TRANSIENT RANDOM WALKS ON GRAPHS AND METRIC SPACES WITH APPLICATIONS TO HYPERBOLIC ... · 2012-11-29 · TRANSIENT RANDOM WALKS ON GRAPHS AND METRIC SPACES WITH APPLICATIONS TO HYPERBOLIC

TRANSIENT RANDOM WALKS ON GRAPHS ANDMETRIC SPACES WITH APPLICATIONS TO

HYPERBOLIC SURFACES

STEEN MARKVORSEN, SEAN McGUINNESS,and CARSTEN THOMASSEN

[Received 1 October 1990]

ABSTRACT

We introduce an (r, /?)-net (0 < 2r < R) of a metric space M as a maximal graph whose vertices areelements in M of pairwise distance at least r such that any two vertices of distance at most R areadjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, theproperty of transience of a net and the property of the net carrying a non-constant harmonic functionof bounded energy is independent of the choice of the net. We give a new necessary and sufficientcondition for a graph with bounded degrees and satisfying an isoperimetric inequality to have nonon-constant harmonic functions. For this purpose we develop equivalent analytic conditions forgraphs satisfying an isoperimetric inequality. Some of these results have been discovered recently byothers in more general settings, but our treatment here is specific and self-contained. We use graphtransience to prove that Scherk's surface is hyperbolic, a problem posed by Osserman in 1965.

1. Introduction

Consider a random walk defined on a connected, infinite graph such that, at eachvertex v, the walk proceeds to a neighbour chosen with probability l/deg(u)where deg(t>) is the number of neighbours of v. The graph is transient if there is apositive probability that the walk never returns to the origin. There are severalequivalent versions of this definition. In this paper we focus on the network flowdefinition (described in the next section).

We say that a graph G satisfies an e-isoperimetric inequality (where £ is anon-negative real number) if there exists a positive real constant c such that, foreach finite vertex set V, there are at least c |V|5+e vertices of V which haveneighbours outside V. The 2-dimensional grid Z2 satisfies an e-isoperimetricinequality for e = 0 and it is recurrent (i.e. non-transient). Varopoulos [22]proved by analytic methods (and a more general result by network flow methodsis presented in [20] and a related result in the continuous case is proved in [7])that every graph with bounded degrees satisfying an e-isoperimetric inequality,for some e>0 , is transient. For e = \ this was proved by Dodziuk [3], andrecently Kaimanovich [8] has also observed this relation in general for Markovoperators on measure spaces. In this paper, we present an analytic result on lp

spaces defined on graphs satisfying an e-isoperimetric inequality. We derive fromthis the transience result of Varopoulos and apply the results in two directions.

Kanai [10] and others (Lyons and Sullivan [13], and Varopoulos [21,22])showed that there is a close relation between hyperbolicity of certain Riemannianmanifolds and transience of their associated nets (defined in the abstract). Weextend the concept of a net to metric spaces in general and show that for certainmetric spaces which we call slim and well-behaved (see the next section), each net

The second author is a postdoctoral fellow from Canada.1991 Mathematics Subject Classification: 05C75, 60J15, 51K10.Proc. London Math. Soc. (3) 64 (1992) 1-20.

2 S. MARKVORSEN, S. McGUINNESS AND C. THOMASSEN

is transient if and only if some net is transient. We prove that the nets of Scherk'ssurface are transient. Combined with Kanai's results, this shows that Scherk'ssurface is hyperbolic. This settles a problem of Osserman [17].

A harmonic function on a graph G is a real function <j) defined on the vertex setsuch that, for each vertex v, <p(v) equals the average of <f>(u), taken on allneighbours of v; that is, </> is harmonic at v for all v. The energy of <f> is the sumE (0(w) - 0(^))2 taken over all edges uv of G. A harmonic function on a graphis a discrete version of a harmonic function on a manifold. We prove that if G andG' are nets on a slim and well-behaved metric space, then G satisfies an£-isoperimetric inequality (e>0) if and only if G' does. If, in addition, G has anon-constant harmonic function of finite energy, then also G' has such a function.This is in contrast to a result of Lyons [14] which shows that such a result does nothold for 'bounded harmonic functions' instead of 'harmonic functions of finiteenergy'.

2. Definitions and terminology

A graph G is a pair V(G), E(G) where V(G) is a set of elements calledvertices and E(G) is a set of unordered pairs uv of vertices called edges. If theedge uv is present, we say that it joins u and v and that u and v are neighbours.The number degc(w) (or just deg(w)) of neighbours of u is called the degree of uin G. The set of neighbours is denoted N(u, G) or just N(u). We shall assumethat G is locally finite; that is, all degrees are finite. A finite path (from u, tovk+l) consists of vertices v1)v2,...,vk+1 and edges i>,u/+1 (1 *£/«=/:). Thenumber k is called the length of the path. A one-way infinite or two-way infinitepath is defined analogously. A graph is connected if any two vertices areconnected by a path. A (connected) component of a disconnected graph is amaximal connected subgraph. The distance dist(w, v, G) (or just dist(u, v)) is thelength of a shortest path from u to v. Clearly, distance defines a metric on V(G).

Let v be a vertex in G. A flow from v is an assignment of a direction and anon-negative real number to every edge such that, at each vertex distinct from v,the incoming flow equals the outgoing flow. The value of / i s the net flow leavingv. The square sum

taken over all edges uv is called the energy of the flow.More generally, if V is a real function on the edge set, then the energy of %p is

the square sum £ ip(e)2 taken over all edges of G and if 0 is a real function onV(G), then the energy of <f) is the energy of ij> where ip(u, v) = \<f>(u) — <j>(v)\ foreach edge uv.

The graph G is called transient from the vertex v, if G admits a flow from v ofpositive value and finite energy. If G is connected, this definition is easily seen tobe independent of v and we say that G is transient. A connected graph which isnot transient is recurrent.

