|Xn |
lim
≤ lim
n→∞ n
k→∞
∗−
|Xηk | H(TXηk )
+
ηk
ηk
!
= 0.
The subcritical tree decomposes into three distinct parts; the backbone Y, the buds
{ρi,ij : ij = 1, ..., ξi∗ − 1, i = 0, 1, ...} and the traps attached to the buds {Tρ∗i,i : ij = 1, ..., ξi∗ −
j
1, i = 0, 1, ...}. The number of buds attached to a single vertex on Y has distribution ξ ∗ − 1
where E[ξ ∗ ] = µ2 /µ (3.12). Up to now we have made no assumptions on the variance of the
offspring distribution however the cases σ 2 = ∞, σ 2 < ∞ yield vastly different behaviour for
the random walk. In particular, we will show that when σ 2 = ∞ the expected time spent in
a shrubbery is infinite whereas when σ 2 < ∞ the expected time spent in a shrubbery is finite
(for β < µ−1 ). Furthermore, when σ 2 = ∞ the almost sure convergence of |Xn |/n is non-trivial
since bounding the distance Xn moves away from the backbone between movements along the
backbone becomes more technical. This isn’t the case with the supercritical tree because we have
37
the simple bound (3.9) on the expected number of children from a special vertex E[Z1 ] ≤
µ
1−q .
Lemma 5.1.5 shows that if the expected time between movements along the backbone is infinite
then the speed is zero.
Lemma 5.1.5. If β > 1, E[ξ log(ξ)] < ∞ and E[ηk − ηk−1 ] = ∞ then rβ = 0 P-a.s.
Proof. Since E[ηk − ηk−1 ] = ∞ we have that E[θk − θk−1 ] = ∞. The increments {θk+1 − θk }k≥1
are P-i.i.d. hence by the strong law of large numbers limk→∞ θk /k = ∞ P-a.s. For θk ≤ n < θk+1
|Xθk+1 |−1
X
|Xθk | ≤ |Xn | ≤ |Xθk+1 | +
H(Tρ∗−
).
j
j=|Xθk |
|Xθ
|−1
k+1
{H(Tρ∗−
)}j=|X
j
θ |
are independent with finite expectation hence since for β > 1 we have that
k
by Corollary 3.4.5 E[|Xθk | − |Xθk−1 |] < ∞ so it follows that P-a.s.
θk+1 |−1
|X
E
X

H(Tρ∗−
) = E[|Xθ2 | − |Xθ1 |]E[H(T ∗− )] < ∞.
j
j=|Xθk |
It therefore follows that P-a.s.
P|Xθk+1 |−1
lim
j=|Xθk |
H(Tρ∗−
)
j
θk
k→∞
= 0.
By the strong law of large numbers and E[|Xθk+1 | − |Xθk |] < ∞ P-a.s.
|Xθk+1 |
= 0.
θk
lim
k→∞
The result then follows from
|Xθk+1 |
|Xn |
lim
≤ lim
+
n→∞ n
k→∞
θk
Let
P|Xθk+1 |−1
j=|Xθk |
H(Tρ∗−
)
j
θk
= 0 P − a.s.
+
N :=
τρ
X
χ{Xn ∈c(ρ)\ρ1 }
n=1
be the number of time the walk hits a bud before returning to the root. Notice that by Corollary
38
2.3.2 when βµ < 1 we have
ETρ
∗−
[τρ+ |ξ ∗ 6= 1] = 2 + ETρ [τρ+ ]E[N ]
∞
=2+
2 X
P(ξ = k)E[N |ξ = k]
1 − βµ
k=0
=2+
=2+
Furthermore
∗−
ETρ [τρ+ ]
2
1 − βµ
∞
X
kpk
k=0
2µ
.
1 − βµ
= 2+
2µ
1 − βµ
P(ξ ∗ 6= 1)
It therefore follows that
ETρ
∗−
[τρ+ ] = ∞
⇐⇒
ETρ [τρ+ ] = ∞
⇐⇒
β ≥ µ−1 .
(5.2)
Using this and Lemma 5.1.5 we have the following corollary which shows that the walk
is sub-ballistic when β ≥ µ−1 = f 0 (1− ).
Corollary 5.1.6. If β ≥ µ−1 , E[ξ log(ξ)] < ∞ then rβ = 0 P-a.s.
Proof. Xηn = ρi for some i and P(ξρ∗i > 1) > 0 so P(ηn 6= ηn−1 + 1) > 0 hence by (5.2)
E[ηn − ηn−1 ] ≥ P(ηn 6= ηn−1 + 1)ETρ
∗−
[τρ+ |ξ ∗ 6= 1] = ∞.
By Lemma 5.1.5 we therefore have that rβ = 0 P-a.s.
Let
N 0 :=
N 00 :=
η2
X
χ{Xn =ρ1 }
(5.3)
n=η1 +1
η1
X
χ{Xn =ρ}
n=1
denote the the number of excursions into a trap between movements along the backbone. Notice
that since N 0 is defined away from the root this is equal in distribution to the number of
excursions into a trap from any vertex on the backbone other than the root. Both N 0 , N 00
follow a geometric distribution with mean p(1 − p)−1 where p is the probability of a excursion
into a trap. Away from the root, given the trap has k − 1 buds the probability of an excursion
is (k − 1)β/(1 + kβ) hence
E[N 0 |ξ ∗ = k] =
39
(k − 1)β
(1 + β)
and from the root the probability of an excursion is (k − 1)/k hence
E[N 00 |ξ ∗ = k] = k − 1.
The following corollary of Lemma 5.1.5 shows that if the offspring distribution has infinite
variance then the walk is sub-ballistic. This is, of course, very different to the walk on the
supercritical tree.
Corollary 5.1.7. If β > 1, E[ξ log(ξ)] < ∞ and σ 2 = ∞ then rβ = 0 P-a.s.
Proof. Let T ∗− denote a rooted tree equal in distribution to Tρ∗− (the root has offspring distributed according to ξ ∗ − 1 and every other vertex independently has offspring according to
the offspring distribution ξ). Since E[N 00 |ξ ∗ = k] ≥ E[N 0 |ξ ∗ = k] we have that
E[ηn − ηn−1 ] =
≥
∞
X
k=0
∞
X
k=1
=
∗
P(ξX
η
∗
= k)E[ηn − ηn−1 |ξX
η
n−1
= k]
kpk
∗−
E[N 0 |ξ ∗ = k]ETρ [τρ+ ]
µ
∗−
ETρ [τρ+ ]
∗−
=
n−1
∞
X
kpk (k − 1)β
k=1
+
[τρ ]
µ
1+β
βEρT
(µ2 − µ)
µ(1 + β)
=∞
since µ2 = ∞ ⇐⇒ σ 2 = ∞. Therefore, the result follows by Lemma 5.1.5.
