Percolation and Minimal Spanning Forests in Infinite Graphs
Percolation and Minimal Spanning Forests in Infinite Graphs Author(s): Kenneth S. Alexander Source: The Annals of Probability, Vol. 23, No. 1 (Jan., 1995), pp. 87-104 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2244781 . Accessed: 09/09/2013 14:22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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The Annals ofProbability 1995, Vol. 23, No. 1, 87-104
PERCOLATION AND MINIMAL SPANNING FORESTS IN INFINITE GRAPHS' BY KENNETH S. ALEXANDER Universityof Southern California The structureof a spanningforestthat generalizesthe minimal spanningtreeis consideredforinfinite graphswitha value f(b) attached to each bond b. Of particularinterestare stationaryrandomgraphs; values f(b) and theVoronoior examplesincludea latticewithiid uniform completegraphon thesitesofa Poissonprocess,withf(b) thelengthofb. The corresponding percolation modelsare Bernoullibondpercolation and the "lilypad" modelof continuumpercolation, It is shown respectively. that under a mild "simultaneousuniqueness"hypothesis, withat most one exception, each treein the foresthas one topologicalend,thatis, has no doublyinfinite paths.Ifthereis a treein theforest, necessarilyunique, withtwotopologicalends,it mustcontainall sitesofan infinite clusterat the criticalpointin the corresponding model.Treeswithzero, percolation or threeor more,topologicalends are not possible.Applicationsto invasion percolationare given.If all trees are one-ended,thereis a unique fromeach site. optimal(locallyminimaxforf) pathto infinity
1. Introduction. For a finiteset V c d a Euclidean minimalspanning tree(MST) of V is a tree withsite (that is, vertex)set V and minimaltotal lengthofall bonds(thatis, edges).More generally,givena finitegraphwith site set V and bond set A, and a labeling functionf: M
-*
[0, oo),a minimal
spanningtree of (V, M, f) is a tree in (V, M) spanningV with EbEf(b)
minimal among all such trees; (V, A, f) determines a labeled graph. It is
natural to ask whetherthere is a structureanalogous to the MST when (V, A, f) is an infinitelabeled graph,and if so to considerits properties, especiallyforrandomlabeled graphs.Two particularcases ofinterestare: the lattice/ uniformmodel: (V, M) is a latticein Rd and {f(b), b EA } are iid random variablesuniformin [0,1] and the Poisson/ Euclidean model: (V, M) is the complete graphon the set ofsites of a Poissonprocessand f is Euclideanlength. ReceivedDecember1993;revisedApril1994. DMS-92-06139. 'ResearchsupportedbyNSF Grant, 'AMS 1991 subjectclassifications. Primary60K35; secondary82B43,60D05, 05C80. invasion continuum Keywordsand phrases.Minimalspanningtree,percolation, percolation, percolation.
87
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K. S. ALEXANDER
In the latter case one can allow more general stationarypoint processes, / Euclidean model. yieldingthe stationary The natural way to definesuch an MST analog is to finda propertyof bondsin a labeled graphwhich(i) in finitegraphs,characterizesmembership then graphsas well.This property in theMST and (ii) makes sense in infinite of membershipin the MST analog. One such definibecomesthe definition in [16] of the MST in finite tion, based on Prim's inductiveconstruction graphs,was used by Aldous and Steele [1] for the stationary/Euclidean model;it yieldsa "minimalspanningforest"(MSF) in whicheverycomponent is equivadespiteappearingquite different, tree.Our definition, is an infinite lent to theirsin a large class of models,whichincludesthe lattice/uniform and Poisson/Euclidean. The main structureofan infinitetreeis givenbyits numberoftopological paths fromany fixed ends, which is the numberof infiniteself-avoiding vertex.Thus a zero-endedinfinitetreemustcontainsites ofinfinitedegree,a a two-endedtree one-endedtreeis like an infinitesystemofrivertributaries, consistsof a single doublyinfinitepath (the trunk)with finitebranches emanatingfromit and a treewiththreeor moreends mustcontainat least one branchpoint,thatis, a pointfromwhichthereare at least threedisjoint paths. Aldous and Steele [1] conjecturedthat forthe infiniteself-avoiding Poisson/Euclideanmodel,theirMSF consistsofa singleone-endedtree. to any infiniterandomlabeled graph X = (V, M, f), there Corresponding is a percolationmodel as follows.We say a bond b is occupiedat levelr if f(b) < r. Let X, x, y ED=d, with( x, y> and ( y, x > identified. Such a graph can include sites of infinitedegree,but a.s. has onlyfinitely manysites in each boundedregion;the asymptoticdensityof sites may be infinite. Multiplebondsbetweena fixedpair ofsites are notallowed,but one can obtainresultsforgraphswithsuch multiplebondsby deletingall bonds betweeneach fixedpair x and y except the one with the smallest label. Bonds ofform( x, x>, called loop bonds,are allowed.We let V denotethe set of sites, M the set of bonds and f the labelingfunction,so that X can be Rd maybe replacedthroughwiththe triple(V, A, f). Alternately, identified out by a lattice L, withtranslationallowedby elementsof L only;without furthermention,we use this formulationwhen appropriate,as in the model. lattice/uniform To ensurethatthe graphsG = (V, A, f) we deal withhave a well-defined (1) all bond unique MST or MSF, we will make twoassumptionsthroughout: labels are distinctand (2) foreverycomponentC and everyfiniteproper bondamongall bondsthat subset A of V n C, thereis a unique f-minimizing conditionfor(2) connectsitesin A to sitesin V \ A. Assuming(1), a sufficient
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K. S. ALEXANDER
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is thateverysite has finitedegreein G, r foreveryr; thisis satisfieda.s. in modelprovided model and in the stationary/Euclidean the lattice/uniform the stationarypointprocessofsites is locallyfinite.We call G locallyfiniteif V n R is finitefor all bounded regions R and call a labeled graph that satisfies (1) and (2) ambiguity-free.