A random walk on a graph is such that at each vertex the walk proceeds to aneighbour with probability l/deg(w). A random walk is called recurrent if thewalk returns to the origin with probability 1. It can be shown, using the theory ofelectrical networks (see [4]), that this probabilistic definition is equivalent to our

TRANSIENT RANDOM WALKS 3

combinatorial definition. One immediate consequence of our definition is that anysubgraph of a connected recurrent graph is recurrent.

If / is a flow of value / from a vertex v in a graph G and V is a finite vertex setin G, then an easy counting argument shows that the net flow out of V is / ifv eV, and zero otherwise.

A contraction of G is a graph G' obtained from G by partitioning V(G) intofinite sets Vx, V2,... and contracting each Vt into a single vertex vh Each multipleedge arising in this way is replaced by a single edge. If / is a flow from v e V, inG, then we obtain a flow / ' from vx in G' by letting the edge from t>, to uy carrythe sum of flows in edges from Vt to Vj (where the flow values from Vy to Vt arecounted negative). Suppose G is transient and/is a flow of finite energy on G. IfG' is a contraction such that, for some natural number m, |V |̂=Sra for each/ = 1, 2,.. . , then the flow / ' is easily seen to have finite energy and hence G' istransient.

If V is a set and p a positive real number, then lp(V) is the set of real functions(f> on V such that

It is well-known that lp(V) is a Banach space for p ss \t and that 12(V) is a Hilbertspace. The support supp 0 is the set of v in V such that 0(u) =£0. The set lo(V) isthe set of finitely supported functions on V.

If G is a graph and 0 is a real-valued function on V(G), then D<£ is thefunction on E{G) defined by

If 0 e 12(V(G)) and G has bounded vertex degrees, then clearly D<f> e 12(E(G)).If y is a vertex of G such that <f> is harmonic at every vertex (except possibly v)then D<p is a flow from v (where the direction of each edge is towards the vertexwith smallest $ value). The energy of that flow is ||Z>0||i. We denote by HD(G)the set of harmonic functions on V(G) such that

3. Transient nets in metric spaces

In this section we show how the concept of transience of graphs can beextended to a large class of metric spaces, including Riemannian manifolds. For alarge subclass of these, transience turns out to be equivalent with hyperbolicity,by results of Kanai [9,10].

Two metric spaces (M, dM) and (AT, dM) are called roughly isometric if thereexist a map </>: M^>M' and positive constants a, b, c such that

a~xdM{p, q)-b^dM.{<t>{p), <j)(q)) =sadM(p, q) + b

for any two p, q in M and such that each element of M' is within distance c fromsome element in <\>{M). Assuming the Axiom of Choice, we see that roughisometry is clearly an equivalence relation among the metric spaces.

We have previously noted that graphs may be regarded as metric spaces andthus we may speak of rough isometry of graphs. We shall give a new proof thattransience is invariant under rough isometry. For that we need the concept ofthe fcth power Gk of a graph G: V(Gk) = V(G) and two vertices in Gk are

4 S. MARKVORSEN, S. McGUlNNESS AND C. THOMASSEN

neighbours if and only if distc(jc, y) *£ k. We shall use the following result ofDoyle and Snell [4]. For the sake of completeness we include a simple proof.

LEMMA 3.1. Let G be a connected graph with maximum degree A. / / Gk istransient for some k, then G is transient.

Proof. It is sufficient to prove Lemma 3.1 for k = 2 (because {Gm)k = Gmk andhence Gk is contained in a Gq where q is a power of 2).

Thus we consider a flow / in G2 of value / ss 0 and of finite energy W(f) fromthe vertex v, say. We shall modify the flow / to a flow g in G. Consider any edgexy in G2 but not in G, directed from x to y. Then G has a path xzy. We delete theedge xy from G2 and add or subtract the flow in xy to (from) the flow in xz and inzy according as the directions on xz or zy agree or disagree with xy. We do thisfor each edge in G2 but not in G. As G has maximum degree A, there will be atmost 2(A - 1 ) flow values added to (subtracted from) any fixed edge. Thus thenew flow g is well-defined. For each vertex x in G2, let q(x) be the maximumvalue of / on the edges in G2 incident with x. Then the value of g on any edge uvis at most &[q(u) + q(v)]. Hence

xeV(G)

THEOREM 3.2. Let G and H be roughly isometric graphs of bounded degrees.Then G is transient if and only if H is transient.

Proof. Let a, b and <p: V(G)—>V(H) be as in the definition of roughisometry. Let k be a natural number such that kâ + b. If uw is an edge in G,then either 0(M) = 0(w) or 0(a)0(w) is an edge of Hk. Let G' be the contractionobtained from G by identifying any two vertices u, w which are mapped to thesame vertex of H by 0. For two such vertices u, w we have

distG(w, w) =£ ab.

If A denotes the maximum vertex degree of G, then fewer than A"6 + 1 verticesof G are mapped to the same vertex of H. Now, if G is transient, then G' istransient, by a remark in § 2. As G' is isomorphic to a subgraph of Hk, Hk is alsotransient. By Lemma 3.1, H is transient.

Consider a metric space (M, dM) and let r, R be real numbers satisfying0<2r<R. An (r, R)-net in M is a graph G defined as follows. Let V{G) be anymaximal set in M such that dM(x, y)>r for any distinct x, y in V(G). Such a setexists by Zorn's lemma. Let xy be an edge of G if dM(x, y)<R. We nowgeneralize Kanai's definition of an £-net in a Riemannian manifold. A net in M isan (r, R)-net for some r, R where 0 < 2r < R. The graph above depends stronglyon the particular choice of V(G) and r, R. However, for a large class of metricspaces the property of G being transient does not depend on the choice of V(G)and r, R.