Proposition 5.1.8 shows that in the remaining case we have that the walk is ballistic. In
Section 5.2 we will show the exact speed in this ballistic region.
Proposition 5.1.8. If σ 2 < ∞ then rβ exists P-a.s. and is positive if and only if 1 < β <
µ−1 < ∞.
Proof. We have shown that rβ = 0 P-a.s. when β ∈
/ (1, µ−1 ) hence it suffices to show that rβ
exists and is a positive constant when 1 < β < µ−1 . Since we can view the random walk Xn
on T ∗ as a walk Yn on Y taking excursions into subcritical branches Tρ∗−
we wish to determine
i
the distribution over the length of the excursions between ηn and ηn+1 .
By (5.2) ETρ
∗−
[τρ+ ] < ∞ since β < µ−1 . Between times ηn and ηn+1 the walk makes a
random number of excursions into the bush at Yn . This has a shifted geometric distribution
with mean either
E[N 0 |dρ = dYn ] =
β(dYn − 1)
.
β+1
or
E [N 00 |dρ = dYn ] = dYn − 1.
40
We can therefore bound the expected time between movements along the backbone as
E[ηn+1 − ηn |dYn ] ≤ 2 +
2µ
1 − βµ
(dYn − 1) ≤ cdYn
where c = 2 + 2µβ/(1 − βµ). In particular since E[dYn ] = µ2 /µ we have that
E[ηn+1 − ηn ] ≤ c
µ2
.
µ
(5.4)
Since, by Lemma 3.4.4, regenerations occur with positive frequency on Y it follows that
E[θk − θk−1 ] = E[θ̃k − θ̃k−1 ]E[ηn − ηn−1 ] < ∞.
We therefore have that {θk+1 − θk }∞
k=1 are P-i.i.d. with E[θk+1 − θk ] < ∞. Furthermore,
{θk+1 − θk }∞
k=1 are independent of θ1 and E[θ1 ] < ∞ (although not equally distributed to the
other times between regeneration epochs). By the strong law of large numbers limk→∞ θk /k =
E[θ2 − θ1 ]. Since the walk is independent between regeneration points we have that by the
strong law of large numbers
|Xθk |
lim
= lim
k→∞
k→∞
k
Pk
j=1 |Xθj |
Therefore
|Xθk |
= lim
lim
k→∞
k→∞ θk
− |Xθj−1 |
k
Pk
j=1 |Xθj |
= E[|Xθ2 | − |Xθ1 |] < ∞ P − a.s.
− |Xθj−1 |
k
· lim Pk
k→∞
k
j=1 θj
− θj−1
which exists P-a.s.
Moreover for θk ≤ n < θk+1 we have that
|Xθk | ≤ |Xn | ≤ |Xθk | + n − θk ≤ |Xθk | + θk+1 − θk .
Since θk+1 − θk
`
θk , θk+1 − θk has finite first moment and θk → ∞ as k → ∞ we have that
P-a.s.
lim
k→∞
and so
θk+1 − θk
=0
θk
|Xθk |
|Xn |
k
1
= lim
≥ lim
=
>0
n→∞ n
k→∞ θk
k→∞ θk
E[θ2 − θ1 ]
lim
P-a.s. so indeed we have positive speed when σ 2 < ∞ and 1 < β < µ−1 < ∞.
5.2
Speed in the ballistic regime
In this section we determine the speed of the random walk in the ballistic regime. This is much
simpler on the subcritical tree than on the supercritical tree because the movement along the
backbone is much more straightforward. Since the walk along the backbone has independent
41
increments (away from the root), we can use an argument that allows us to neglect the walk up
to the last hitting time of ρ and the strong law of large numbers to say that
|Xηk |
|Yk |
= lim
k→∞
k→∞ k
k
k
1X
= lim
|Yj | − |Yj−1 |
k→∞ k
lim
j=1
= E[|Y2 | − |Y1 |].
We would like to use a similar argument to say that
ηk
= E[η2 − η1 ].
k→∞ k
(5.5)
E[|Xη2 | − |Xη1 |]
E[η2 − η1 ]
(5.6)
lim
This would give us that
rβ =
which is what we will eventually obtain in Lemma 5.2.8. Unfortunately the distribution of the
times between movements along the backbone is not independent because traps are entered
multiple times and the structure of the trap at ρ differs slightly. In order to calculate the speed
we consider a two-sided tree, determine the speed for this tree using ergodic theory techniques
and show that the random walk has the same speed on the one-sided tree.
We write T∗∗ := (Tf )Z to be the space of two-sided sequences of finite trees which we
call two-sided trees. We define a probability measure over such trees as the product measure
Q
P∗∗ (dT ) = i∈Z P∗− (dTi ) where T = (Ti )i∈Z and P∗− is the distribution over branches T ∗− in
the subcritical GW tree conditioned to survive. We write T ∗∗ to be a tree distributed according
to P∗∗ .
T ∗∗ can be seen as a graph in which each element of Z gives rise to a tree according
to the measure P∗− . Write ρi := ρTi and Y := (..., ρ−1 , ρ0 , ρ1 , ...) to be the backbone for the
graph. We define the β-biased random walk X ∗ = (Xn∗ )n≥0 on T = (Ti )i∈Z ∈ T∗∗ as the Markov
process starting from X0∗ = ρ0 with transition probabilities
∗
P T (Xn+1
= y|Xn∗ = x) =

1


 1+βdx

β
1+βdx



0
−
y=←
x
y ∈ c(x)
otherwise
for x ∈ Ti \ ρi and
P
T
∗
(Xn+1
=
y|Xn∗
= x) =

1



 1+β+βdx
y = ρi−1


0
otherwise
β
 1+β+βdx
42
y ∈ c(x) ∪ ρi+1
for x = ρi . Here for x ∈ Ti we write dx to be the degree of x in Ti hence when x = ρi ∈ Y this
does not include ρi+1 as it would in the regular tree which motivates the unusual form of the
transition probabilities.
