Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs

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www.elsevier.com/locate/anihpb

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs

^✩

Sara Brofferio

¹

, Wolfgang Woess

^∗

Institut für Mathematik C, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria Received 5 April 2004; accepted 22 December 2004

Abstract

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graphDL(q, r), whereq, r2. The latter is the horocyclic product of two homogeneous trees with respective degreesq+1 andr+1. Whenq=r, it is the Cayley graph of the wreath product (lamplighter group)ZqZwith respect to a natural set of generators. We describe the full Martin compactification of these random walks onDL-graphs and, in particular, lamplighter groups. This completes previous results of Woess, who has determined all minimal positive harmonic functions.

Résumé

On détermine le comportement asymptotique précis (dans l’espace) du noyau de Green de la marche aléatoire simple avec dérive sur le graphe de Diestel–LeaderDL(q, r), oùq, r2. Ce graphe est le produit horocyclique de deux arbres homogènes de degrésq+1 etr+1, respectivement. Quandq=r, il s’agit du graphe de Cayley du produit en couronne (« lamplighter group »)ZqZpar rapport à un ensemble naturel de générateurs. On décrit la compactification de Martin complète de ces marches aléatoires sur les graphesDL, et en particulier, les groupes du « lamplighter ». Ceci complète les résultats précédents de Woess, qui a déterminé les fonctions harmoniques minimales.

MSC: 60J50; 05C25; 20E22; 31C05; 60G50

Keywords: Lamplighter group; Wreath product; Diestel–Leader graph; Random walk; Martin boundary; Harmonic functions

✩ Supported by European Commission, Marie Curie Fellowship HPMF-CT-2002-02137 and partially by FWF (Austrian Science Fund) project P15577.

* Corresponding author.

E-mail address: woess@weyl.math.tu-graz.ac.at (W. Woess).

1 Current address: Laboratoire de Mathématiques, Université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France.

doi:10.1016/j.anihpb.2004.12.004

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1. Introduction

Consider the additive groupZof all integers as a two-way-infinite road where at each point there is a lamp that may be switched on in one ofqdifferent intensities (states)∈Zq= {0, . . . , q−1}, the group of integers moduloq. At the beginning, all lamps are in state 0 (switched off), and a lamplighter starts at some point ofZ. He chooses at random among the following actions (or a suitable combination thereof): he can move to a neighbour point inZ, or he can change the intensity of the lamp at the actual site to a different state. As the process evolves, we have to keep track of the positionk∈Zof the lamplighter and of the finitely supported configurationη:Z→Zq that describes the states of all lamps. The setZqZof all pairs(η, k)of this type carries the structure of a semi-direct product ofZwith the additive groupCof all configurations, on whichZacts by translations. This is often called the lamplighter group; the underlying algebraic construction is the wreath product of two groups.

Random walks on lamplighter groups have been a well-studied subject in recent years, see Kaimanovich and Vershik [18] and Kaimanovich [17] (Poisson boundary≡bounded harmonic functions), Lyons, Pemantle and Peres [20], Erschler [12], Revelle [24], Bertacchi [3] (rate of escape), Grigorchuk and ˙Zuk [13], Dicks and Schick [7], Bartholdi and Woess [2] (spectral theory), Saloff-Coste and Pittet [22,23], Revelle [25] (asymptotic behaviour of transition probabilities), and Woess [28] (positive harmonic functions).

Here, we shall deal with Green kernel asymptotics and positive harmonic functions. Let us briefly outline in general how this is linked with Martin boundary theory of Markov chains. Consider an arbitrary infinite (connected, locally finite) graph X (e.g., a Cayley graph of a finitely generated group) and the stochastic transition matrix P =(p(x, y))_x,y_∈_Xof a random walkZ_nonX. That is,Z_n is anX-valued random variable, the position of the random walker at timen, subject to the Markovian transition rule

Pr[Zn+1=y|Zn=x] =p(x, y).

Then-step transition probability

p⁽ⁿ⁾(x, y)=^Pr[Z_n=y|Z₀=x], x, y∈X,

is the(x, y)-entry of the matrix powerPⁿ, withP⁰=I, the identity matrix. The Green kernel is G(x, y)=^∞

n=0

p⁽ⁿ⁾(x, y), x, y∈X.

This is the expected number of visits in the pointy, when the random walk starts atx. We always consider random walks that are irreducible and transient, which amounts to

0< G(x, y) <∞ for allx, y∈X.

Renewal theory in a wide sense consists in describing the asymptotic behaviour in space ofG(x, y), whenx is fixed andytends to infinity (or dually,y is fixed andxtends to infinity). If we fix a reference pointo∈X, then the Martin kernel is

K(x, y)=G(x, y)/G(o, y), x, y∈X.