We say that a metric space M is slim, if, for any two real numbers r, R suchthat 0 < 2r < R, there exists a natural number a(r, R) such that M does notcontain a set of a(r, R) elements each pair of which has distance between r and


R. The property of M being slim guarantees that all nets of M have boundeddegrees. We also say that the metric space M is well-behaved if for any two realsr, R with 0<r<R, there exists a natural number j3(r, R) such that, for any twoelements u, v with dM{u, v)^R there exist elements vo,vu..., v^r.R) s u c n thatu = v0, v = vPirtR) and dM(vt.u vs) < r for 0 < i ^ /3(r, R). We put j3(fl, r) = 1. IfG is an (r, /?)-net in a slim and well-behaved metric space (A/, dM), then for anytwo vertices u, v in G we clearly have

dM(u, v)^R distc(w, v).

If we put k - fi(R — 2r, dM(u, v)), then there exist elements p0, px, ..., pk in Msuch that dM(pi_i, pi)<R— 2r for i = l,2, ..., k. (Here po = u, pk = v.) Foreach / = 1, 2 , . . . , / c - 1 , G has a vertex u, such that dM(yhpi)^r. Thus^M(^ , , U / + I ) < / ? , and U/U.-+! is an edge. The path uu1u2 ••• ^ - i w shows that

distc(u, v) ^ P(R - 2r, dM(u, v)).

So, if for any fixed r, /5(r, R) is bounded above by a linear function of R, then Gand Af are roughly isometric. It is easy to give examples where this is not the case.However, we have the following

THEOREM 3.3. / / (M, dM) and (A/', dM) are roughly isometric slim andwell-behaved metric spaces and G and G' are nets of M and M', respectively, thenG and G' are roughly isometric. In particular, G is transient if and only if G' istransient.

Proof. Let G be an (r,/?)-net in M and G' an (r', /?')-net in M'. Let0: M-+M' be a rough isometry and let a, b, c be as in the definition of roughisometry.

Consider any vertex v of G. Then G' has a vertex v' such that

Select any such vertex v' and put ip(v) = v'. We claim that i/;: V(G)—> V(G') isa rough isometry.

If u' is any vertex in G', then there exists a w in M such that dM(u', 4>(w)) =s c.There exists ave V(G) such that dM(t>, w) =s r. Now

Hencedistc.(n', V(v)) ^ /3'(/?' - 2r', flr + b + c + r').

(Denote the right-hand side by y.) So ip(G) is 'full' in G'.If M and v are two neighbours in G, then

^ flrfM(«, v) + 6 + 2r' ^ a/? + b + 2r'.Hence

distc.(t//(u), V(^)) ^ /5'(i?' - 2r', aR + b + 2r').


It follows that, for any vertices u, v in G,

distc.(V("), V00) ^ P'(R' ~ 2r', aR + b + 2r')d\stG(u, v).

Now let u, v be any vertices of G. Let ty{u)u\u'2... u'm_x\p(v) be a shortest pathin G' between rp(u) and t/;(u). By the definition of y, there exists a w, in G suchthat distC'(uJ, V(Ui)) ̂ 7 for i = 1, 2, ..., m - 1. Then

for i = 1, 2, ..., m (where uo = u and wm = i>). Since

a-iduiu,.!, ut) -b^rf

we conclude that dM{Ui_x, «,), and hence also distc(w,_1, «,), is bounded aboveby a constant a' (depending only on r',R', y, r,R). So, there also exists aconstant b' such that

distG(w, v) =s a'm + b = a' distc-(i/;(w), %l>(v)) + b'.

The constant b' takes care of the case where ip(u) = T//(V). It follows that x\) is arough isometry.

A slim and well-behaved metric space is called net-transient if all its nets aretransient. Otherwise it is net-recurrent.

We say that a well-behaved metric space is very well-behaved if there exists apositive constant y such that fi(r, R)<yR/r for 0<r<R. By the remarkpreceding Theorem 3.3, a slim and very well-behaved metric space is roughlyisometric to all its nets. Important examples of very well-behaved metrics are thelength metrics defined below. They include the metrics on the Riemannianmanifolds.

An arc J from x to y in M is the image of a continuous function g: [0, l]-> Msuch that g(0)=x, g(l)=y. The length of / is the supremum of all numbersEfoi4K(g(0»S('/+i)) w h e r e 0 = fo<' i<-. .< '*+i = l. K (M,d) is metricallyconnected in the sense that any two elements of M are joined by an arc of finitelength, then the length metric dM(x, y)'\s the infimum of lengths of arcs between xand y. The metric dM is a very well-behaved metric.

Although it may be complicated to describe a net in a given slim and verywell-behaved metric space, it is often easy to describe a graph which is roughlyisometric to every net. For example, the rf-dimensional grid Zd is roughlyisometric to the real space Ud. However, Zd does not correspond to any net inUd. (In 1d two elements are neighbours if and only if they differ by 1 at onecoordinate and agree on all other coordinates.) It is well-known that Z2 isrecurrent and that Z3 is transient [4]. Therefore metric spaces 'between' U2 and1R3 are particularly interesting. Kanai [9] used the concept of rough isometry toreduce the problem of deciding if a given manifold is hyperbolic to determiningwhether a certain graph is transient. In order to describe this importantconnection we need some terminology. For those who are not familiar withRiemannian geometry we point out that every n-dimensional Riemannianmanifold (M, d) can be thought of as a subset of IR* (for k sufficiently large) suchthat, for every point x e M the following condition is satisfied: there are an openset UczUk containing x, and an open set V c:Uk, and a diffeomorphism


h: U-* Vsuch that

h(UDM) = Vn (Un x {0}) = {yeV\ y"+x = . . . = / = 0}.

(Thus U C\M can be 'straightened out' by a diffeomorphism.)Let rf' be the metric induced on M from the usual Euclidean metric in the

ambient space Uk. Then the metric d is the length metric d'. As previously noted,d is very well-behaved.

Now the Riemannian manifold M has bounded geometry if the Ricci curvatureis bounded from below and the injectivity radius i{M) for the exponential map ispositive. (For this and other notions of Riemannian geometry we refer to [2]).