Intuitively, the tree can be constructed by fixing two subcritical GW trees, reversing
the orientation of the backbone of one of the trees and connecting the roots by an edge. Thus
the two-sided tree is essentially a subcritical GW tree conditioned to survive which has been
extended by replacing the one-sided infinite backbone (ρ0 , ρ1 , ...) with the two-sided infinite
backbone (..., ρ−1 , ρ0 , ρ1 , ...).
For x ∈ T = (Ti )i∈Z ∈ T∗∗ we write |x| = i + |x|Ti where x ∈ Ti and | · |Ti denotes the
generation number of x in Ti . It follows that |x| ∈ Z; in particular | · | can take negative values
and hence is not a norm.
The random walk Xn∗ on T ∈ T∗∗ is β times more likely to move deeper into a sub-tree
Ti or right along Z as it is to move towards the root of its current sub-tree or left on Z. It is
therefore clear that the process |Xn∗ | is equal in distribution to the process |Xn | on T ∗ started
from the same arbitrary backbone vertex up until the first hitting time of ρ0 . Similarly the
two-processes are equally distributed after the last hitting time of the roots ρ0 . This property
will be important as well shall use it to extend properties of the walk Xn∗ onto Xn .
Lemma 5.2.1 shows that the speed
|Xn∗ |
n→∞ n
rβ∗ = lim
exists and is constant which follows trivially from the walk on T ∗ .
Lemma 5.2.1. For σ 2 < ∞ and 1 < β < µ−1 < ∞ rβ∗ exists P-a.s. and is a positive constant.
Proof. After the first regeneration time θ1∗ := inf{n > 0 : Xn∗ ∈
/ {Xj }n−1
j=0 , Xm 6= Xn−1 ∀m ≥ n}
we have that the transitions of the random walk Xn∗ are equal in distribution to the transitions
of the random walk Xn after θ1 . Since E[θ1∗ ] < ∞ it follows that Proposition 5.1.8 extends
trivially to Xn∗ .
Lemma 5.2.7 shows that the corresponding form of (5.6) for the two-sided tree holds.
In order to obtain this we need to show the convergence of ηk /k which we do by showing that
the environment seen from the particle together with the walk is ergodic in Proposition 5.2.6.
Environment seen from the particle techniques are instrumental for random walk in random
environments because they allow the process which is not Markov under the annealed measure
to be considered as a Markov process and hence limiting results can be proven (see De Masi,
Ferrari, Goldstein & Wick [8] for more detail).
Define ζ : T∗∗ → T∗∗ to be the shift (ζT )i = Ti+1 to be the shift from the environment
seen from one vertex of the backbone Y to the environment seen from the particle to the right.
Let F ∗∗ denote the Borel σ-field over T∗∗ then (T∗∗ , F ∗∗ , P∗∗ ) is a probability space. By a
cylinder set C ∈ F ∗∗ we mean an element of the form C = {T ∈ T∗∗ : Tn ∈ An ∀n ∈ F } where
F ⊂ Z is a finite set and {An }n∈F are measurable sets of Tf .
43
Lemma 5.2.2. ζ is an ergodic measure preserving transformation on (T∗∗ , F ∗∗ , P∗∗ ).
Proof. Since P∗∗ is a product measure and ζ a bijective shift we clearly have that ζ is measure
preserving i.e. for T = (Ti )i∈Z we have that
P∗∗ (dζ −1 T ) =
Y
=
Y
P∗− (dTi−1 )
i∈Z
P∗− (dTi )
i∈Z
= P∗∗ (dT ).
Suppose that A ∈ F ∗∗ such that ζ −1 A = A then we want to show that P∗∗ (A) ∈ {0, 1}. Let
ε > 0 then we can find some collection {Cj }N
j=1 of disjoint cylinder sets such that

P∗∗ A4
N
[

Cj  < ε.
j=1
Write E =
SN
j=1 Cj .
M such that F :=
Since each Cj depends only on finitely many coordinates we can find some
ζ −M E
is independent from E. This gives us that
P∗∗ (F ∩ E) = P∗∗ (F )P∗∗ (E)
= P∗∗ (E)2
by invariance of ζ under P∗∗ . Since A = ζ −M A we have that
P∗∗ (A4F ) = P∗∗ (ζ −M A4ζ −M E)
= P∗∗ (ζ −M (A4E))
= P∗∗ (A4E)
< ε.
In particular
P∗∗ (A4(E ∩ F )) ≤ P∗∗ (A4E) + P∗∗ (A4F ) < 2ε.
44
We therefore have that
|P∗∗ (A) − P∗∗ (A)2 | ≤ |P∗∗ (A) − P∗∗ (E ∩ F )| + |P∗∗ (E ∩ F ) − P∗∗ (A)2 |
≤ 2ε + |P∗∗ (E)2 − P∗∗ (A)2 |
= 2ε + |P∗∗ (E) + P∗∗ (A)||P∗∗ (E) − P∗∗ (A)|
≤ 4ε.
ε was chosen arbitrarily hence this holds for all ε > 0; in particular, P∗∗ (A) = P∗∗ (A)2 hence
P∗∗ (A) ∈ {0, 1}.
Write Ω = (T∗∗ )Z and F to be the Borel σ-field over Ω. In general we will write
(n)
T = (T (n) )n∈Z for the elements of Ω where each T (n) = (Ti )i∈Z ∈ T∗∗ is a two-sided tree. For
T, T 0 ∈ T∗∗ write the transition probabilities
P (T, T 0 ) =

β



 1+β
1
1+β



0
T 0 = ζT
T 0 = ζ −1 T
otherwise.
This yields a probability measure Pn over elements of (T∗∗ )Z∩[−n,n] as
Pn (T (−n) , ..., T (n) ) = P∗∗ (dT (0) )
n−1
Y
P (T (k) , T (k+1) ).
k=−n
By Kolmogorov’s extension theorem we have that this extends to a well-defined probability
measure P∗ over Ω. We therefore have that (Ω, F , P∗ ) is a probability space over two-sided
sequences of two-sided trees where the starting tree T (0) is distributed according to P∗∗ and
transitions along the sequence of trees are the trees seen from a β-biased random walk Xn∗ on
T (0) observed at the times of transitions along the backbone. We define S : Ω → Ω to be the
shift to time of the next movement along the backbone (ST )n = T (n+1) .