If we have precise asymptotic estimates in space of the Green kernel, then we can also determine the Martin compactification. This is the smallest metrizable compactification ofXcontainingXas a discrete, dense subset, and to which all functions K(x,·),x ∈X, extend continuously. The Martin boundaryM=M(P )is the ideal boundary added toX in this compactification. Thus,Mconsists of the “directions of convergence” ofK(x, y), wheny→ ∞. Its significance is that it leads to a complete understanding of the coneH⁺=H⁺(P )of positive harmonic functions. A functionh:X→Ris called harmonic, orP-harmonic, if

h=P h, whereP h(x)=

y

p(x, y)h(y).

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A functionh∈H⁺is called minimal if

h(o)=1 and hh₁∈H⁺ ⇒ h₁/ h≡constant.

The minimal harmonic functions are the extreme points of the convex base B= {h∈H⁺: h(o)=1}of the coneH⁺.

The reader is referred to the excellent introduction to Martin boundary theory by Dynkin [10], based on the pioneering work by Doob [9] and Hunt [16]. A main result of this theory is that

• Every minimal harmonic function is of the form K(·, ξ ), where ξ ∈M, and the set Mmin= {ξ ∈M: K(·, ξ )is minimal}is a Borel subset ofM;

• For everyh∈H⁺there is a unique Borel measureν^honMsuch that ν^h(M\Mmin)=0 and h(·)=

M

K(·, ξ )dν^h(ξ ).

The above is an abstract construction of the Martin compactification. The kind of approach that we have in mind here is the following. The transition matrixP is adapted to the graph structure, and we want to understand and describe the Martin compactification in terms of the specific geometry ofX. Results of this type for random walks on various classes of graphs and groups, along with many references, are presented in the book by Woess [27].

Returning to lamplighter walks, this spirit requires as the starting point a good understanding of the geometry of the wreath productZqZin terms of a suitable Cayley graph of that group. This is the Diestel–Leader graph DL(q, q), a special case of the Diestel–Leader graphsDL(q, r)(q, r 2), which were introduced in [8] and are explained in detail in Section 2. Briefly speaking,DL(q, r)is the horocyclic product of the homogeneous trees TqandTr with degreesq+1 andr+1, respectively. It is precisely this geometric realization of the lamplighter groups in terms of relatively simple objects such as trees, that allows us to perform many computations.

The random walk with transition matrixP_αonDL(q, r)that we consider is the simple random walk (SRW) with an additional drift parameterα∈(0,1). Ifr=q(the case of the lamplighter group), then this random walk can be interpreted in lamplighter terms as follows. Think of the lamps not placed at each vertex of the two-way-infinite pathZ, but at the middle of each edge. Suppose the actual position of the lamplighter isk∈Z. He first tosses a coin. If “head” comes up, which happens with probabilityα, he moves tok+1 and switches the lamp on the transversed edge to a state chosen at random inZq. Otherwise, he moves tok−1 and also switches the lamp on the transversed edge to a random state.

Even whenq =r, the random walkP_α onDL(q, r)may be interpreted as a lamplighter walk in an extended sense. Imagine that on each edge ofZ, there is a green lamp withq possible intensities (including “off”) inZq

plus a red lamp withrpossible intensities (including “off”) inZr. The rule is that only finitely many lamps may be switched on, and in addition, if the lamplighter stands atk, then all red lamps betweenkand−∞have to be switched off, while all green lamps betweenkand+∞must be switched off. The lamplighter tosses hisα-coin.

If “head” comes up, he moves fromktok+1 and switches the green lamp on the transversed edge to a random state, while switching off the red lamp on that edge. Otherwise, he moves tok−1 and switches the red lamp on the transversed edge to a random state, while switching off the green lamp sitting there.

Then the random walkP =P_α (whose definition is formalised in (2.3)) is irreducible and transient. Via our geometric interpretation, we see that it has natural projectionsP₁=P_α,qandP₂=P₁₋_α,ron the two trees used to make up the graph, and alsoP=P_αonZ, which describes just the moves of the lamplighter. A good understanding of these projected walks is crucial for our approach, and in Section 3, we quickly review the necessary facts concerning those random walks onTq(andTr).

In Section 4, we derive our main results concerning the asymptotic behaviour of the Green kernel associated withP_α, subsumed in Theorem 4.2. The asymptotics are different along different directions of moving to infinity.

Also, the drift-free case (α=1/2) is substantially different from the other cases (α =1/2).