f) and a subgraphH, define Givena graphG = (V, AW, dH:= {b = (x, y> Em': x e H, y 4 H}.
we willcall f(b) the lengthof b and referto In a mildabuse ofterminology bonds as shorter,longest,and so forth;this should not cause confusion because we never make use of Euclidean lengthof bonds exceptwhen it coincideswiththe labelingby f. The MST in finitegraphscan be constructedby an inductive"invasion" procedureknownas Prim'salgorithm[16]; the same procedurewas used by Aldousand Steele [11to definetheirMSF forcertaininfinitegraphs.Specifif) be a labeledgraphand let v E V be a site.Let Jo(v) = {v} cally,let (V, AW,
and, given Jn(v), let Jn+ (v) be Jn(v) together with the shortest (that is, f-minimizing) bond Bn 1 in 9Jn(v), including the endpoint of BnI 1 in V \ Jn(v). Let J0,(v) U n 2 0Jn(V). We say a bond b is invaded from v if b E Joo(v). path in a graphand u, v sitesin y, let zag denotethe For y a self-avoiding
path y in a labeled graph segmentof y fromu to v. We say a self-avoiding
(V, IW,f) is locally f-minimax if for every pair u, v of sites in y and every path a fromu to v,
max(f(b): b E ZaJ< max( f(b): b Era). All the equivalences To defineour MSF, we willneed the nextproposition. are well knownin the case offinitegraphs;see [15]. For infinitegraphs,we essentiallyneed onlyverifythatexistingideas forfinitegraphscan establish equivalenceamong(2.1)-(2.5) withoutusingequivalenceto (2.6) or finiteness of the graph. The proofsof.this and all results in this sectionappear in Section3. Let G = (V, A, f) be a (finiteor infinite)ambiguity-free labeled graph and let b = <x, y> _Wwith x = y. The following are equivalent: PROPOSITION 2.1.
(2.1)
There exists a set A of sites such that b is the shortestbond fromAtoV\A.
(2.2) (2.2)
There exists no path from x to y with all bonds strictly shorterthan b.
(2.3)
b is a bond in some locally f-minimaxpath.
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The followingare equivalent: (2.4)
b is invaded eitherfromx or fromy.
(2.5)
x and y are in distinctstrictf(b)-clusters and these clusters are not both infinite.
If strictsimultaneous uniqueness holds for G, then (2.1)-(2.5) are all equivalent. If G is finite,then (2.1)-(2.5) are each equivalent to (2.6)
b is a bond of the MST of G.
Again,shortermeans havinga smallerlabel. Equivalenceof(2.4) and (2.6) the forfinitegraphs is the basis of Prim's algorithm[151 forconstructing MST and showsthatthe set ofinvadedbondsdoes notdependon the starting criterion, by analogyto a site in finitegraphs.We call (2.2) the creek-crossing hikertryingto crossa creekby steppingfromstoneto stone,avoidinggetting wet by nevertakingan available step if a path of all strictlyshortersteps exists. Aldous and Steele [1] defineda spanningforestconsistingof all bonds satisfying(2.4) and provedthat (2.4) and (2.5) are equivalent.We preferto that is, we definethe criterionas our definition, use the creek-crossing minimal spanning forest (or MSF) of an ambiguity-freelabeled graph G
(V, A, f) to be the graphwithsite set V and bondset
=
{b = (x, y> e A: thereis no pathfromx to y
consistingentirelyofbondse withf( e) < f( b)}. Because simultaneousuniquenessplays a majorrole in thiswork,we will conditionforit from[2]. For this we need the notionof give a sufficient positivefiniteenergy:fora full definitionin our context,see [2]; the idea on the appears in [6]. Loosely,positivefiniteenergymeans that conditioning about graph outside a finitebox, togetherwith certainpartial information bonds crossingthe box boundary,a.s. yields a nonzeroprobabilitythat all sites withinthat box are.connectedat a givenlevel r, at least forr large enoughthat percolationoccursat level r. LettingAt denote[- t, tVd, let us definethe site densityof a stationary random graph X = (V, A, f) to be
lim.IV n Ati/lAt i,
t X-.0
where [X]denotescardinalityforfinitesets and volumeforregionsin Stationarityensuresthatthislimitexistsa.s.
Rd.
PROPOSITION2.2 ([2], Theorem 1.8 and Remark 1.10). Suppose X is a stationary random labeled graph in Rd, with positive finiteenergyand finite site density a.s. Then both simultaneous uniqueness and strictsimultaneous uniqueness hold forX, a.s.
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K. S. ALEXANDER
and Poisson/Euclideanmodelsclearlyhave positive The lattice/uniform finiteenergyand finitesite density,so Proposition2.2 showsthat forthese (2.4) used byAldous (2.2) is equivalentto the definition models,ourdefinition and Steele [1]. The distinctionbetweenstrictand nonstrictsimultaneousuniquenessis by way we will explicatethe distinction importantin our results.Therefore, of the followingresult,thoughProposition2.2 makes it nonessentialto our main results. LEMMA 2.3. (i) Suppose G is an infinitelabeled graph with all labels distinct. If strictsimultaneous uniqueness holds for G, then so does simultaneous uniqueness. (ii) Suppose X is a stationary random labeled graph in Rd that a.s. has finitesite densityand all labels distinct. Then withprobability 1, simultaneous uniqueness holds forX if and only if strictsimultaneous uniqueness holds forX.