Lemma 5.2.3. S is an ergodic measure preserving transformation on (Ω, F , P∗ ).
Proof. Let A ∈ F be the cylinder set A = {T ∈ Ω : T (n) ∈ An ∀n = −k, ..., k}, then since
45
ζ, ζ −1 are measure preserving transformations we have that
P∗ (S −1 A) =
P∗∗ (ζA0 )
= P∗∗ (A0 )
β
1
+ P∗∗ (ζ −1 A0 )
1+β
1+β
k−1
Y
k−1
Y
P (An , An+1 )
n=−k
P (An , An+1 )
n=−k
∗
= P (A).
Since cylinder sets generate the Borel σ-field we therefore have that S is a measure preserving
transformation.
Write
N :=
[
{ζ m T (0) ∈
/T}
m∈Z
to be the set consisting of all trajectories in which the entire backbone is not explored. By
transience of the biased random walk on Z we have that P∗ (N ) = 0.
Suppose A ∈ F and S −1 A = A then we want to show that P∗ (A) ∈ {0, 1}. We have
that A0 := A \ N is invariant (i.e. S −1 A0 = A0 ) since N is clearly invariant. Thus, it suffices to
show that P∗ (A0 ) ∈ {0, 1}.
By Hairer [9] there is some measurable collection of two-sided trees A ⊆ T∗∗ such that
Z
A0 = A . In other words A0 contains all collections of sequences of trees from A.
Z
Z
Using that S −1 A = A we want to show that T ∈ A if and only if ζT ∈ A. In other
words A = ζ −1 A. This is true because A0 only contains sequences of trees for which the entire
backbone has been explored and hence the tree T seen from every backbone vertex belongs to
A.
ζ is an ergodic measure preserving transformation on Ω hence P∗∗ (A) ∈ {0, 1}. Since
ζ −1 A = A we have that P (A, A) = 1. It therefore follows that P∗ (A0 ) ∈ {0, 1}.
Write B to be the Borel-σ field over [0, 1]Z and ν the probability measure of the uniform
distribution on ([0, 1]Z , B). Define the shift Sν : [0, 1]Z → [0, 1]Z by (Sν U )i = Ui+1 .
Lemma 5.2.4. Sν is weakly mixing on ([0, 1]Z , B, ν).
(A)
Proof. Fix A, B ∈ B and let ε > 0 then we can find collections of disjoint cylinders {Cj }N
j=1
SN
SM
(B)
(A)
(B)
and {Ci }M
such
that
writing
E
=
C
and
F
=
C
we
have
that
i=1
j=1 j
i=1 i
ν (A4E) , ν (B4F ) < ε.
(5.7)
Since E, F are cylinders we have that ∃K > 0 such that Sν−k E is independent from F
for all k ≥ K. Then for k < K we have that |ν(Sν−k A ∩ B) − ν(A)ν(B)| ≤ 1 and for k ≥ K we
46
have that
|ν(Sν−k A ∩ B) − ν(A)ν(B)| ≤ |ν(Sν−k A ∩ B) − ν(Sν−k A ∩ F )|
+ |ν(Sν−k A ∩ F ) − ν(Sν−k E ∩ F )|
+ |ν(Sν−k E ∩ F ) − ν(E)ν(F )|
+ |ν(E)ν(F ) − ν(A)ν(F )|
+ |ν(A)ν(F ) − ν(A)ν(B)|.
The first, fourth and fifth terms on the right hand side are less than ε by (5.7) and writing
Fk = Sνk F we have that the second term can be written as
|ν(Sν−k A ∩ F ) − ν(Sν−k E ∩ F )| = |ν(Sν−k (A ∩ Fk )) − ν(Sν−k (E ∩ Fk ))|
= |ν(A ∩ Fk ) − ν(E ∩ Fk )|
< ε.
Here, the second equality holds because Sν is measure preserving and the inequality holds by
(5.7). Sν−k E and F are independent hence we can write ν(Sν−k E ∩ F ) = ν(Sν−k E)ν(F ) then,
since Sν is measure preserving, we have that |ν(Sν−k E ∩ F ) − ν(E)ν(F )| = 0. It therefore follows
that for k ≥ K we have that |ν(Sν−k A ∩ B) − ν(A)ν(B)| < 4ε.
We then have that
n−1
1X
K + 4nε
lim
|ν(Sν−k A ∩ B) − ν(A)ν(B)| ≤ lim
n→∞ n
n→∞
n
k=0
= 4ε.
Since ε was arbitrary we indeed have that
n−1
1X
lim
|ν(Sν−k A ∩ B) − ν(A)ν(B)| = 0
n→∞ n
k=0
hence Sν is weakly mixing.
A result by De La Rue [7] says that if (X, FX , µX , SX ) is an ergodic probability space and
(Y, FY , µY , SY ) is a weak mixing probability space then we have that (X × Y, FX ⊗ FY , µX ×
µY , SX × SY ) is an ergodic probability space where (SX × SY )(AX , AY ) := (SX AX , SY AY ).
From this, we have Corollary 5.2.5.
Corollary 5.2.5. (Ω × [0, 1]Z , F ⊗ B, P∗ × ν, S × Sν ) is an ergodic probability space.
Using Corollary 5.2.5 we can prove Proposition 5.2.6 which gives us that (5.5) is indeed
true for the two-sided tree.
47
Proposition 5.2.6. If 1 < β < µ−1 < ∞ and σ 2 < ∞ then P∗ -a.s. we have that
ηn∗
= E∗ [η1∗ ].
n→∞ n
lim
Proof. Write f : Ω × [0, 1]Z → N as f (T , U ) = FT−1
(0) (U0 ) where FT is the c.d.f.
FT (t) = PρT0 (η1∗ ≤ t).
∗
We then have that f (S k T , Sνk U ) = ηk+1
− ηk∗ thus, f ∈ L1 (Ω × [0, 1]Z , F ⊗ B, P∗ × ν) since it
d
∗
is F ⊗ B-measurable and E∗ [ηk+1
− ηk∗ ] < ∞ for all k ∈ N.
By Birkhoff’s ergodic theorem we have that
Pn−1
∗
ηk+1
− ηk∗
n
Pn−1
k
k
f
k=0 (S T , Sν U )
= lim
n→∞
n
∗
= E [f (T , U )]
η∗
lim n = lim
n→∞ n
n→∞
k=0
= E∗ [η1∗ ].