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These results are used in Section 5 to describe the full Martin compactification. In the drift-free case, this is the “natural” geometric compactification in terms of the two underlying trees. Namely,DL(q, r)is a subgraph of Tq×Tr, and the Martin compactification is the closure ofDL(q, r)inTq×Tr, whereTq andTr are the well- known end-compactifications of the respective trees. However, whenα =1/2, the Martin compactification is larger than (≡surjects non-trivially onto) the geometric one. The minimal Martin boundary, previously described in [28]

without elaborating the directions of convergence, is recovered.

These results can also be adapted to obtain the Martin compactification for all positivet-harmonic functions, that satisfy P h=t ·h, where t ρ(P_α)=lim sup_np⁽ⁿ⁾(x, x)^1/n, the “bottom of the positive spectrum”. The picture at the bottom is that of the drift-free case, while the caset > ρ(P_α)corresponds to non-vanishing drift. See Section 6.

In the short Section 7, we present another little by-product of our Green kernel estimates, namely, we illustrate their use for showing directly that SRW onDL(q, q)(the lamplighter group) does not satisfy the elliptic Harnack inequality.

In conclusion, let us remark that in general it is significantly harder to determine the whole Martin compactification than to determine the minimal harmonic functions associated with a random walk, since the former contains more detailed analytic-geometric informations than the latter, whose computation often has rather an algebraic than an analytic flavour. Let us also remark that our results provide the first case where one can successfully determine the whole Martin compactification of a class of random walks on finitely generated groups that are solvable, but do not have polynomial growth.

2. The geometry of Diestel–Leader graphs and lamplighter groups

We now explain the structure of the DL-graphs and their relation with the wreath productsZqZ. This section is a short and slightly modified version of §2 in [28], included here for the sake of completeness.

LetT=Tq be the homogeneous tree with degreeq+1,q 2. A geodesic path, respectively geodesic ray, respectively infinite geodesic inTis a finite, respectively one-sided infinite, respectively doubly infinite sequence (x_n)of vertices ofTsuch thatd(x_i, x_j)= |i−j|for alli, j, whered(·,·)denotes the graph distance.

Two rays are equivalent if their symmetric difference is finite. An end ofTis an equivalence class of rays. The space of ends is denoted∂T, and we writeT=T∪∂T. For allw, z∈T,w =z, there is a unique geodesicwzthat connects the two. In particular, ifx∈Tandξ∈∂Tthenxξis the ray that starts atxand representsξ.

Forx, y∈T,x =y, we define the coneT(x, y)= {w∈T: y∈xw}. The collection of all cones is a sub-basis of a topology which makesTa compact, totally disconnected Hausdorff space withTas a dense, discrete subset.

We fix a root vertexo∈T. Ifw, z∈T, then their confluentc=w∧zwith respect toois defined byow∩oz= oc. Similarly, we choose and fix a reference endω∈∂T. Forz, v∈T\ {ω}, their confluentb=vzwith respect toωis defined byvω∩zω=bω. We write

zv ifzv=z.

Forx, y∈T, we describe their relative position by the two numbers u(x, y)=d(x, xy) and d(x, y)=d(y, xy).

Thus,d(x, y)=u(y, x). In Fig. 1,u(x, y)andd(x, y)correspond to the numbers of steps one has to take upwards (in direction ofω), respectively downwards, on the geodesic path fromxtoy. We haved(x, y)=u(x, y)+d(x, y).

The Busemann functionh:T→Zand the horocyclesHk with respect toωare h(x)=d(o, x)−u(o, x) and Hk=

x∈T: h(x)=k .

Every horocycle is infinite. We write H (x)=H_k if x∈H_k. Every vertex x in H_k has one neighbour x⁻ (its predecessor) inH_k₋₁andqneighbours (its successors) inH_k₊₁. Thusis the transitive closure of the predecessor relation. We set∂^∗T=∂T\ {ω}.

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Fig. 1.

We label each edge ofTby an element ofZqsuch that for each vertex, the “downward” edges to itsqsuccessors carry labels 0, . . . , q−1 from left to right (say), see Fig. 1. Thus, for eachx∈T, the sequence(σ (n))_n₀of labels on the geodesicxωhas finite support{n: σ (n) =0}. We write Σ_q for the set of all those sequences. On every horocycle, there is exactly one vertex corresponding to eachσ ∈Σq. Thus,Tq is in one-to-one correspondence with the setΣq×Z, and thek-th horocycle isHk=Σq× {k}.

Now consider two treesT¹=TqandT²=Tr with rootso₁ando₂and reference endsω₁andω₂, respectively.

2.1. Definition. The Diestel–Leader graphDL(q, r)is DL(q, r)=

x₁x₂∈Tq×Tr: h(x₁)+h(x₂)=0 , and neighbourhood is given by

x₁x₂∼y₁y₂ ⇐⇒ x₁∼y₁ and x₂∼y₂.