The assumptionofall labels distinctcannotbe eliminatedin Lemma2.3. It is easilyverifiedthatin the stationaryrandomlabeled graphofExample 1.9 of[2], strictsimultaneousuniquenessholdsa.s. butnotsimultaneousuniqueness. See also Example 5.2 below. (2.2) thatthelongestbond criterion It is immediatefromthecreek-crossing in any cyclein a labeled graphG is notin the MSF. Thus the MSF is acyclic. connectedlabeled Criterion(2.1) ensures that in an infiniteambiguity-free graph,everycomponentof the MSF is infinite.In particular,there are no meaningtheMSF spans thesite set V. This provesthe one-pointcomponents, whichjustifiesthe name "minimalspanningforest." following, LEMMA 2.4. In an ambiguity-freelabeled graph G = (V, A, f) with all componentsinfinite,the MSF is a forestthat spans V and consists of infinite trees.
way of obtaininga random For an infinitelattice,a completelydifferent spanningforestis consideredin [141. Let C0,(G,r) denotethe unionofall infiniter-clustersin the graphG and let C.t(G, r) be the union of all infinitestrictr-clusters.Here is our main result. THEOREM 2.5. Suppose X is a stationaryrandom labeled graph in Rd that a.s. is ambiguity-free,has finitesite densityand has all componentsinfinite. Then withprobability 1:
(i) The MSF contains no zero-ended treesor trees with threeor more ends. (ii) If simultaneous uniqueness holds forX, then the MSF includes at most one two-endedtree; all othertrees are one-ended. If a two-endedtree T exists, then thereis percolation at level r (X) in the correspondingpercolation model,
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PERCOLATION AND FORESTS IN GRAPHS
T contains all sites of CQt(X, rc(X))
and the trunk of T is contained in
CQ'(X9 rc(X)).
If X is ergodic,thenthereis a nonrandomalmost-surevalue ofthe critical
point rj(X), which we denote rc.
Considera planar graph G, withoutloop bonds, embeddedin R2. The bonds, viewed as curves in R2, divide the plane into faces, which are connectedcomponents ofthe complement ofthe graph.We call G latticelikeif (i) G is locallyfinite,(ii) everysite in G has finitedegree,(iii) G is connected, (iv) G is planar,(v) G has no loopbondsand (vi) everyfaceis bounded.For a lattice-likegraph, a dual graph, also planar, is obtainedby selectingan arbitrary "facesite"in each faceand then,foreach bond b thatformspartof the boundarybetweentwodistinctfaces,puttinga dual bond b* betweenthe face sites in these faces.For G labeled,a dual bond b* is said to be strictly occupiedat level r ifand onlyif f(b) 2 r. THEOREM2.6.
Suppose G is an ambiguity-free lattice-likelabeled graph in
R2. If the MSF of G consists of more than one tree,then thereis percolation of
strictlyoccupied dual bonds at level rc(G) in the correspondingpercolation model.
EXAMPLE lattice,it 2.7. For the lattice/uniform modelon the hypercubic large d [91thatthere is knownfordimensionsd = 2 [11] and forsufficiently percolationmodel; is no percolationat the criticalpointin the corresponding large"[10]. Therefore, thebestresultat presentis that d 2 19 is "sufficiently all treesin the MSF are one-ended.For d = 2, wherethe latticeand its dual thereis also no percolationin the dual graphat level rc [111, are isomorphic, so the MSF consistsofa singleone-endedtree,as was provedin [4]. EXAMPLE 2.8. For the two-dimensional model,in stationary/Euclidean place ofthe completegraphon the site set V, one may considerthe Voronoi graph, definedas follows.Let d(, ) denote Euclidean distance.Given V, dividethe plane intothe polygonalcells Q(v) := y e R2: d(y, v) < d(y, x) forall xEVV},
vEV.
The Voronoigraph has site set V and bondset
{ (u, v:
u, v E V, Q(u) and Q(v) have an edgein common),
labeled by Euclideanlength.It is easily checkedthatonlyVoronoibondscan satisfy(2.1) or (2.2), so the MSF is the same fortheVoronoigraphas forthe completegraph.This is well knownforfinitegraphs;see [15]. The Voronoi so Theorem2.6 can be applied.Percolationat level r graphis a.s. lattice-like, in the graphdual to the Voronoigraphis equivalentto percolationofvacant blobmodelat level r (radius r/2). In the Poisson space in the corresponding case, it is provedin [3] thatthereis no percolationofvacantor occupiedspace
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at level rc. Therefore,the MSF consists of a single one-endedtree, as conjecturedbyAldousand Steele [1]. 3. Proofs of the main results. PROOF OF PROPOSITION2.1. We will show (2.1) < (2.2), (2.2) < (2.3) and, understrictsimultaneousuniqueness,(2.2) < (2.5). The equivalence(2.4) < (2.5) is provedin [1]. The equivalence(2.1) < (2.6) forfinitegraphscan be foundin [15]. Suppose firstthat(2.1) holds and x E A, y E V \A. If y is a path fromx to y, theny includesa bond e fromA to V\A. By (2.1), f(e) 2 f(b). Since y is arbitrary,(2.2) follows.Converselysuppose (2.2) holds and let A be the strict f(b)-clustercontainingx. By (2.2), y E V\A and by definitionof "strictf(b)-cluster," thereis no bond shorterthan b fromA to V\A. Thus (2.1) holdsand we have (2.1) (2.2). Next suppose (2.3) holds. Suppose a is a path fromx to y. Then by definitionof "locallyf-minimax," max{f(e): e Eca) 2 f(b). Since a is arbitrary,(2.2) holds.Conversely,(2.2) says that b by itselfconstitutesa locally f-minimax path.Thus (2.3) < (2.2). Now (2.2) says that x and y are in distinctstrictf(b)-clusters.Under strictsimultaneousuniqueness,these clustersare necessarilynot bothinfinite,so (2.2) and (2.5) are equivalent. [1
For a graphG in Rd and x E Rd., let OG denotethe translationof G by -x; thusforeverysite v of G0 QvGhas a site at 0. The proofof Theorem2.5 is based principallyon the idea that certain and possiblestructuresin labeled graphsare prohibiteda.s. by stationarity finitesite density.We beginwithfourpropositionson this theme.The first says,loosely,thatanythingthathappensonlyfinitely manytimesperinfinite clusteractuallyneverhappens. PROPOSITION 3.1 ([2]). Suppose X = (V, ~, f) is a stationary random labeled graph in Rd withfinitesite density,and A is a set oflabeled graphs G in which the origin is a site in an infinitecomponentof G. Suppose that with probability 1, thereare onlyfinitelymany sites v in each infinitecomponentof X for which QvX e A. Then e V] =0. P[QvXeAforsomev
PROOFOF LEMMA2.3. (i) Suppose G includestwo disjointinfiniter-clusters forsome r. Since thereis at mostone bond in G withlabel r, each of these two r-clusterscontainsan infinitestrictr-cluster,so strictsimultaneous uniqueness fails.