For ease of notation we now use P, E for the annealed probability and expectation respectively of the random walks Xn , Xn∗ since it is clear from context to which we refer.
Using Proposition 5.2.6 we can now obtain a convergence result for ηk /k and hence a
form of the speed on the two-sided tree in terms of the expected position of the walk at the first
movement along the backbone and the expected time taken to reach this point.
Lemma 5.2.7. For 1 < β < µ−1 < ∞ and σ 2 < ∞ we have that P-a.s.
rβ∗
=
E[|Xη∗∗ |]
1
E[η1∗ ]
.
Proof. ηi∗ is a strictly increasing sequence taking values in N hence since rβ∗ exists P-a.s. and
belongs to (0, 1] we have that
∗
rβ∗
|Xη∗ |
|Xn∗ |
k
= lim
= lim
∗ .
n→∞ n
k→∞ ηk
By the strong law of large numbers we have that P-a.s.
lim
k→∞
|Xη∗∗ |
k
k
= lim
k→∞
k |X ∗∗ | − |X ∗∗ |
X
ηj
ηj−1
k
j=0
= E[|Xη∗1∗ |]
since movements along the backbone are P-i.i.d. By Proposition 5.2.6 the environment seen
48
from the particle together with the walk ((T ∗∗ , Xη∗n∗ ), (Xηn∗ , Xηn∗ +1 , ...))∞
n=0 is an ergodic Markov
chain with respect to P so for ω = ((T ∗∗ , X0∗ ), (X0∗ , X1∗ , ...)) write
∗
∗
∗
}| = ηm+1
− ηm
f (S m (ω)) = |{Xη∗m
∗ , ..., Xη ∗
m+1 −1
then by Birkhoff’s ergodic theorem P-a.s.
k
∗
X ηj∗ − ηj−1
ηk∗
= lim
k→∞ k
k→∞
k
lim
j=1
= lim
k
X
f (S m (ω))
k→∞
k
j=1
= E[η1∗ ].
So indeed P-a.s.
rβ∗
=
limk→∞
|Xη∗∗ |
k
k
η∗
limk→∞ kk
=
E[|Xη∗∗ |]
1
E[η1∗ ]
.
We have now obtained a computable form of the speed of the walk on the two-sided tree.
Since transient, the walk is only in T ∗∗ \ Tρ∗∗ for a finite period of time hence we would expect
the speed to be the same on the one-sided tree. Lemma 5.2.8 shows that this is indeed true.
Lemma 5.2.8. For 1 < β < µ−1 < ∞ and σ 2 < ∞ we have that rβ = rβ∗ P-a.s.
Proof. It suffices to show that for independent walks Xn , Xn∗ P-a.s.
lim
|Xη∗∗ | − |Xηk |
k
k
k→∞
and
=0
(5.8)
ηk∗ − ηk
=0
k→∞
k
lim
(5.9)
since the result then follows by the fact that
rβ =
|Xηk |
k
ηk .
k
limk→∞
limk→∞
Write ψ := sup{i ≥ 0 : Xηi = ρ1 } and ψ ∗ := sup{i ≥ 0 : Xη∗∗ = ρ1 } be the last times
i
that the processes on the backbone are at ρ1 and ψk := ψ ∧ k, ψk∗ := ψ ∗ ∧ k.
49
lim
k→∞
|Xη∗∗ | − |Xηk |
k
k
Pk
∗
j=1 (|Xηj∗ |
= lim
− |Xη∗∗ |) − (|Xηj | − |Xηj−1 |)
j−1
k
k→∞
Pψk∗
∗
j=1 |Xηj∗ |
= lim
−
|Xη∗∗ |
j−1
k
k→∞
Pk
∗
j=ψk∗ +1 |Xηj∗ |
+
Pψk
j=1 |Xηj |
−
k
|Xη∗∗ |
j−1
−
− |Xηj−1 |
k
Pk
j=ψk +1 |Xηj |
−
− |Xηj−1 |
k
.
Since ψ ∗ , ψ < ∞ P-a.s. and for j < ψ ∗ , ψ respectively we have
||Xη∗j∗ | − |Xη∗j−1
||, ||Xηj | − |Xηj−1 | ≤ 1
∗
it follows that P-a.s.
Pψk∗
∗
j=1 |Xηj∗ |
lim
j−1
k
k→∞
and
− |Xη∗∗ |
Pψk
j=1 |Xηj |
lim
− |Xηj−1 |
k
k→∞
For each fixed j we have that Xη∗∗ ∗
ψ +j
Pk
lim
∗
j=ψk∗ +1 |Xηj∗ |
− |Xη∗∗ |
j−1
k
k→∞
= 0.
d
− Xη∗∗ ∗
= Xηψ+j − Xηψ+j−1 hence by P almost
ψ +j−1
sure convergence of |Xn |/n, |Xn∗ |/n we have that
=0
Pk
= lim
j=ψk +1 |Xηj |
− |Xηj−1 |
k
k→∞
.
(5.8) therefore follows hence it remains to show (5.9).
η ∗ − ηk
lim k
= lim
k→∞
k→∞
k
Pk
∗
j=1 (ηj
∗ ) − (η − η
− ηj−1
j
j−1 )
k
Pψk∗
∗
j=1 ηj
= lim
−
∗
ηj−1
k
k→∞
Pk
+
∗
j=ψk∗ +1 ηj
k
−
Pψk
−
∗
ηj−1
j=1 ηj
− ηj−1
k
Pk
−
j=ψk +1 ηj
− ηj−1
k
Since ψ ∗ , ψ < ∞ P-a.s. and for j < ψ ∗ , ψ respectively we have that by (5.4)
E[ηj∗
−
∗
ηj−1
], E[ηj
− ηj−1 ] ≤ 2 1 +
50
µβ
1 − βµ
µ2
.
µ
.
It follows that P-a.s.
Pψk∗
∗
j=1 ηj
lim
k
k→∞
and
Pψk
lim
∗
− ηj−1
j=1 ηj
− ηj−1
k
k→∞
=0
= 0.
d
Since for each fixed j we have that ηψ∗ ∗ +j − ηψ∗ ∗ +j−1 = ηψ+j − ηψ+j−1 , by Proposition
5.2.6 we have that P-a.s.
Pk
lim
∗
j=ψk∗ +1 ηj
∗
− ηj−1
k
k→∞
Pk
= lim
j=ψk +1 ηj
k→∞
− ηj−1
k
hence indeed we have (5.9) and so rβ = rβ∗ P-a.s.