For complete surfaces in U3, such as the helicoid and Scherk's surface (whichwe will consider in the following), the Ricci curvature is just the Gauss curvatureof the surface. Let p be a point on the surface and consider any smooth curve yon the surface through p. Let k(y) denote the curvature vector of y at p in IR3 andlet N be a unit vector orthogonal to the surface at p. The principal curvatures ofthe surface at p are defined by

A,(p) = min k(y) • N, A2(p) = max k(y) • N,y y

and the Gauss curvature k{p) at p is then

Surfaces which have A,(p) = —X2(p) everywhere are called minimal. (Thiscondition can be shown to be equivalent to the property of local area-minimization, but we shall not need that.) The helicoid and Scherk's surface areexamples of minimal surfaces.

For surfaces with non-positive Gauss curvature (such as the minimal surfaces)the injectivity radius is

Z(M) = iinf{length(y)},^ y

where the infimum is taken over all closed curves y on the surface which are nothomotopic to a point.

For minimal surfaces it is therefore not difficult to check for boundedgeometry. We note that bounded geometry always implies slimness. For this weneed the following well-known consequences of bounded geometry.

Let BR{p) = {x e M\ d(x, p)^R}. If (M,d) has bounded geometry, thenthere exist positive real functions V(R) and v(r) such that

(i) Vo\(BR(p)) ^ V(R) for all p e M, r > 0, and

(ii) Vol(flr(p)) ^ v(r) for all p e M, r < i(M).Now let /?i,..., pk be a set of k points in M such that d(pit p,) e [r, R] for all i =£/.Then pt e BR(px) for all / and

Vo\(BR+>r(Px)) > 2

and thus k < V(R + \r)lv{mm{\r, \i(M)}), which implies that M is slim.According to the Kelvin-Nevanlinna-Royden criterion, a complete Rieman-

nian manifold M is hyperbolic if and only if there is a (flow) vector field t// on M


with finite energy (JM \\ip\\2dV <<x>), and finite but non-zero net source or sink(Jwldiv V l ^ < ° ° and ]M div xp dV =£ 0). We refer to [13] for the history anddiscussion of this.

If we take this criterion as a definition of hyperbolicity, it becomes quitereasonable to expect a close relationship between hyperbolicity of (A/, d) andtransience of nets in (A/, d). Kanai proved the following in [10].

THEOREM 3.4. If a Riemannian manifold (M, d) has bounded geometry, thenthe inclusion map 0 of any (r, 3r)-net G into M is a rough isometry. Moreover, Mis hyperbolic if and only if G is transient.

REMARKS, (i) As in the case of graphs it can be shown that hyperbolicity ofcomplete Riemannian manifolds is actually equivalent to the probabilistic notionof transience of Brownian motion generated by the Laplace-Beltrami operatoron M.

(ii) The term 'hyperbolic' stems from the fact that a 2-dimensional simplyconnected Riemann surface is hyperbolic if and only if it is conformal to the unitdisc with the complete Poincare metric of constant curvature — 1.

(iii) In view of Theorem 3.3 we conjecture that if (M, d) is a slim completeRiemannian manifold, then (M, d) is hyperbolic if and only if (A/, d) admits atransient net, that is, if and only if (A/, d) is net-transient.

Theorem 3.4 can be used to translate results on manifolds to results on graphsand vice versa. We illustrate this by the helicoid which can be obtained by movinga horizontal line upwards and rotating it around a fixed axis (the z axis). Moreprecisely, we fix two positive real numbers y, (o and consider all points in U3 ofthe form (r cos(atf), r sin(a>t), yi) where r and t are independent real parameters.For each choice of y, co we thus obtain a surface which can be shown to beminimal (cf. [17]). It has infinite injectivity radius and the (Gauss-) curvature isbounded from below by -y~2. Thus every helicoid has bounded geometry and istherefore slim. We describe a graph which is roughly isometric to the helicoid.Without loss of generality, let us put y = 1 and co = 2JT. We first consider thesemi-helicoid; that is, the part of the helicoid where rÔ. We let H denote thegraph obtained from Z3 by deleting all edges parallel to the z-axis and also theedges from (/, 0, k) to (/, — 1, k) where i and k are integers and / >0. Then addall edges of the form (/, — 1, k), (i, 0, k + 1) where k and i are integers, / > 0. Weobtain a rough isometry from H to the semi-helicoid simply by mapping eachvertex v of H to the unique point of the semi-helicoid obtained by moving vupwards (that is, parallel to the z-axis) until we hit the semi-helicoid. The graphH is redrawn in Fig. 1.

A graph roughly isometric to the helicoid is then obtained by adding the mirrorimage of H (obtained by reflecting H in the bottom line). It is known that thehelicoid is parabolic [17]. Hence its nets are recurrent. We have a combinatorialproof of this non-trivial fact. It should be noted that it does not follow fromNash-William's criterion, [16], as H grows too fast. More precisely, the number ofvertices of distance n from a fixed vertex in H grows like n2. In particular, thehelicoid is not roughly isometric to U2. As far as growth rate of distance classes isconcerned, the (recurrent) helicoid behaves like the (transient) U2.

TRANSIENT RANDOM WALKS

FIG. 1. A graph roughly isometric to the semi-helicoid.

In the next section we apply Theorem 3.4 in the opposite direction. We answera question of Osserman [17] by showing that Scherk's surface is hyperbolic.

4. Hyperbolicity of Scherk's surface

Scherk's surface 5 consists of all points {x, y, z) in U3 such that eithercosxcos>'>0 and z — log(cos_y/cosx), or cos x = cos y = 0. The surface S isdrawn in Fig. 2.

FIG. 2. Scherk's surface.