(ii) We may assume the randomgraphis ergodic,so rc(X) = rc a.s. Suppose that in the graph X, simultaneousuniqueness holds. We may
assume all labels are distinct. For each r > rc, M(r) := U 8< rCoo(X,s) is an infinitestrictr-cluster;we call any otherinfinitestrictr-clusterextraneous.
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95
For r > rc we call an infinitestrictr-clusterhollowifit containsno infinite s-cluster,for every s < r. Thus every extraneousinfinitestrictclusteris hollow.For each site v of X let r.(v) = infIr> rc: v E CO(X,r)}. If v is a site ofan extraneousinfinitestrictr-clusterforsome r > rc,thenv t CO(X,s) for any s < r, so r (v) = r; in particular,thereis at mostone such r foreach site v ofX. Therefore, distinctextraneousinfinitestrictclustersare disjoint,even ifthe corresponding values of r are distinct. If C is an extraneousinfinitestrictr-clusterforsome r > rc,then since C M(X, theremustbe a bond b = (x, y) E dC withx E C and r) is connected, = f(b) r. Since labels are distinct,thereis at mostone suchchoiceofb, x and y; we call b the attachmentbond and x the attachment site of C. To detach all extraneousclustersfromeach other,we wish to replace the attachment bond <x, y) withthe loop bond <x, x). We call <x, x) the alteredattachment bond of C and giveit the same label r as the attachmentbond.Let Y be the stationaryrandom labeled graph whose componentsare the extraneous infinitestrictclustersin X togetherwiththeiralteredattachmentbonds.In each componentof Y, the altered attachmentbond is the unique longest bond. Let A be the set of labeled graphsin which0 is an endpointof the unique longestbond in its connectedcomponent.If v is a site of Y, then OvYe A if and onlyif v is the alteredattachmentsite of some extraneous infinitestrictclusterin X. Hence O Y E A forexactlyone site v in each infiniteclusterin Y. By Proposition3.1, we may assume thereare no sites v with OvYE A; this means Y is empty,so there are no extraneousinfinite strictclustersin X. It remainsto establishuniquenessofthe strictrc-cluster. AnotherapplicationofProposition3.1 and the distinct-labels propertyshowsthat thereare a.s. no bonds b in X withf(b) = rc.However,thismeans everyinfinitestrict is also an infiniterc-cluster, and thereis at mostone ofthe latter. rc-cluster El
If F is a finiteset of sites in a singlecomponentof a graph G, we write C(G, F) forthis componentand let G \ F denotethe subgraphof G obtained by deleting F and all bonds emanating from F. We write C(G, v) for C(G, {v}). We call such a finiteF a core if thereare infinitely manyfinite componentsin G \F that are containedin C(G, F); that is, removingF splitsoffinfinitely manynew finiteclusters.An exampleis furnishedby the "infinite-spoked bicyclewheel,"in whicha countablenumberof "rim sites" are locatedon somecircle,and thereare twoother"hub-end"sitesnoton this circle;thereis a bondbetweeneach hub-endsite and each rimsite.The two hub-endsites thenforma core.Neitherhub-endsite is a coreby itself. PROPOSITION 3.2 ([2]). SupposeX is a stationary randomlabeledgraph in Rd withfinitesitedensity.Then withprobability1, X containsno core.
The easy proofofthe following lemmais containedin the proofofLemma 2.3 of[2].
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LEMMA3.3. Suppose G is an infiniteconnectedgraph that containsno path core, and v is a site of G. Then G containsan infiniteself-avoiding startingat v. PROPOSITION3.4. Suppose T is an infinitetreethat has at mostfinitely manytopologicalends and containsno core. Then all sites of T have finite degree.
PROOF. Suppose T is an infinitetreethathas k topologicalends (O < k
0, k ? 1 and v a branch pointin a componentC of X, let C(')(t, k), i = 1,..., nv(t,k), be a listingof ofC \ {v) thathave at least k sitesin thetranslatev + At. thosecomponents large; let ofbranchpoint,nv(t,k) 2 3 if t is sufficiently Fromthe definition A(t, k) be the set of graphsin which0 is a branchpointwith no(t, k) 2 3. Then fors > 0, the hypothesesofLemma 2 of [6] are satisfiedforV n At+s (in place of S), VA(tk) n As (in place of R) and C ()(t, k) (in place of C()) yielding IVA(tk)
n AsJ? k'IV n At+sJ.