To visualiseDL(q, r), drawTq in horocyclic layers withω₁at the top and∂^∗Tq at the bottom, and right to it Tr in the same way, but upside down, with the respective horocyclesHk(Tq)andH₋k(Tr)aligned. Connect the two originso₁,o₂by an elastic spring. It is allowed to move along each of the two trees, may expand infinitely, but must always remain in horizontal position. The vertex set ofDL_q,r consists of all admissible positions of the spring. From a positionx₁x₂withh(x₁)+h(x₂)=0 the spring may move upwards to one of ther successors of x₂inTr, and at the same time to the predecessor ofx₁inTq, or it may move downwards in the analogous way.

Such a move corresponds to going to a neighbour ofx1x2. We see thatDL(q, r)is regular with degreeq+r. As the reference point inDL(q, r), we chooseo=o₁o₂. Fig. 2 illustratesDL(2,2).

The position ofy=y₁y₂with respect tox₁x₂∈^DL(q, r)is described by the four numbersu(x₁, y₁),d(x₁, y₁), u(x₂, y₂),d(x₂, y₂), see below in Section 4, (4.1) and Fig. 3. The random walks that we are going to deal with are all such that the transition probabilitiesp(x₁x₂, y₂y₂)depend only on those four parameters – a crucial prerequisite for our approach.

We now recall in more detail the construction of the lamplighter groupZqZ. The group of all finitely supported configurations is

C=

η:Z→Zq,supp(η)finite

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Fig. 2.

with point-wise addition moduloq. The groupZacts onCby translationsk→Tk:C→CwithTkη(m)=η(m−k).

The resulting semi-direct productZCis ZqZ=

(η, k): η∈C, k∈Z

with group operation (η, k)(η, k)=(η+Tkη, k+k).

We identify each (η, k)∈ZqZ with the vertexx₁x₂∈^DL(q, q), where according to the identification Tq↔ Σq×Z, the verticesxi are given by

x₁=(η_k⁻, k) and x₂=(η⁺_k,−k), where

η⁻_k =η|(−∞,k] and η⁺_k =η|_[k+1,∞), (2.2)

both written as sequences over the non-positive integers.

This is a one-to-one correspondence betweenZqZandDL(q, q), and that group acts transitively and fixed- point-freely on the graph. Namely, the action of m∈Z is given byx₁x₂=(σ₁, k)(σ₂,−k)→y₁y₂=(σ₁, k+ m)(σ2,−k−m), and the action of the group of configurations is point-wise addition moduloq. Writeδ_k^l for the configuration in C with valuel atk and 0 elsewhere. ThenDL(q, q)is the (right) Cayley graph ofZqZwith respect to the symmetric set of generators

δ^l₁,1 ,

δ₀^l,−1 : l∈Zq

,

i.e., an edge corresponds to multiplying with a generator on the right. (This is precisely the set of generators considered in [13] and [7] when computing the spectrum of the associated SRW-operator.)

Returning toDL=^DL(q, r), the transition matrixP_α of the random walk that we have described in the Intro- duction is given as follows. Forx=x₁x₂,y=y₁y₂∈^DL(q, r)

p_α(x, y)=





α/q ify₁⁻=x₁andy₂=x₂⁻, (1−α)/r ify₁=x₁⁻andy₂⁻=x₂,

0 otherwise.

(2.3)

3. Simple random walk with drift on a homogeneous tree

In general, ifP is a transition matrix over a setXand{X_i: i∈I}is a partition ofXwith the associated quotient mapπ:X→I, then one says thatP factorises (or projects) with respect toπ, ifp(i, j )˜ :=

y∈Xjp(x, y)does not depend on the specific choice of x ∈X_i. In this case, the Green kernel G associated with P=π(P ) also satisfies

G(i, j ) =

y∈X_j

G(x, y), x∈X_i. (3.1)

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In our case, we have three natural, neighbourhood preserving projections π₁:DL→Tq, π₂:DL→Tr, and

˜

π:DL→Z, given by

π₁(x₁x₂)=x₁, π₂(x₁x₂)=x₂, and π (x˜ ₁x₂)=h(x₁).

P_α factorises with respect to each of them. Letπ₁(P_α)=P₁,π₂(P_α)=P₂andπ (P˜ _α)=P. ThenP₁=P_α,q on T¹=Tq,P₂=P₁₋_α,ronT²=Tr, andP=P_α onZ, where

p_α,q(x₁, y₁)=





α/q ify⁻₁ =x₁, 1−α ify₁=x₁⁻, 0 otherwise,

˜

p_α(k, l)=

α ifl=k+1, 1−α ifl=k−1, 0 otherwise.

(3.2) The projected random walks are well understood. Everybody is familiar with the gambler’s processP_α onZ. We outline the basic features ofP_α,q onTq(or, equivalently,P₁₋_α,ronTr).