Theorem 5.2.9 is the main result of the report and gives an exact form of the speed in
the ballistic regime in terms of the bias of the walk and the first two moments of the offspring
distribution.
Theorem 5.2.9. If 1 < β < µ−1 < ∞ and σ 2 < ∞ then P-a.s.
rβ =
µ(β − 1)(1 − βµ)
.
µ(β + 1)(1 − βµ) + 2β(σ 2 + µ2 − µ)
Proof. By Lemmas 5.2.7 and 5.2.8 it suffices to calculate
E[|Xη∗∗ |]
1
E[η1∗ ]
.
E[|Xη∗1∗ |] = P (Y1∗ = ρ1 |Y0∗ = ρ) − P (Y1∗ = ρ−1 |Y0∗ = ρ)
β
1
−
β+1 β+1
β−1
.
=
β+1
=
(5.10)
Recall from (5.3) that N 0 is the number of excursions into a trap between movements
along the backbone.
E[η1∗ − 1] =
∞
X
T ∗−
P(ξ ∗ = k)E[N 0 |ξ ∗ = k]Eρ ρ [τρ+ ]
k=1
=
T ∗−
Eρ ρ [τρ+ ]
∞
X
P(ξ ∗ = k)E[N 0 |ξ ∗ = k].
(5.11)
k=1
The second term is the expected number of excursions into a trap. Using properties of geometric
51
random variables we have E[N 0 |ξ ∗ = k] = (k − 1)β/(1 + β) hence
∞
X
P(ξ ∗ = k)E [N 0 |ξ ∗ = k] =
k=1
∞
X
kpk (k − 1)β
µ
k=1
1+β
∞
=
X
β
pk k 2 − kpk
µ(1 + β)
k=1
β
(µ2 − µ).
=
µ(β + 1)
The first term in the product (5.11) is the expected time taken to return to the root of Tρ∗− .
←
−
Notice that conditional on ξ ∗ > 0 this is independent of the value of ξ ∗ . Letting T denote an
←
−
PT ←
−
ρ
f -GW tree with an artificial parent of the root and N 000 := n=1
χ{Xn =ρ} be the number of
←
−
excursions before hitting ρ by the branching property we have that
←
−
T ∗−
Eρ ρ [τρ+ ] = 2 + ETρ [τρ+ ]EρT [N 000 ]
∞
=2+
←
−
2 X
P(ξ = k)EρT [N 000 |ξ = k]
1 − βµ
k=1
=2+
=2+
So we have that
E[η1∗ ]
2
1 − βµ
∞
X
kpk β
k=1
2βµ
.
1 − βµ
β(µ2 − µ)
=
·
µ(β + 1)
2βµ
+ 2 + 1.
1 − βµ
Combining this with (5.10) it follows that
rβ =
β−1
β+1
β(µ2 −µ)
µ(β+1)
·
2βµ
1−βµ
+2 +1
which simplifies to the given form.
For a fixed offspring distribution with 1 < β < µ−1 < ∞ and σ 2 < ∞ the speed is
continuous and monotonic either side of its maximum. In particular, it is unimodal with the
maximum speed obtained at
β
Figure 5.1 shows a plot of
opt
p
µσ 2 (σ 2 − µ(1 − µ)) − µ2
=
.
µ(σ 2 − µ)
µ(β−1)(1−βµ)
µ(β+1)(1−βµ)+2β(σ 2 +µ2 −µ)
52
(for specific values µ = 0.5 and σ 2 = 1 on
the region β ∈ [−0.3, 6.5]). This function has asymptotes at
p
2σ 2 − µ(1 − µ) ± (2σ 2 − µ(1 − µ))
β =
2µ2
±
and takes the value 0 at β ∈ {1, µ−1 } which is what we would expect since these are the
thresholds of the ballistic region. This is a rational function hence is smooth away from its
asymptotes. Furthermore, the one-sided derivatives of rβ at the edges of the ballistic region are
lim
β→1+
lim
β→(µ−1 )−
drβ
µ(1 − µ)
=
,
dβ
2σ 2
drβ
µ2 (1 − µ)
=
.
dβ
2(µ(1 − µ) − σ 2 )
These are both positive showing that the speed is not differentiable at the thresholds. Figure
5.2 shows a magnified version of the ballistic region from the plot in Figure 5.1.
Figure 5.1: Speed diagram.
Figure 5.2: Magnified speed diagram.
53
5.3
Escape in the sub-ballistic regime
In Chapter 4 we saw that in the sub-ballistic regime for the walk on the supercritical tree
conditioned to survive, when E[ξ 2 ] < ∞ there is an exponent γ such that |Xn |n−γ is tight and
the sequence log(|Xn |)/ log(n) converges P-a.s. to γ. In this section we study the argument
behind this in more detail and investigate the extension of this result to the subcritical GW
tree conditioned to survive in order to formulate a final conjecture.
Recall that ∆n := inf{i ≥ 0 : |Xi | = n} is the first hitting time of the nth level and γ :=
− log(f 0 (q))/ log(β) is our exponent where for µ < 1 we have q = 1 hence γ = − log(µ)/ log(β).
Conjecture 5.3.1. For 1 < µ−1 < β < ∞, when E[ξ 2 ] < ∞ the laws of ∆n n−1/γ and |Xn |n−γ
are tight with respect to P. Furthermore, we have that P-a.s.
log |Xn |
= γ.
n→∞ log n
lim
The proof of Theorem 4.1.3 (by Ben Arous, Fribergh, Gantert & Hammond [2]) can
be broken down in to a sequence of stages which investigate different aspects of the walk
and the tree. This is ideal for extending the result onto the subcritical tree because many of
these behavioural properties will be very similar for the walk on the subcritical tree due to
the similarity of the traps. In particular, the only major differences between the walk on the
subcritical tree and an associated supercritical tree are the time taken to traverse the backbone
and the number of excursions into traps therefore it will become clear that most of these
steps should transfer very straightforwardly to the subcritical tree and hence we should expect
Conjecture 5.3.1 to hold.
We fix ε > 0 and write
log(n)
hn := −(1 − ε)
log(f 0 (q))
which will be the critical height of a trap. We call a trap Tρ∗i,i an hn -trap if H(Tρ∗i,i ) ≥ hn .
j
j
Roughly, the proof can be broken down into the following steps.