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PERCOLATION AND FORESTS IN GRAPHS
Dividing by IA81and letting s -3
00, we
97
see that
P(VA(tk) X) < kjp(V, X) a.s. (3.2) Now A(t, k) increases to A as t -) o0, and p(V., X) is a.s. a measure, so lettingt -0oo in (3.2), we obtain p(VA,X) < k-p(V, X) a.s. Since k is arbitrary,this shows P(VA,X) = 0, so by (3.1) VA = O a.s. E1
in the lemmais relatedto the criterion(2.3) formembership The following MSF. labeledgraph withMSF F. Then LEMMA 3.6. Let G be an ambiguity-free path in F is locallyf-minimax. everyself-avoiding path in F, let u, v be sitesin y and let a PROOF. Let y be a self-avoiding be anotherpath fromu to v in G. Followingyuvfromu to v, thenfollowing a backwardfromv to u producesa circuit.If the longestbond b in this circuitappears in onlyone of yu and a, then eitherb is a loop bond or b b is criterion (2.2), so b is nota bondofF. Therefore, failsthe creek-crossing a bondof a and the lemmafollows.E1 LEMMA3.7. Suppose T is a componentof theMSF of an ambiguity-free labeledgraph G and b = (x, y> E dT, withx E T. Thenf(b) 2 r,(G). infinite self-avoiding If all sitesof T have finitedegreein T, thenthereis an infinite < e bonds in y. path y in T startingat x such thatf(e) f(b) forall PROOF. Let bo = b, x0 = x and yo = y, and let F be the MSF of G. Then bo 0 F, so thereexists a path yo in G fromx0 to yo consistingentirelyof bonds e withf(e) < f(b0). Let b1 = (x1, y1>be the firstbondin yo thatis in dT, withx1 E T; such a bondnecessarilyexistssince x0 E T and yo 0 T. By path fromx0 to xl lies in T and Lemma 3.6, the unique locallyf-minimax also consistsofbondse withf(e) < f(bo), so we mayassume the sectionofyo fromx0 to x1 is preciselythis f-minimax path;in particularthismeans this sectionof y0 lies in T. Similarly,thereexistsa path y1 in G fromx1 to y1 consistingentirelyofbondse withf(e) in y1 that is in dT, with the section of y1 fromx1 to x2 lyingin T; Let y be the path in T thisprocesscan be continuedindefinitely. inductively, that followsyo from x0 to x1, then yj from x1 to x2 and so on. Let S = y U {b1,b2,... }. Now thebonds bi are distinct,since f(b0) > f(b1) > so S is infiniteand connected,and all bonds e in S have f(e) < f(b). Therefore, f(b) ? rc(G). If all sitesin T have finitedegree,thensinceall bi are distinct,theremust be infinitely y containsan infiniteself-avoidmanysites xi in y. Therefore, ing path in T startingat x, consistingofbonds e withf(e) < f(b). :1
Suppose T is a two-endedtree.Then foreach site v in T thereis a unique trunksite, denotedz(v), such that everyinfinitepath in T startingfromv firstmeetsthe trunkat z(v).
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PROOFOF THEOREM2.5. Let F denotethe MSF of X. We may assume X is ergodic,so r,(X) = r, a.s. (i) The absenceofzero-endedtreesfollowsfromProposition3.2, appliedto F, and Lemma3.3. The absenceoftreeswiththreeor moreends followsfrom Proposition3.5 appliedto F. (ii) Suppose T is a two-endedtreeofthe MSF withtrunky. Let the bonds ofy be labeled as { ... ,e_1,eo, ej, ... } in the orderin whichtheyappear when followingy in an arbitrarily chosendirection, and let l+(T)
= limsupf(en), n-
+-o
I-(T)
=
limsupf(en). -X n
o
By Proposition3.1, foreach rational q thereis a.s. no firstor last bond en with f(en) > q, so we musthave l+(T) = I-(T); thus we denotethe common value by l(T). If f(en) > 1(T) forsome n, thenthereis an f-maximizing bond in y. However,by Proposition3.1 again, using the factthat all labels are thereis a.s. no f-maximizing distinct, bondin y and nobondwithf(en) = 1(T). < for all Therefore, 1(T) n; thus y is partofa infinitestrictl(T)-cluster f(en) in X and l(T) ? rc. By strictsimultaneousuniqueness,the infinitestrictl(T)-clusterin X is unique; we claim that T containsall sites ofthis cluster.If T = F, thereis nothingto prove,so suppose T # F and let b e dT. By Propositions3.2 (applied to F) and 3.4, all sites of T have finitedegreein T. Therefore, by Lemma 3.7 thereis an infiniteself-avoiding path in T consistingofbonds e withf(e) < f(b). An infiniteself-avoiding pathin T mustinclude{en: n ? m} or {en: n < m} forsome m, so it followsthat l(T) < f(b). Since b E dT is arbitrary,the infinitestrict l(T)-cluster cannot cross dT and the claim follows. Let us show that l(T) = rc. Suppose not and fix l(T) > r > rc. Since T containsall sites of C.t(X, l(T)), T also containsall sites of C.t(X, r). By Propositions3.2, 3.4 and 3.5, z-'(v) is finiteforeach trunksite v, so there must exist x, y E C.t(X, r) with z(x) # z(y). By uniqueness,C.t(X, r) is connected,so forany such x, y thereis a path f3in C.t(X, r) fromx to y. Thereis also a self-avoiding path a in T thatgoes fromx to z(x), thenvia y to z(y), thento y. By Lemma3.6, a is f-minimax. Since all bondsin 13have label less than r, the same mustbe trueforall bondsin a, so a is contained in C.t(X, r). It followsthat y n C.t(X, r) is a nonemptyconnectedsubsetof y. By Proposition3.1, this connectedsubset a.s. has no firstor last bond,so must be all of y. However,this would mean l(T) < r, contraryto our assumption.Hence no such r exists,thatis, l(T) = rc,so T containsall sites of an infinitestrictrc-cluster. Since thereis at most one such cluster,the theoremfollows.El PROOF OF THEOREM2.6. Suppose T is a tree of the MSF F of G, with T 0 F. Let d*T be the set ofdual bonds{b*: b E daT}.Since G is lattice-like, dT is infiniteand every componentof d*T is infinite.By Lemma 3.7,
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f(b) 2 r,(G) forall b E daT,so everycomponentof d*T is part ofan infinite clusterofoccupieddual bondsat level r,(G). EJ 4. Invasion percolation and optimal paths to infinity. The following is immediatefromProposition2.1, Lemma 3.6 and Theorem2.5. PROPOSITION4.1. For X as in Theorem2.5 with strictsimultaneous uniquenessholdinga.s., withprobability1 foreach site v thereexisteither If one or two self-avoiding locallyf-minimax paths in X fromv to infinity. percolation thereis no percolationat the criticalpoint in the corresponding model,thereis a uniquesuchpath foreach v.