Scherk's surface is also a complete minimal surface and it has Gauss curvaturebounded from below by - 1 . Furthermore, the injectivity radius is bigger than In(since this is certainly a lower bound for half the length of every closed curve onthe surface which is not homotopic to zero). Again we conclude that the surfacein question has bounded geometry so that Theorem 3.4 applies. Now consider theset P of points (x, y, z) on S where COSJC = cosy = 0 and z is a multiple of n. LetG denote the subgraph induced by P in a (\n, §jr)-net containing P. The curve onS joining {\JZ, \n, kn) and (\JZ, -\n, kjz) parallel to the xy-plane has length at


most W 2 with equality for k = 0. So {\TC, \n, kri) and {{JZ, - \n, kri) areneighbours in G. Using the symmetry of S, we see that G contains a subgraphwhich is isomorphic to the following graph H. The vertices are the points in U3

with non-negative integer coordinates. Two vertices (x, y,z) and (x',y', z') areadjacent if x=x' and \y -y'\ + \z - z'\ = 1 or z = z' = 0 and \x -x'\ + \y -y'\ =1. The vertices with z-coordinate zero induce a quadrant of Z2 which we call B.The vertices with a fixed x-coordinate k induce a quadrant of Z2 which we call Hk.We let Lk denote the path in Hk with vertex set V(Hk) D B.

THEOREM 4.1. The graph H is transient and hence Scherk's surface S ishyperbolic.

Proof. Fix two real numbers 6, e such that 0 < 3 e < 6 < l . We are going todescribe a flow / of value 1 and of finite energy from the origin. We say that avertex is used for the flow if some of its incident edges have non-zero flow. Inother words, if we delete the unused vertices we still have a transient graph. Weare going to describe a flow / such that the vertices in Hn which are used all havey-coordinate at most n1+3e and z-coordinate at most n6. The subgraph of H whichis inside this 'wedge' is therefore itself transient and is hence an example of atransient 'fattening' of the recurrent Z2. We refer to [12] for similar—thoughmore 'dense'—fattenings of Z2. Our construction shows that we can cut asubstantial part off Scherk's surface and preserve hyperbolicity. We also describe/ such that all flow values in edges from Ln_x to Ln are at most n~x~2e. Thisimplies that the contribution to W(f) by the edges from Ln_x to Ln is at most

and hence the contribution to W(f) of the edges not in some Hn is at most

Suppose now that we have already defined / on the edges in //, and on theedges from L, to L,+, for i = 0, 1,..., n — 1, and we assume that n is large. Weshall then define/on the edges in Hn and on the edges from Ln to Ln + l such thatthe above requirements are satisfied. If a vertex v in Ln has ^-coordinate at mostn1+3e, we let q(v) denote the flow that v receives from Ln_,. Now putA/1 = n - 1 - 2 e - (n + l)-1-2 e . Then An<2n~2-2e. If q{v) < (n + I ) " 1 " 2 6 - An, wecall v a potential receiver in Ln. If q{v) > (n + I)"1"26, we call u a transmitter in

If v is not a transmitter, we send the flow q(v) directly in the edge from v toLn+l. Otherwise, we send the flow (n +1)"1"26 through that edge and we sendthe flow q(v) - (n + l)~x~2e to a potential receiver using Hn as explained below.First we note that there are at most (n + l)1+2e<4n1+2e transmitters, since thenet flow from Ln-X to Ln is 1. We divide the transmitters into [n6] classes suchthat any two of them differ in cardinality by at most 1. Then the cardinality ofeach such class is at most 4n1+2e~6. With each such class K we associate preciselyone of the \n6] paths, say PK, in Hn which has a fixed z-coordinate in{0,1,. . . , [n6] — 1} and which has _y-coordinates between 0 and n1+3e. For eachtransmitter v in K we now choose a potential receiver r(v) and send the flowq(v) - (n + l ) " 1 "^ from v to PK (using the shortest path), then through a part ofPK, and finally through the shortest path from PK to r(v). We shall prove that

TRANSIENT RANDOM WALKS 1 1

there are more potential receivers than transmitters (hence we can assume thatr(v) =£ r(v') whenever v =£ v'). If v is not a potential receiver then

q(v) >(n + I)"1"26 - AH > in-1-2'.

As the total flow entering Ln is 1, the number of non-potential receivers is lessthan 2n1+2e. Hence the number of potential receivers is at least n1+3e — 2nl+2e >2«1+3e. In particular, we can assume that r (u )#r ( i / ) whenever v ^v'.

It only remains to calculate the flow energy in Hn. Consider again thetransmitter class K, the path PK, the corresponding potential receivers, and theshortest paths from K (and the corresponding potential receivers) to PK.

The maximum flow in the shortest paths is at most A,, and the maximum flowvalue in PK is at most \K\ A,,. Hence the contribution to W(f) in Hn from the flowfrom K to the corresponding potential receivers is at most

Since \K\^4nl+2e~6 and there are \n6] such classes K, the total contribution toW(f) in Hn is at most

Since 3e < 6, this shows that W(f) is finite and the proof is complete.

5. Recurrence, transience, and isoperimetric inequalities

In the next section we provide a general sufficient analytic condition for a graphto be transient, known also to Kaimanovich [8] and Varopoulos [22]. However,the treatment here is self-contained and is needed for results in the last section.This analytic condition will imply the sufficiency of the combinatorial conditiondescribed in terms of isoperimetric inequalities [7,20,22]. In order to indicate thestrength of the result, we first make a few remarks on recurrence of graphs.

Consider a graph G whose vertex set has a partition V(G) = Vo U V, U V2 U...into finite subsets such that there is no edge between Vt and Vj when |/ — j \ 5=2.Let an be the number of edges between Vn and Vn_x for n = 1, 2, ... . Suppose/isa flow of value / from a vertex v in Vo. As the total flow from Vn_x to Vn is /, thetotal flow energy in the edges between Vn-X and Vn is at least an{l21a2

n) = I21an.So, if E^=i I/a* = °°» then G is recurrent. This is a special case of N'ash-Williams'criterion for recurrence [16]. This shows, in particular, that the 2-dimensional gridZ2 is recurrent. More generally, if an<en logn, then G is recurrent. Thiscriterion cannot be applied to the nets of the helicoid.