Spectral radius. The spectral radius of any irreducible transition matrix is defined as ρ(P )=lim sup

n

p⁽ⁿ⁾(x, y)^1/n.

It is independent ofx, y. In the specific case of our random walks with drift parameterα, we have ρ(Pα)DL=ρ(Pα,q)_T_q =ρ(P1−α,r)_T_r =ρ(Pα)_Z=2

α(1−α). (3.3)

(The subscript refers to the respective underlying graph.) ForP_α onZ, this is well known. ForP_α,q onTq, it can be easily computed in various ways. See e.g. Saloff-Coste and Woess [26], Example 1.

Green kernel. The – simple – computations of the Green kernelG1=Gα,q associated with Pα,q can be done following the method of §1.D in [27], see also [28], (3.9). The main point is that we have a nearest neighbour random walk on a tree (transition probabilities are positive only between neighbours). Thus, ifF₁(x₁, y₁)is the probability that the random walk starting atx₁ever hitsy₁(x₁, y₁∈Tq), then

F₁(x₁, y₁)=F₁(x₁, w₁)F₁(w₁, y₁) for allw₁∈x₁y₁. (3.4) Furthermore, since p₁(x₁, y₁) depends only on u(x₁, y₁) and d(x₁, y₁), the same is true for F₁(x₁, y₁) and G₁(x₁, y₁). In particular,

F₁⁻=F₁(x₁, x⁻₁) and F₁⁺=F₁(x₁⁻, x₁)

are independent ofx₁∈Tqas well asG₁(x₁, x₁). Using these facts, and settingα⁺=max{α,1−α}, one computes G₁(x₁, y₁)=F₁(x₁, y₁)G₁(y₁, y₁)=

F₁⁻u(x₁,y₁)

F₁⁺d(x₁,y₁) q (q+1)α⁺−1, where

F₁⁻=







 1−α

α ifα1 2,

1 ifα1

2,

F₁⁺=







 1

q ifα1

2, α

(1−α)q ifα1 2.

(3.5)

Martin compactification. By (3.4), the Martin kernelK₁=K_α,qassociated withP_α,q satisfies K₁(x₁, y₁)=F₁(x₁, y₁)

F₁(o₁, y₁)=F₁(x₁, c₁)

F₁(o₁, c₁), wherec₁=x₁∧y₁

(the confluent with respect too₁). From here, the following is almost immediate.

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3.6. Proposition. The Martin compactification of Tqwith respect toP_α,q is the end compactificationTq, and for ξ₁∈∂Tq, the Martin kernel is given byK₁(x₁, ξ₁)=K₁(x₁, c₁), wherec₁=x₁∧ξ₁.

Furthermore, each functionK₁(·, ξ₁),ξ₁∈∂Tq, is minimal harmonic forP_α,q.

For general transient nearest neighbour random walks on arbitrary locally finite trees, this is due to by Cartier [4], and in the specific case of free groups (which is close to, but not identical with our situation), it was shown previously by Dynkin and Malyutov [11].

The analogous results forP₁₋_α,ronTr are obtained from the above by exchangingαwith 1−αandqwithr.

Whenα =1/2, the same computations are also valid forP_α onZ, settingq=1. When α=1/2 then P_α is of course recurrent, i.e., the associated Green kernel diverges.

Below in Section 5, we shall also need the following functions onTq×Tq, which we call (generalised) spherical functions. We setu=u(x₁, y₁),d=d(x₁, y₁)andβ=(1−α)/α.

φ_α,q(x₁, y₁)=









 1 q^d

q+1

q−1 +d(x₁, y₁)

, ifα=1

2, 1

(qβ²)^d

qβ+1

qβ²−1+β^u−1

β−1 +β^d−1 β−1

ifα <1 2, β^d(x¹^,y¹⁾

q^d

qβ⁻¹+1

qβ⁻²−1+β⁻^u−1

β⁻¹−1 +β⁻^d−1 β⁻¹−1

ifα >1 2.

(3.7)

(Recall thatd(x1, y1)=u+d.) Thenφα,q(·, y1)isPα,q-harmonic onTqfor eachy1∈Tq.

4. Green kernel asymptotics

We now embark on the main computational part of this paper. We considerPα onDL=^DL(q, r), and we shall always assume thatα1/2, since all results in the caseα1/2 are obtained from the former by exchanging the role of the two trees (i.e., exchangingrwithq).