path in X For X as in Theorem2.5, let yx. denotethe locallyf-minimax fromx to infinity whenthereis onlyone, and the unionofbothwhenthere makes yo, or are two.In the lattice/uniform model,the f-minimax property the treecontaining0, candidatesto be called an incipientinfinitecluster,but we have not investigatedhow their propertiesrelate to those of other candidatesthat have been put forthin the literature;see [7], Section7.4. In the Poisson/Euclideanmodel,the same holdswith0 replacedby the closest site to 0. Invasion percolation,introducedin the mathematicalliteraturein [5], is definedas followsin an ambiguity-free labeled graph. Let IO(x) = {x for somesite x. GivenIn(x), let An(x) be the set ofbondsnotin In(x) butwithat bond in An(x), least one endpointin In(x), let En+ 1(x) be the f-minimizing let In,1(x) = In(x) U {En+,(x)) and let I(x) = Un In(x). (Here bonds are viewedas containingthe sitesthatare theirendpoints.)Notethatthisdiffers fromPrim'salgorithm,the invasionproceduredescribedin Section2. More precisely,we call En+1(x) a backfillbond (withrespectto x) if it has both endpointsin In(x), and a breakoutbond ifit has onlyone endpointin In(x). The breakout bonds in I(x) are preciselythe bonds invaded in Prim's thatis, the bondsin J0(x). algorithm, THEOREM 4.2. Let X be as in Theorem2.5, with strictsimultaneous uniquenessholdinga.s. and letF denotetheMSF ofX. Thenwithprobability one:
(i) An invaded bond is a breakoutbond if and onlyif it is a bond ofF. (ii) For everyx, I=(x) containsylxo. I=(x)&IA,(y) is finiteifand onlyifx and y are (iii) The symmetric difference in thesame treeofF. For the square lattice, (iii) reproducesTheoremA.1 of [5], where,by Example 2.7, F consists of a single tree. Nonrigorousargumentsin [13] suggestthat forthe integerlattice in dimensiongreaterthan 8 thereis a forx # y that I4O(x)and I4(y) are disjoint,muchas the positiveprobability
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paths ofindependentrandomwalks startedat x and y can be disjoint.By (iii), thiswouldimplythatthe MSF is notconnectedin highdimensions. For A c G, let A = {b E G: b has an endpointin A}. PROOF OF THEOREM 4.2. We may assume that X is ergodicand, by Propositions3.2 and 3.4, that all sites have finitedegreein F. (2.1) that b is If b = is a breakoutbond,it followsfromthecriterion a bond of F. Conversely,if b is a backfillbond, then there is a path of breakoutbondsfromu to v. Since this path lies in F and F is acyclic,b is not a bondof F. This proves(i). If the treeT ofF thatcontainsx is one-endedand b is a bondofyz, then onlyfinitely manybondsof F can be reachedfromx via paths in F without passing throughb. Hence by (i) onlyfinitelymanybreakoutbonds can be b E Ij,(x) and (ii) followsforone-endinvadedwithoutinvadingb. Therefore, ed T. By Theorem2.5 and Proposition3.4, to prove(ii) in generalit remainsto considertwo-endedT, withall sites offinitedegreein T. The trunkconsists Let ao = z(x) and oftwo disjointpaths,say a and A, fromz(x) to infinity. let a,, a2, ... be the bonds of a and bl, b2, ... the bonds of 13,each listed startingfromz(x). From(i), I.(x) mustcontainat least one ofthese paths, say 13,and In(x) n (a U 18) is a singleintervalof a U 13foreach n. Suppose ao,..., ak1 are in I4x), with ak 1 invadedat some time T> 0 and I,(x) n (a U /3) = {akl, ... , a, ao, bl, ..., b1}forsome j. By Theorem 2.5, f(ak) < r, so thereis a.s. no infinitef(ak)-cluster;in particular,f(bm) > f(ak) forsome m > j. However,this means ak mustbe invadedbeforebi. Thus by induction,ak n I.(x) forall k. Hence I.(x) containsa U /8and (ii) followswhenT is a two-endedtreeas well. treesof F, thenit followsfrom Turningto (iii), if x and y are in different (i) that I.(x) n I.(y) = 4. Thus suppose x and y are in the same treeT of F. We considertwocases. Case 1: T is one-ended.Thereexists a site sxy,where x-> firstmeets yy., startingfromx or y. If sxyis neitherx nor y, then T\ {sxyl includesfinite Let Tx(or TY)be 4 if components Tx and Tycontainingx and y,respectively. x (or y) is sxy.Let 8n(x) be the nthbondinvaded,startingfromx, whichis bondin notin TxU Ty.Then 13n(x)is the f-minimizing b e G: b
OTXU TyU ({1(X),,...,
1(n-X)}
b has an endpointin Isxy, 131(x), **,
/3n-1( X) )}
if pi(x) = pi(y) forall 1 < i < n - 1, then 13n(x) = Pn(y). It folTherefore, lows that 13n(x)= 13n(Y)forall n and (4.1)
I4(X) t& 4(y) C TxU Ty,
Note that,althoughTx is finite,dTx and thus Tx may be infiniteif Tx has sites ofinfinitedegree.