Now we turn to transience. If V is a vertex set in a graph G, then we let 5Vdenote the set of vertices in V which have neighbours in V(G)\V. We say that Gsatisfies an e-isoperimetric inequality (where £ is a non-negative real number) ifthere exists a positive constant c such that |<5V| 2*c |V|^+e for each finite vertexset V. If G has bounded vertex degrees, then clearly, e =£ \. If e = 5 we speak of astrong isoperimetric inequality.

Clearly Zd (d^2) satisfies an e-isoperimetric inequality for e = { — l/d and forno smaller e. So, for e = 0 we may have recurrence. For any e > 0 we gettransience:


THEOREM 5.1 [7,20,22]. If G is a graph whose vertices have bounded degreesand, for some positive e, G satisfies an e-isoperimetric inequality, then G istransient.

Dodziuk [3] proved Theorem 5.1 for e = i (See also Gerl [6].) In [20] it isshown that we get transience provided G satisfies an inequality of the form\6V\ >g(\V\) for each finite vertex set V, where g is an increasing real functiondefined on the natural numbers such that E"=ig(«)~2<°°- Thomassen [20] alsoshows that it is sufficient to verify the e-isoperimetric inequality for those finitevertex sets V that induce connected subgraphs. These strengthenings of Theorem5.1 cover almost all those transient graphs that we know of. The nets of theScherk surface do not satisfy an e-isoperimetric inequality for e > 0.

We have constructed the following class of graphs whose transience does notseem to follow from any modification of an e-isoperimetric inequality. Let or be apositive real number. Consider the graph obtained from an infinite path UiU2u3...by adding a path of length n" from vn to v2n. The new paths are pairwise disjointexcept that each v2n is the end of two new paths. Then the resulting graph can beshown to be transient if and only if a<\. This, in turn, can be used to proveTheorem 4.1.

6. Analytic conditions for transience

In this section we introduce vertex energies and Sobolev constants, and werelate them to isoperimetric inequalities. A more general development can befound in [8], but we give a concise, analytic proof of Theorem 5.1, which we uselater for the treatment of harmonic functions in § 7.

If v is a vertex in a graph G, we define the vertex energy

W(v,G) = inf\\D<p\\22

where the infimum is taken over all finitely supported functions 0 over V(G) suchthat <j)(y) = 1. This definition is motivated by the following well-knownobservation.

PROPOSITION 6.1. G is transient if and only if W(v, G) > 0.

We shall only sketch the proof which goes via the theory of electrical networks.(For definitions the reader is referred to [4] or [19] which contains a conciseintroduction of the relevant part of the theory.) Let Vn be the set of vertices ofdistance n from v. Let Gn be the finite graph obtained from G by contractingVn+1UVrt+2U... into a single vertex vn. Let r(v, vn, Gn) be the effectiveresistance between v and vn in Gn. Then r(v, vn, Gn) is a non-decreasing functionand we define the effective resistance from v as r(v, G) = limw_>Oo r(f, vn, Gn). Wealso let Wn be the energy dissipated when we send an electric current from v to vn

with voltage drop one Volt. The theory of electrical networks implies that

Wn = \lr(v, vn, Gn).


It is not difficult to prove that

W(v, G) = lim Wnn—»»

and that G is transient if and only if r(v, G) < °°. Hence W(v, G) may be thoughtof as the energy dissipated when a voltage generator of one Volt sends electriccurrent from v to infinity. Recurrence means that r(v, G) = °° and that the abovecurrent is the zero current and hence W(v, G) = 0.

If G is a graph and e is a positive real number, we define the Sobolev constant

where the infimum is taken over all finitely supported non-vanishing functions <j>on V(G). We also define the isoperimetric constant

where the infimum is taken over all finite vertex sets V. Then G satisfies ane-isoperimetric inequality if and only if / e(G)>0. Theorem 6.3 below relatesSE(G) and Ie(G). In the proof we shall use the Hardy-Littlewood-Polyainequality:

PROPOSITION 6.2. / / F{i) is a non-negative decreasing function defined on thenon-negative real numbers, and ar2= 1, then

oc \ ta-xF{t)adt^{\ F(t)dt) .

Proof. Since tF{i) ^ $'o F(x) dx, we have

The result follows by integration.

THEOREM 6.3. Let G be a graph of maximum degree A, and let \^ e > 0. Then

REMARK. This relation is the discrete analogue of a corresponding identity forcomplete Riemannian manifolds due to Federer and Fleming [5] and Maz'ya [15].We also refer to Varopoulos [22] and Kanai [11] for a discussion of the relationbetween Ie and Se and the robustness of IB > 0 (respectively Se > 0) under roughisometries.


Proof of Theorem 6.3. If V is a finite vertex set of G and (f> is the characteristicfunction for V, then

This implies the second inequality of Theorem 6.3.We shall now prove the first inequality of the theorem. Put a = ({ + e)~\ Let

0 be any non-vanishing non-negative finitely supported real function on V(G).Let M0 be any vertex such that (p(u0) = 0, and let uu u2, ...,«„ be the vertices insupp (f> such that

0 = <p(u0) < 0 ( W l ) ^ <p{u2) =s ^ tf>(*O-

For each i = 0, 1, 2, ..., n - 1, let Vt = {ui+l, ui+2) ...,un) and put F{t) = \Vi\a =

(n-i)a for 0(«l)^f<0(M/+i) and F(0 = 0 for t^(f>(un). We shall prove that^\\D<t>yie(G). Now

ueV(G) i = l

1=0

n-\n-\ r<t>(.ui+i)

= or [ F(t)Uat"-'1 dtJo

J=0

,=o

It only remains to prove thatn-\

Let Uj be any vertex which contributes to the left-hand side of (*) in the sense thatUj is in some dV(. Then there is a smallest k (k <j) such that G has an edge UjUk.(If Uj is joined to a vertex not in Vo, we assume that u0 is chosen as a neighbour ofUj.) The contribution of ut to the left-hand side of (*) is 0(w;) - <t>(uk). This isalso the contribution of the edge to the right-hand side of (*). Hence (*) holdsand the proof is complete.