We want to derive asymptotic estimates of the associated Green kernelG(x, y)=G_α(x, y), wherex=x₁x₂and y=y₁y₂∈DLand the graph distanced(x, y)→ ∞. The latter means that at least one ofd(x₁, y₁)andd(x₂, y₂) (distances in the respective trees) tends to∞. We remark here that

d(x, y)=d(x₁, y₁)+d(x₂, y₂)−h(y₁)−h(x₁),

d(x_i, y_i)=u_i+d_i, h(y_i)−h(x_i)=d_i−u_i, (i=1,2), and u₁+u₂=d₁+d₂, whereui=u(xi, yi)anddi=d(xi, yi).

(4.1)

(Cf. Bertacchi [3] for the distance formula.) In terms of the lamplighter moving alongZ(with the lamps – possibly red and green – sitting on the edges, as described in the Introduction), u1 is the minimal number of steps the lamplighter has to walk in the negative direction in order to obtain the new position and configuration encoded in the vertexy=y1y2ofDL, andu2is analogous in the positive direction.

We setci=xi yi. See Fig. 3. We also chooseai, bi∈Tⁱ withxi ai,yi bi, such thath(a₁)=h(b₁)=

−h(c₂)andh(a₂)=h(b₂)= −h(c₁), i.e., the pairsa₁c₂,b₁c₂,c₁a₂,c₁b₂belong toDL. In particular,d(x, y)→ ∞ means thats→ ∞, where

s=s(x, y)=u1+u2=d1+d2= −h(c₁)−h(c₂) is the span ofxandy.

The following is the first main result of this paper.

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Fig. 3.

4.2. Theorem. Referring to (4.1) and Fig. 3, suppose thatd(x, y)→ ∞, and hences=s(x, y)→ ∞. Then we have the following.

(a) Ifα <1/2 andβ=(1−α)/αthen G(x, y)∼ A_β

(qβ)^d¹r^d²

B_ββ^s−β^u¹ β^s−1

β^s−β^d¹ β^s−1

1

β^s +β^s−β^u¹ β^s−1

β^d¹−1 β^s−1 +β^u¹−1

β^s−1

β^s−β^d¹

β^s−1 +B_β^∗β^u¹−1 β^s−1

β^d¹−1 β^s−1

, where

Aβ=G₁(o₁, o₁)G₂(o₂, o₂)

G(0, 0) = qr(β²−1) (qβ−1)(qr−1), B_β=(β−1)(qβ+1)

qβ²−1 , and B_β^∗=(β−1)(rβ+1) rβ²−1 . (b) Ifα=1/2 then

G(x, y)∼ A₁ s⁴q^d¹r^d²

q+1

q−1u2d2+su2d1+su1d2+r+1 r−1u1d1

, where

A₁=G1(o1, o1)G2(o2, o2)

2 = 2qr

(q−1)(r−1).

According to the way howy tends to infinity geometrically (when we think ofx being fixed), one or more of the four terms will dominate the others, as we shall see below.

As mentioned at the beginning, the caseα >1/2 is obtained by exchangingr↔qandα↔1−α. Equivalently, we may use Lemma 4.12 and apply statement (a) of Theorem 4.2 toG^∗(x, y), withβ^∗=1/βin the place ofβ.

We now start our (laborious) way towards the proof of Theorem 4.2. The following is obvious, but crucial.

4.3. Lemma. The Green kernelG(x, y)depends only onu₁,d₁,u₂,d₂.

Let Z_n be the random position of theP_α-walk. This is a DL-valued random variable defined on a suitable probability space (trajectory space). We writePr_x=^Pr[· |Z₀=x]andE_xfor the associated expectation. Also, 1_[···]

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will denote the indicator function of an event[· · ·]in the trajectory space. The projectionZ_nⁱ =π_i(Z_n)represents the random position at timenof theP_i-walk onTⁱ,i=1,2, and the random variableZ_n= ˜π (Z_n)realizes then-th position of theP_α-walk onZ.

We shall use several stopping times. Ifx=x₁x₂∈^DL,x_i∈Tⁱ (i=1,2), resp.k∈Z, then we set t(x)=inf

n0: Z_n=x

, t_i(x_i)=inf

n0: Z_nⁱ =x_i

(i=1,2), and

˜

t(k)=inf

n0: Z_n=k .

Note that these random variables are all defined on the same trajectory space associated withP_α. 4.4. Lemma. Referring to the situation of Fig. 3, we have

t1(c1)= ˜t

h(x1)−u1

and t2(c2)= ˜t

h(x2)+u2

Pr_x-almost surely.

Furthermore, in order to reachy starting inx, bothZⁱnhave to pass throughci,i=1,2, i.e., max

t₁(c₁),t₂(c₂)

t(y) Pr_x-almost surely.