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Fix b = (u, v) E dTX, with u e T,. Suppose b E I.(x); we claim that thereis a b E Ic,(y). By (i), since b t F, b mustbe a backfillbond.Therefore, path A fromu to v consistingentirelyofbreakoutbonds(withrespectto x); criterion (2.2) that b by(i), Ais a path in T. It followsfromthe creek-crossing is the longestbond in A U {b). Further,since v 0 Tx, A must pass through sxysand thereforeb is invadedaftersy. Letting 80(x) := {sxy},we let n 2 0 be such that b is invaded(startingfromx) afterpn(x) is invadedbut before 8n+1(x) is invaded.Then (4.2) f( fn+1(Ax))> f( b) > f(e) forall bondse in A. Now startingfromy, sy is invaded beforepn+ (y) = n+ 1(x) is invaded. Since sxyis a site of A,it followsfrom(4.2) thatstartingfromy, all of A U {b} will be invadedbeforein+ 1(Y),and our claimthat b E Ij(y) follows. Conversely,forthe same b E dTx, if b E Ic0(y),then virtuallythe same we have proofshows b E Ij(x). Therefore, (I(x)A I4(y)) n (dTxu dTy)= 4. With(4.1) this establishes(iii) forone-endedtrees. Case 2: T is a two-endedtree with trunk-y.Let AV denote the (finite) componentofa site v in T \ {z(v)), togetherwill all bondsof X whichhave bothendpointsin this component(so A, = (Aif v E y). We claimthat
(4.3) Qs'(X, rj) c I.(v) c CQt(Xj rj) U;AV. Let o- be the least n such that In(v) containsz(v); such an n always exists by (i). Since, by Theorem2.5 z(v) e C~t(X,rc), at all futuretimes i 2 or, Ai(v) includesa bond of C~t(X, rc). Thus everybond b outside A, invaded aftertime oahas f(b) < r, so in factall suchbondsare in C~t(X, r,), and the secondinclusionin (4.3) follows.For the firstinclusion,supposethereexistsa bond b E C~t(X, rc)\ I4o(v).Since by uniqueness Ct(X, r,) is connected,we can findsuch a b withat least one endpointin Ij(v), so b E An(v)forsome n. However,then f(e) < f(b) forall e E I(v) \ In(v). The graph consistingof such e has onlyfinitely manyconnectedcomponentsso it includesan infinite f(b)-cluster.Since f(b) < r, thereis no such infiniteclustera.s. It follows that CQt(X,rd)\ 1(v) = 4 a.s. This proves (4.3), which in turn shows oo(X)A 4(y)
is finite. El
COROLLARY 4.3. In the lattice/uniform model,foreveryr > rc and every sitex, at mostfinitely manybondsof yx. are outsidethe infiniter-clusterof thecorresponding percolationmodel,a.s. PROOF. The analogous fact for I4o(x)was proved in [5], so the result followsfromTheorem4.1(ii).In [5], rc is replacedby a percolationthreshhold fromrc,but it was provedin [8] thatthetwothresholdsare possiblydifferent the same. El
5. Stability of the MSF under local changes. We considernext the extentto whichlocal changesin a graphcan produceglobal changesin the
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labeled graphG in Rd, let us define MSF. Specifically, foran ambiguity-free the functionSG on Rd X Rd by ( x y) 1 if x and y are sitesin thesame treeoftheMSF ofG,
{0:otherwise.
Local changesin G maycause bondsto be added to or deletedfromtheMSF, and as a resultthe function8G maychange. To begin, we define some modificationsof the graph G = (V, a, f ). For F C Rd let Dr(G) = {x E V nl: (x, y > E- forsomey E FC}. Definethe restriction GIr of G to F by = { <x, y> EG: x, y e F), O~r Glr =(V n rF, r, f) off to Sire To Here,in a slightabuse ofnotation,f is actuallytherestriction of G to F U DrC(G). add in bondscrossingdF, let Gj denotethe restriction We say twolabeled graphsG1 and G2 agree outside F if Gljc = G2 rc. THEOREM 5.1. (i) Suppose G1 and G2 are ambiguity-freelocally finite labeled graphs in Rd that agree outside At for some t > 0, and let Fi be the MSF of Gi. Then the symmetricdifferenceF1 A F2 is a finiteset of bonds. (ii) Let X be as in Theorem 2.5, with strict simultaneous uniqueness holding a.s. There exists a set A of labeled graphs such that (a) P(X e A) = 1 and (b) if t > 0 and Gi = (Vi, i, fi) are labeled graphs in A that agree outside At, then there exists s > t such that 8Gl(X, y) = 8G2(X,y) for all x, y e As
Roughlyspeaking,(ii) says that changesin a finitebox can onlyshifta treesoftheMSF. Finitechangescannot,for finitenumberofsitesto different treeintotwoone-endedtreesorgluetwoone-ended example,splita two-ended treesintoa two-endedtree,exceptpossiblyby creatinga labeled graphofa typethata.s. does notoccur,thatis, one notin A. PROOFOF THEOREM5.1.
(i) For u, v distinct sites in DAt(Gi),define
:=inffr> 0: u is connectedto v at level r in GilAtK} Note DA(Gi) and GiLct do not dependon i. Suppose b = <x, y> is a bond of criterion (2.2), thereis GiIct thatis in F1 but not F2. Fromthe creek-crossing a path in G2 fromx to y consistingofbonds e withf(e) < f(b), but no such this path in G2 path existsin G1 and hence none existsin Gilt. Therefore, musthave a bondin G21At, so forsome pair u, v ofdistinctsites in DAI(Gi), thereexistdisjointpaths in G I+c fromx to u and fromy to v at somelevel less than f(b). Thus rUV< f(b). However,therecan be no pathfromu to v at a level less than f(b) in GiI ct,forotherwisewe wouldalso have such a path fromx to y. Thus ruv= f(b); since labels are distinctthereis at mostone such b foreach pair u, v. Since G1 and G2 are locallyfinite,(i) follows.