We are going to relate the numbers W(v, G) to the Sobolev constants and theisoperimetric constants. For that we need the following lemma.

LEMMA 6.4. Let G be a graph and P a real number, /3 2s 2. / / <f> is anon-negative real function defined on V = V(G), then

ueV )

Proof. Each edge uv contributes 2 \4>p{u) — 0**(u)| to the left-hand side andx(u) + (pp~\v)) \<t>(u) - (f)(v)\ to the right-hand side. So it suffices to prove

the following: for any real numbers x and y such that x ^ y ^ O w e have

2(x/?-yp)**p(xp-1 + / - ' ) ( * - y ) .

This is an easy exercise which we leave for the reader.

The next theorem is essentially a special case of [8, Theorem 4.3].

THEOREM 6.5. Let G be a graph of maximum degree A satisfying an e-isoperimetric inequality for some e (0 < e =? 2)- If <t> G /()(V{G)), then

Proof. Put p = 1 + l/(2e). Then p s»2. By the definition of 5e(C) and Lemma6.4 we have (assuming without loss of generality, that <f> is non-negative)

(/l"1(M) 2 |^(u)-0(vueV \ veN(u)

The last inequality follows from Schwartz's inequality since every edge contrib-utes twice and every vertex at most A times to the sum. Since

\\4>%l+Br> = \\<l>Ve\\\+e

andn ^ - 1 i i 2 = i i 0 1 / E i i i

we conclude that

THEOREM 6.6. Let G be a graph with maximum degree A satisfying ane-isoperimetric inequality, where 0 < e ̂ |- F°r each vertex v,

W(v, G)^ e2A-lSe(G)2^ £2A-'/£(G)2.

In particular, G is transient.

Proof. In the definition of W(v, G) we consider functions <f> such that<j>(v) = 1. Then ||0||i/e3= 1 and the first inequality follows from Theorem 6.5. Thesecond inequality follows from Theorem 6.3.

Further applications of Theorem 6.5 will be given in the next section.


7. Applications to discrete potential theory

In this section we address the problem of deciding whether a given graph has anon-constant harmonic function of bounded energy (that is, a function inHD(G)). If G has such a function, then G is transient. The converse is far fromtrue. The following result will appear in a forthcoming paper of Thomassen:

THEOREM 7.1. Let a be a positive real number and G a graph and Vx U V2 U... adecomposition ofV(G) into finite sets satisfying the following:

(i) there is no edge between Vt and V, when \i—j\^2;

(ii) if u, v are vertices in Vn having neighbours in Vn+l, then G has a path oflength less than cm from u to v in Vn.

Then HD(G) has no non-constant function.

Theorem 7.1 applies to the d-dimensional grids Zd. It also shows that theparticular graph G that we constructed on the Scherk surface has no non-constantfunctions in HD(G). If G satisfies the assumption of Theorem 7.1, then everypower Gk of G satisfies the assumption of Theorem 7.1 (using the decompositionV£U V[ U... where V\ = VUU Vki+l U ... U Vft(/+1)_,).

Also, if G2 (or, more generally, Gk) satisfies the assumption of Theorem 7.1,then the proof of Theorem 7.1 shows that G has no non-constant function inHD(G).

If some net of a slim and well-behaved metric space satisfies the assumptions ofTheorem 7.1, then it is not difficult to modify the proof of Theorem 7.1 to showthat no net G of M has a non-constant function in HD(G).

We now relate e-isoperimetric inequalities to the existence of non-constantharmonic functions of finite energy. We have previously defined D<f> for a realfunction on V{G) where G is a graph. We assigned directions to the edges suchthat D4> is always non-negative. Now we change the definition slightly in that wefix the directions on E(G) once and for all. Then D<p taken on a directed edge uvis defined as (f)(u) - (p(v). It is easy to see that a real function ip defined on E(G)is of the form D<f> (where <f> is defined on V(G)) if and only if the sum of valuesof ip around any cycle (where a i//-value is counted negative when an edge isdirected against the direction of the cycle) is zero. (With the terminology fromthe electric network theory, ip satisfies Kirchhoff s voltage law.) If this is the case,then <f> is uniquely determined up to an additive constant (provided G isconnected).

Let FV(G) be the space of real functions on V(G) and let DFV(G) consist ofall D<f> where <f) e FV(G). Then DFV(G) n l\E(G)) is closed in 12(E(G)) and istherefore itself a Hilbert space (with the induced inner product from 12(E(G))),and hence it can be rewritten as a direct sum L® LL whenever L is a closedsubspace. We shall consider the case where L is the closure of

A function \\) defined on V(G) is harmonic if and only if Dip is perpendicular to


each D<f> where the support of <p is precisely one vertex. Hence

DHD(G) = D(lo(V))±

where\ \j)eHD(G)}.

It follows that

DFV(G) fl 12(E(G)) = closure(D(/0(VO)) 0 DHD(G).

In other words, we get the following which is a discrete version of a result ofRoyden [18].

PROPOSITION 7.2. If G is a graph, then every function D<j> in 12(E(G)) has aunique decomposition

D(f>=f' + Dg

where g e HD(G) andf e closure(D(/0(V))).

Clearly/' is of the form Df. For graphs satisfying an e-isoperimetric inequalitywe can go a little further. Cartwright and Woess (private communication) provedthe following for e = \-

PROPOSITION 13. If G is a graph with maximum degree A and G satisfies ane-isoperimetric inequality (0<e^2)> then each function f in the closure ofD(lo(V)) (taken in l\E(G))) is of the form

f' = Dfwhere felVe(V(G)).