1. Show that time up to ∆n is essentially spent in hn -traps.
We have that, asymptotically, the probability that the height of a trap is at least
hn is αf 0 (q)hn for some known constant α. Letting χ(n) denote the time spent in hn -traps
before ∆n it can be shown that for all t > 0 we have that P(|∆n − χ(n)|n−1/γ ≥ t) → 0
as n → ∞. This is done by showing that the walk Yk on the backbone spends less than
C1 n time before reaching level n, sees less than C2 n traps and enters any trap at most
C3 log(n) times with high probability. Here we say that Yk sees a trap Tρ∗i,i by time n
j
if Yk = ρi for some k ≤ n. It can then be shown that on this event the amount of time
spent out of hn -traps doesn’t exceed tn1/γ with high probability because the expected
time spent in a ‘small’ trap on a single visit is less than Cn(1−ε)(−1+1/γ) .
54
For the walk on the subcritical tree it is easy to see that Yn will take less than C1 n
time to reach level n by comparing with a biased random walk on Z. Combining this with
the fact that the size-biased distribution has finite mean when E[ξ 2 ] < ∞ by the laws of
large numbers it follows that Yn sees fewer than C2 n traps with high probability. The
number of visits to a trap is dominated by a geometric random variable and consequently
on the event that there are fewer than C2 n traps seen it follows that none of them are
entered at more then C3 log(n) times with high probability.
2. Show that hn -traps are far enough away from each other that any correlation is negligible.
In order to show the asymptotic independence of the hn -traps we use regeneration
times. The probability that there is an hn -trap before θ1 converges to 0 as n → ∞. To
see this one shows that the number of backbone vertices visited between regeneration
epochs is less that nε , the number of buds seen before the first regeneration is less than
n2ε and finally that there are no hn -traps before the first regeneration. A similar structure
can then be used to show that there is at most one branch between any two consecutive
regeneration points which contains an hn -trap.
It is easy to see that this extends to the subcritical tree because the number
of backbone vertices visited between regeneration epochs is of the same order due to
exponential moments of the times between regenerations. Then conditional on the number
of backbone vertices visited between regenerations, showing that the number of hn -traps
from these vertices only depends on a bound on the mean of the size-biased distribution.
3. Show that the number of hn -traps by level n is approximately Cnε .
It can be shown that the number of backbone vertices reached up to ∆n is approximately cn for some known constant c and hence that the number of backbone vertices
containing an hn -trap is approximately Cnf 0 (q)hn = Cnε . This follows because the probability that a given trap has height at least hn is asymptotically Cf 0 (q)hn and the height
of the traps are independent. The maximum number of hn -traps in a single branch up to
time ∆n is 1 by the previous part hence there are approximately Cnε hn -traps up to time
∆n .
By Theorem 3.3.1 the probability that a given trap is an hn -trap is asymptotically
Cµhn
on the subcritical tree and since there are precisely n backbone vertices reached by
∆n this result follows straightforwardly with
h∗n
log(n)
:= −(1 − ε)
log(µ)
on the subcritical tree.
4. Write ∆n as the sum of Cnε i.i.d. random variables.
From the previous parts we have seen that ∆n is essentially comprised of χ(n). This
can be broken down into individual visits to hn -traps. We know that there approximately
55
Cnε branches with hn -traps seen up to ∆n and that each such branch has only one hn -trap
with high probability. We therefore have that asymptotically
ε
χ(n) ≈
Cn
X
χi (n)
i=1
where χi (n) are i.i.d. copies of the time spent in the first hn -trap entered χ0 (n).
This only uses results from the previous steps and hence trivially extends to the
subcritical tree.
5. Decompose time in hn -traps into the number of excursions and show that the time spent
in a trap can be neglected if the excursion doesn’t reach a certain depth.
Each χi (n) approximately consists of a geometric number of excursions into an hn trap hence the problem largely reduces to evaluating the time spent on a single excursion
into an hn -trap. Let δi (n) denote the leftmost deepest point of the ith hn -trap. It can be
shown that the time spent in big traps is essentially spent at the bottom of the deep traps
i.e. that the time taken to reach δn from the root of the branch can be neglected. We let
χ∗i (n) denote the time spent in the ith hn -trap after reaching δi (n) for the first time on
the given excursion. This is simply χi (n) without the total time spent to reach δi (n) in
each excursion. It can then be shown that for any t > 0 we have that


Cnε
X
P 
χj (n) − χ∗j (n) n−1/γ ≥ t → 0
j=1
as n → ∞.
On the subcritical tree the number of excursions into a trap follows an approximate
geometric distribution similarly to the number of excursions into a trap on the supercritical
tree and the distribution of time spent in an hn -trap on a single excursion is the same for
the walk on the subcritical tree as it is for the walk on the supercritical tree since both
traps are subcritical GW trees.
6. Determine the time spent in an hn -trap.
We say a trap has been explored if the leftmost deepest point has been reached
then let Hn denote the maximum height of a trap that has been explored by time ∆n .
It can be shown that χ∗1 (n)/β Hn converges in distribution to a proper random variable Z
whose distribution is known. From this we see that the time spent during all excursions
into hn -traps clusters around powers of β. Moreover, Z can be written approximately as a
constant multiplied by the sum of a binomial number of i.i.d. exponential rate one random
variables. This represents the number of deep excursions to the hn -trap which follows an
approximate binomial distribution. More simply, it is shown that the time spent on a
deep excursion is asymptotically proportional to β Hn .
56
This only depends on the time spent in an hn -trap which is equal in distribution
between supercritical and subcritical trees as we have already noted hence this result
extends trivially to the subcritical tree. It is in this step we see the value of γ arise and
hence why it should be the same for the walk on the subcritical tree. Heuristically, because
for large n there are positive constants c1 , c2 such that
c1 µn ≤ P(Hn ≥ n) ≤ c2 µn ,
it follows that
log(n)
log(n)
c1 µ log(β) ≤ P(β Hn ≥ n) ≤ c2 µ log(β) .
Since ∆n is asymptotically proportional to β Hn it therefore follows that γ is the correct
scaling.
7. Use standard tightness results and properties of sums of i.i.d. random variables to show
the desired conclusion.