Proof. TheP_α-walk onDLas well as the projected random walks are of nearest neighbour type. Thus, starting inx, the first point in the set{v=v₁v₂∈^DL:h(v₁)=h(c₁)}visited byZnmust be of the formc₁v₂. This translates into t₁(c₁)= ˜t(h(x₁)−u₁), and exchanging the roles of the two trees, also into t₂(c₂)= ˜t(h(x₂)+u₂). The same type of argument shows the last statement. 2

The last lemma leads us to the identities Pr_x

t₁(c₁) <t₂(c₂)

=ϕ₁(u₁,u₂) and Pr_x

t₂(c₂) <t₁(c₁)

=ϕ₂(u₁,u₂), (4.5)

where for k, l0, the probability that thePα-walk on Zstarting in 0 reaches−k beforel isϕ₁(k, l), and the probability that it reaches l before−k is ϕ₂(k, l)=1−ϕ₁(k, l). It is a well-known exercise to compute these functions, and they are given by

ϕ₁(k, l)=β^k⁺^l−β^k

β^k⁺^l−1 and ϕ₂(k, l)= β^k−1

β^k⁺^l−1, withβ=1−α

α , ifα =1 2; ϕ₁(k, l)= l

k+l and ϕ₂(k, l)= k

k+l, ifα=1 2.

(4.6)

See e.g. Kemeny and Snell [19], §7.1, in particular (5) and (6) on p. 153. Next, let us introduce the function ψ (k)=

αr (1−α)q

k

, k∈Z. (4.7)

If we setm(x)=ψ (h(x₁)), wherex=x₁x₂∈^DL, then we havem(x)p_α(x, y)=^m(y)p_α(y, x)for allx, y∈^DL. That is,P_α ism-reversible, and we also get

G(x, y)=ψ

h(y₁)−h(x₁)

G(y, x) for allx=x₁x₂, y=y₁y₂∈^DL. (4.8) 4.9. Proposition. Referring to the situation of Fig. 3, we have the following decomposition.

G(x, y)=ϕ₁(u₁,u₂)ϕ₁(d₁,d₂)ψ (d₁)G(c₁b₂, c₁a₂) (I) +ϕ₁(u₁,u₂)ϕ₂(d₁,d₂)ψ (d₁)G(b₁c₂, c₁a₂) (II) +ϕ₂(u₁,u₂)ϕ₁(d₁,d₂)ψ (−d₂)G(c₁b₂, a₁c₂) (III) +ϕ₂(u₁,u₂)ϕ₂(d₁,d₂)ψ (−d₂)G(b₁c₂, a₁c₂). (IV)

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Fig. 4.

Proof. By (4.4) and (4.5), we have G(x, y)=^Ex

1_[_t₁_(c₁_)<t₂_(c₂₎_]G(Z_t₁_(c₁₎, y) +^Ex

1_[_t₂_(c₂_)<t₁_(c₁₎_]G(Z_t₂_(c₂₎, y) .

If t₁(c₁) < t₂(c₂) and Z_t₁(c₁) =c₁w₂, then we must have c₂ w₂ and furthermore w₂b₂=c₂. Thus, u(w2, y2)=u(a2, y2)andd(w2, y2)=d(a2, y2). Lemma 4.3 impliesG(Zt₁(c₁), y)=G(c1a2, y). In the same way, G(Z_t₂_(c₂₎, y)=G(a₁c₂, y). Thus,

G(x, y)=^Prx

t1(c₁) <t2(c₂)

G(c₁a₂, y)+^Prx

t2(c₂) <t1(c₁)

G(a₁c₂, y)

=ϕ₁(u₁,u₂)G(c₁a₂, y)+ϕ₂(u₁,u₂)G(a₁c₂, y).

Using (4.8), we getG(c₁a₂, y)=ψ (d₁)G(y, c₁a₂). Applying once more (4.4) and (4.5), G(y, c₁a₂)=^Ey

1_[_t₁_(c₁_)<t₂_(c₂₎_]G(Z_t₁_(c₁₎, y) +^Ey

1_[_t₂_(c₂_)<t₁_(c₁₎_]G(Z_t₂_(c₂₎, y) .

We can repeat the above argument withyin the place ofxandc₁a₂in the place ofy, and we have to replacea₁, a₂ withb₁, b₂. Therefore

G(y, c₁a₂)=ϕ₁(d₁,d₂)G(c₁b₂, c₁a₂)+ϕ₂(d₁,d₂)G(b₁c₂, c₁a₂).

Analogously,G(a₁c₂, y)=ψ (−d₂)G(y, a₁c₂)and

G(y, a₁c₂)=ϕ₁(d₁,d₂)G(c₁b₂, a₁c₂)+ϕ₂(d₁,d₂)G(b₁c₂, a₁c₂).

Combining these formulas, we obtain the proposed decomposition. 2

Thus, in order to understand the asymptotics ofG(x, y)in the general case of Fig. 3, we can reduce our computations to the following four basic cases of relative positions ofxandy.