Proof. Let (f>n be a sequence of functions in IQ(V) such that

By Theorem 6.5, the sequence <f>n forms a Cauchy sequence in lVe(V(G)). Hencethere is an / in lVe(V(G)) such that

lim \\f-4>n || 1/e = 0.

As lp-convergence implies pointwise convergence, / ' = Df.

This simple observation turns out to have remarkable consequences describedbelow.

If / is a real function defined on a countably infinite set V and )3 is a realnumber, then we write

if, for each sequence vx, v2, ... of distinct elements in V we have


THEOREM 7.4. Let G be a graph with maximum degree A satisfying ane-isoperimetric inequality ( 0 < £ ^ | ) . Then HD(G) has non-constant functions ifand only if there exists a real function 0 on V(G) such that

D(f> e 12(E(G))

and limu_00 <f)(v) does not exist.

Proof. Since a non-constant harmonic function has no maximum and nominimum, any non-constant function of HD(G) can play the role of <£. Supposeconversely that <p exists. Let g, f and / be as in Propositions 7.2 and 7.3. As\imv_aof(v) = 0, limu_»oog(i») does not exist. In particular, g is not constant.

COROLLARY 7.5. Let G and H be graphs such that G has maximum degree A,V{H) = V{G), E(H)Ê(G) and both G and H satisfy an e-isoperimetricinequality, e>0 . If there is a non-constant function in HD(G), then there is anon-constant function in HD(H).

Proof. Let 0 eHD(G), with <f> non-constant. As \imvôo <f)(v) does not exist,and Dcj) (taken in H) is in l\E{H)), Theorem 7.4 applies.

COROLLARY 7.6. Let G be a graph of maximum degree A satisfying ane-isoperimetric inequality, with e>0 . Let k be a natural number. Then HD{G)has a non-constant function if and only if HD(Gk) has a non-constant function.

Proof. Clearly Gk satisfies an e-isoperimetric-inequality. So, if there exists anon-constant function in HD(Gk), then, by Corollary 7.5, there exists anon-constant function in HD(G). Suppose conversely that <f> is a non-constantfunction in HD(G). Then l i n v ^ (j>{v) does not exist. It is easy to see thatD<f> e l\E{Gk)) (because G has bounded degrees). Now apply Theorem 7.4.

PROPOSITION 7.7. / / G is a graph of maximum degree A and Gk is a powerof G satisfying an e-isoperimetric inequality, with e 5= 0, then G satisfies ane-isoperimetric inequality.

Proof. It is sufficient to verify the statement for k = 2. If V is a finite vertex setof G, then \6V\ taken in G2 is at most A times \6V\ taken in G.

PROPOSITION 7.8. / / G is a graph of maximum degree A satisfying ane-isoperimetric inequality (£2=0) and G' is a graph which is roughly isometric toG, then G' satisfies an e-isoperimetric inequality.

Proof. By definition of rough isometry, there exist a map 0: V(G)—>V(G')and positive constants a, b, c such that, for any u, v e V{G),

a~x dist(«, v,G)-b^ dist(0(w), <p(v), G')â dist(w, v, G) + b,

and such that each vertex u' in G' has distance at most c from some <f>(u), whereu e V(G).

We note first that G' has maximum vertex degree A' depending on a, b, c andA. If V is a finite vertex set in G', then we let

V = {ue V(G)\ dist(<K«), V, G')^ c}.


Each u in dV is adjacent to some v not in V. Hence

dist(0(u), V, G')^cbut

(u), V, G')>c.

Also dist((j)(u),(j)(v),G')â + b. Let \p(u)eV' be such thatdist(0(u), V(w), G')^c. Then dist(V/(u), <t>(v), G')â + b + c. Let H e anatural number, with kâ + b + c. Then both edges ii)(u)<j)(v) and ip(u)<p(u)belong to (G')fc. It follows that {V(M)| " 6 <5V} c 6V (taken in (G')*)- For somenatural number m, depending only on a, b, c, A, m \5V'\^\dV\, and since G'has maximum vertex degree A', there exists a constant / depending on a, b, c andA' such that |V| 2=/|V|. Hence (G')k satisfies an e-isoperimetric inequality andby Proposition 7.7, G' does too.

Kanai [9] proved Proposition 7.8 by different arguments.

THEOREM 7.9. Let G be a graph of maximum degree A satisfying an e-isoperimetric inequality, with 0< £^\. Let G' be any graph roughly isometric toG. Then HD(G) has non-constant functions if and only if HD(G') hasnon-constant functions.

Proof. By Proposition 7.8, G' satisfies an £-isoperimetric inequality. Supposeip' is a non-constant function in HD(G'). Put ip = ip'°<j>. Let A: be a naturalnumber such that kâ + b. If uv is an edge in G such that 0(w)# 4>{v), then(f>(u)<f>(v) is an edge in (G')k. There exists a natural number m (depending onlyon a and b and A) such that at most m edges uv are mapped onto the same edge0(K)0(U) of {G'f. Let 0 denote | |D^' | | 2 calculated in 12(E((G')*))• It is easy tosee that /? is finite. Moreover,

By Theorem 7.4, HD(G) has a non-constant function.

THEOREM 7.10. Let M be a slim and well-behaved metric space such that somenet G in M satisfies an e-isoperimetric inequality. Let M' be roughly isometric toM. Then each net in M' satisfies an e-isoperimetric inequality. Moreover, ifHD{G) has a non-constant function, then, for each net G' of M', HD(G') has anon-constant function.

Proof. Since M and M' are roughly isometric, Theorem 3.3 implies that G andG' are roughly isometric. Now apply Theorem 7.9.

References

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20 TRANSIENT RANDOM WALKS

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Mathematical InstituteThe Technical University of Denmark

Building 303DK-2800 Lyngby

Denmark

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