It can be shown that the scaled process ∆n n−1/γ converges in distribution to a
continuous infinitely divisible law using properties of sums of i.i.d heavy tailed random
variables. The tightness of ∆n n−1/γ then follows by showing
lim sup P ∆n n−1/γ ∈
/ [M −1 , M ] = 0
n
which holds because the infinitely divisible law is continuous and hence doesn’t have an
atom at 0. The tightness of |Xn |n−γ follows using a similar argument and the additional
fact that since times between regenerations (for the backbone process) have exponential
moments the process doesn’t backtrack too far. The convergence of log(|Xn |)/ log(n) then
follows by the previous two tightness statements, Fatou’s lemma and the previous steps
in the proof.
These final steps largely consist of proving results for sums of i.i.d. random variables which have been shown by [2] and hence extend to the subcritical tree.
The limiting behaviour of the transient random walk in the sub-ballistic regime is still
unknown on both the supercritical and subcritical tree when E[ξ 2 ] = ∞. The main problem
that arises when E[ξ 2 ] = ∞ concerns the number of traps seen since the number of buds from a
backbone vertex has infinite mean. Fundamental to the proof by Ben Arous, Fribergh, Gantert
& Hammond [2] is that the traps of critical height are sufficiently far apart. This cannot be
obtained using hn as a critical depth on the subcritical tree hence this critical threshold would
have to increase to follow this style of proof. The problem with this is that we would still expect
the walk to spend the majority of time in hn -traps due to the geometric nature of entering traps.
Recall that Croydon, Fribergh & Kumagai [6] have shown the convergence of log(∆n )/n when ξ
is in the domain of attraction of an α-stable law for the walk on the critical GW tree conditioned
to survive. The critical tree also has buds distributed according to the size-biased distribution
57
and this has infinite mean when ξ belongs to the domain of attraction of an α-stable law for
α ∈ (1, 2). We would, therefore, expect the behaviour to be very similar albeit with smaller
traps. On the critical tree, the critical depth of a sub-tree is n log(n)−1 . This is significantly
larger than hn for the supercritical tree because the traps are formed by critical GW trees hence
won’t be the correct critical depth. Moreover, the proof of the convergence of log(∆n )/n in [6]
also relies on showing that the critical traps are sufficiently far apart hence the major obstacle
of proving the convergence without this for the subcritical tree still remains.
58
Chapter 6
Conclusion
In disordered media the speed of a biased random walk is not necessarily monotonic in the bias.
In fact, strong biases can cause a sub-ballistic regime where a smaller bias may yield a ballistic
regime. The reason for this is that traps form in the environment which slow down the walk
and this slowing down effect is more extreme when the bias is large because, in order to escape
the trap, the walk must move against the bias. In the case that the random environments are
GW trees conditioned to survive these traps are formed from collections of subcritical or critical
GW trees.
In Chapter 3 we considered how a GW tree can be conditioned to survive; in particular,
we have seen that it makes sense to condition critical and subcritical GW trees to survive and
that they have a very interesting structure. Unlike the supercritical GW trees, the subcritical
and critical GW trees conditioned to survive have a single infinite line of descent. It is this backbone structure that determines whether the walk will be transient or recurrent; more specifically
it is the branching number of this backbone that determines this property. For the critical and
subcritical tree this backbone is isomorphic to N hence the walk is transient when the bias is
directed away from the root. The backbone vertices of the supercritical tree on average have
µ descendants on the backbone hence, since it suffices that the expected drift is directed away
from the root, the walk is transient whenever β > µ−1 .
It is the trapping phenomena which determine the upper bound for the bias in the
ballistic regime. Both the supercritical and subcritical tree have traps as independent GW trees
with offspring distribution with mean f 0 (q) (which is µ in the subcritical case). The expected
time spent in such a trap on a single excursion only depends on the bias and the mean of the
offspring distribution hence the two models have very similar trapping behaviour. The main
difference is the number of traps which in the supercritical case has a complicated distribution
depending on the number of offspring onto backbone vertices, whereas in the subcritical case
the number of such traps follows a size biased-distribution P(ξ ∗ = k) = kpk µ−1 . A main
distinction between these two distributions is that the expected number of offspring from a
backbone vertex on the supercritical tree can be bounded above by µ(1 − q)−1 (3.9) whereas
the expected number of vertices from a backbone vertex in the subcritical tree is µ2 µ−1 (3.12).
This means that, unless σ 2 < ∞, the expected number of traps in a single branch is infinite.
59
In terms of the walk this means that when σ 2 = ∞, the walk on the subcritical tree expects to
spend an infinite amount of time in a single shrubbery and hence the walk will be sub-ballistic
whereas this isn’t necessarily true on the supercritical tree.
In Chapter 5 we extended the known results from the supercritical tree onto the subcritical tree to distinguish the cases in which the biased random walk is ballistic and sub-ballistic.
The walk is ballistic if and only if 1 < β < µ−1 < ∞ and σ 2 < ∞ in which case
rβ =
µ(β − 1)(1 − βµ)
.
µ(β + 1)(1 − βµ) + 2β(σ 2 + µ2 − µ)
It is noteworthy that an exact form of the speed is unknown for the supercritical tree however
Aı̈dékon [1] has shown a form in terms of the annealed expectation.
Still open is the question as to whether there is an appropriate transformation of |Xn |
which yields non-trivial convergence in the case that the walk on the subcritical GW tree
conditioned to survive is transient and sub-ballistic. Heuristics in Section 5.3 suggest that when
σ 2 < ∞ we have that the correct scaling is |Xn |n−γ where γ = − log(µ)/ log(β) however we
expect that there will be a lattice effect which prevents this from converging. Furthermore,
determining the escape regime when the offspring distribution has infinite variance and the
walk is transient and sub-ballistic remains open for the supercritical and subcritical tree.
A final interesting question is whether there is a central limit theorem for the process.
More specifically, is there some scaling sβ such that the process
|Xbtnc | − rβ n
√
sβ n
t≥0
converges in law to a standard Brownian motion (or the absolute value of a Brownian motion
when rβ = 0). For β < 1, σ 2 < ∞ the walk is positive recurrent and hence the result won’t hold
however the other cases are much less obvious since the walk goes on excursions of arbitrary
length. Peres & Zeitouni [17] have shown various results to this theme for the walk on the
supercritical GW tree with p0 = 0. This condition removes the trapping phenomena however it
is believed that similar results hold without this restraint. When the walk is positive recurrent
we have that the process X2n converges in distribution to a stationary distribution depending on
the tree. Another interesting problem is to determine the rate of convergence to the stationary
distribution.
60
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