In all four cases,s=s(x, y)→ ∞. In case (I),u₁=d₁=0 and u₂=d₂=s. In case (II),u₁=d₂=sand d₁=u₂=0. In case (III),d₁=u₂=sandu₁=d₂=0. In case (IV),u₁=d₁=sandu₂=d₂=0.

We start with an extended version of case II, see Fig. 5.

Fig. 5.

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4.10. Proposition. If, as in Fig. 5,x=x₁x₂andy=y₁y₂withs=s(x, y)are such thatu₁−d₁=d₂→ ∞,u₂=0 andd₁is arbitrary, then

G₁(x₁, y₁)=C(s)r^d²G(x, y)+R(d₁,d2), (4.11)

where

C(s)→ G(0, 0)

G₂(o₂, o₂) ifα =1

2, C(s)∼ 2s

G₂(o₂, o₂) ifα=1 2, and 0< R(d1,d2) < G1(x1, y1)with

lim

d₂→∞R(d₁,d₂)=0.

Proof. Applying (3.1) to the projectionπ1givesG1(x1, y1)=

w₂∈H (y₂)G(x, y1w2).

Letw₂∈H (y₂), where H (y₂)is the horocycle ofy₂inTr. We write v₂=v(w₂)for the unique element in H (x₂)that satisfies v₂w₂. By Lemma 4.4, the random walk has to pass through some point of the form in {u₁v₂: u₁∈H (x₁)}on the way fromxtoy₁w₂, that is,

G(x, y1w2)=^Ex

1_[t₂(v₂)<∞]G(Zt₂(v₂), y1w2)

=^Ex

1_[t₂(v₂)<t₁(c₁)]G(Z_t₂_(v₂₎, y₁w₂) +^Ex

1_[t₁(c₁)<t₂(v₂)<∞]G(Z_t₂_(v₂₎, y₁w₂) .

Now, if starting at x, we have t₂(v₂) <t₁(c₁), thenZ_t₂_(v₂₎=u₁v₂ for some randomu₁∈H (x₁)that must satisfy u(u₁, y₁)=u₁ andd(u₁, y₁)=d₁, sincec₁ cannot lie on x₁u₁. But we also have u(v₂, w₂)=u₂=0 and d(v2, w2)=d2. That is, the pointsu1v2 andy1w2 have the same relative position as the points x andy, and thereforeG(u₁v₂, y₁w₂)=G(x, y)by Lemma 4.3. We get

E_x

1_[_t₂_(v₂_)<t₁_(c₁₎_]G(Z_t₂_(v₂₎, y₁w₂)

=^Prx

t₂(v₂) <t₁(c₁)

G(x, y).

Now, givenv₂∈H (x₂), there are preciselyr^d²elementsw₂∈H (y₂)withv(w₂)=v₂. Combining all these obser- vations,

G₁(x₁, y₁)=

v₂∈H (x₂) Pr_x

t₂(v₂) <t₁(c₁)

r^d²G(x, y)+R(d₁,d₂), where R(d₁,d₂)=

w₂∈H (y₂) E_x

1_[_t₁_(c₁_)<t₂_(v(w₂_))<_∞]G(Z_t₂_(v(w₂₎₎, y₁w₂) .

Let us first consider the error termR(d₁,d₂). Note thatG(·,·)G(o, o) <∞, since our random walk is transient.

(Already the projections ontoTqandTr are transient!) Sinced₂→ ∞, also t₁(c₁)→ ∞almost surely. It follows that

r_d₁_,d₂(w₂):=^Ex

1_[_t₁_(c₁_)<t₂_(v(w₂_))<_∞]G(Z_t₂_(v(w₂₎₎, y₁w₂) ^Prx

t₁(c₁) <t₂ v(w₂)

<∞

G(o, o)→0 whend₂→ ∞. On the other hand,

rd₁,d₂(w₂)G(x, y₁w₂) and

w2∈H (y2)

G(x, y₁w₂)=G₁(x₁, y₁)G₁(o₁, o₁).

Thus, dominated convergence (in the summation) implies thatR(d₁,d2)→0 asd2→ ∞. It remains to show thatC(s)=

v₂∈H (x₂)Pr_x[t2(v₂) <t1(c₁)]has the proposed asymptotic behaviour, whend₂ (ands)→ ∞.

We may suppose without loss of generality thath(x₁)=h(x₂)=0, so thath(c₁)= −s. Then Lemma 4.4 implies Pr_x[t₂(v₂) <t₁(c₁)] =^Prx[t₂(v₂) <˜t(−s)]. Now let the superscript⁽⁻^s)refer to the random walkPonZstopped