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(ii) We may assume X is ergodic,so by Theorem2.5 there exists an n E {0, 1} such thatthe MSF of X has exactlyn two-endedtrees,a.s. Let A be the set of graphsforwhichthe MSF has exactlyn two-endedtrees and in Theorem2.5(ii). If n = 0, thenforeach bond b in satisfiesthe description F1 \ F2, removingb fromF1 splitsthe tree of F1 containingb into a finite and an infinitepiece,and similarlyforb E F2 \ F1. If x and y are sites of X and neither x nor y is in one of these finitepieces, then 8G1(X,y) = 8G2(X,y). Thus forn = 0, (b) followsfrom(i). If n = 1, it followsfrom(i) thatto prove (b) we need onlyeliminatethe possibilitythatforsome site z in a one-ended treeT ofF1,thepath y,,.in T is containedin thetrunkofthetwo-endedtree of F2. However,thiswouldimplythatall bonds e in y%.have f(e) < rc,so T wouldmeetthe infinitestrictrc-cluster, so in G1 the infinitestrictrc-cluster wouldnotbe containedin the two-endedtree,meaningG1 0 A. El Withoutstrictsimultaneousuniqueness,Theorem5.1(ii) is false.The MSF of X maythenincludemultipletwo-endedtrees.Throughchangesin a finite box,it maybe possibleto in effect cutin halfthetrunksoftwosuchtreesand thenglue the fourresultingrays back togetherin a different pairing,as the following example,similarto Example 1.9 of[2], shows. EXAMPLE 5.2. Considera randomlabeled graphX in R2 withsite set Z2. Verticalbondsare nearestneighbor;horizontalbondsare longrange.Let us call the subgraphin the verticalline at m, columnm. Let {Um: m GEZ} be iid uniform[0,1] randomvariablesand let {Tl: j E ZZ}be the arrivaltimesofa timesnotidentically1. Let stationaryrenewalprocessin Z, withinterarrival us referto [?Tj, n Z as block on {Um: m E Z} and {T.: 1) j. Conditionally Tj, j E Z}, we label bonds as follows:For m in block j, verticalbonds ((m, k), in [0, Uj); form in blocki (m,k + 1)> in columnm getlabels Wnkiid uniform and n in blockj, each horizontalbond ((m, k), (n, k)> independently gets a label uniformin (max(Ui,Uj),1]. The discussionin Example 1.9 of[2] establishes that X does notsatisfysimultaneousuniqueness,strictor not,though forfixedr > 0 thereis a.s. a unique infiniteclusterat level r. Uniqueness failspreciselyat the levels Uj. It is easilyverifiedthatthe MSF of X consists manytwo-ended preciselyof all verticalbonds,so is composedof infinitely trees.We nowintroducelocal modifications a crossover to X. We can construct in a square [m, m + 1] x [k, k + 1] by deletingthe verticalbonds ((m, k), (m, k + 1)>, (m + 1, k + 1)> and adding diagonal bonds ((m, k), (m + 1, k + 1)> withlabel Wmkand ((m, k + 1), (m + 1,k)> withlabel Wm+ 1,k. Let {Zmk: m,k E ZZ}be idd takingvalues 0 and 1 withprobability1/2 each. Starting with the graph X, in each square [m, m + 1] x [k, k + 1] we constructa crossoverif m and m + 1 are in the same block,and Z- 1,k = 0, Zmk = 1 and Zm+ 1,k = 0. We denotethe resultinggraphY. Thus no twohorizontally adjacent cubes both have crossovers.The subgraphof Y consistingof premanydistinct ciselythe verticaland diagonalbondsis made up ofinfinitely infinitelines; let us call these lines strands.Note that a horizontalbond emanatingfroma givenstrandis longerthanany ofthebondsofthatstrand.
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Usingthecreek-crossing criterion (2.2) it followseasilythattheMSF ofY a.s. consistspreciselyofthe infinitecollectionofstrands.ComparingY itselfto Y witha singleadded crossover,we see that Theorem5.1 (ii)(b) does not hold. [1
REFERENCES [1] ALDOUS, D. and STEELE,J. M. (1992). Asymptotics forEuclideanminimalspanningtreeson randompoints.Probab.TheoryRelatedFields 92 247-258. random [2] ALEXANDER, K. S. (1994). Simultaneousuniquenessofinfinite clustersin stationary labeledgraphs.Comm.Math.Phys.To appear. K. S. (1994). The RSW theoremforcontinuumpercolationand the CLT for [3] ALEXANDER, Euclideanminimalspanningtrees.Unpublishedmanuscript. K. S. and MOLCHANOV, [4] ALEXANDER, S. A. (1994). Percolationof level sets fortwo-dimensionalrandomfieldswithlatticesymmetry. J. Statist.Phys.To appear. ofinvasion [5] CHAYES, J. T. CHAYES, L. and NEwMAN, C. M. (1985). The stochasticgeometry percolation. Comm.Math.Phys.101 383-407. component [6] GANDOLFI, A., KEANE,M. and NEwMAN, C. M. (1992). Uniquenessofthe infinite in a randomgraphwithapplicationsto percolationand spinglasses. Probab.Theory RelatedFields 92 511-527. G. (1989). Percolation.Springer,New York. [7] GRIMMETT, G. R. and MARSTRAND, J. M. (1990). The supercriticalphase ofpercolation is well [8] GRIMMETT,
behaved.Proc.Roy.Soc. LondonSer.A 430 439-457. in highdimenT. and SLADE,G. (1990. Mean-field [9] HARA, criticalbehaviourforpercolation sions.Comm.Math.Phys.128 333-391. T. and SLADE,G. (1993). Personalcommunication. [10] HARA, in a certainpercolation [11] HARRIS,T. E. (1960). A lowerbound forthe criticalprobability process.Proc.CambridgePhil. Soc. 56 13-20. [12] NEVEU, J. (1977). Processus ponctuels.Ecole d'Et6 de Probabilitesde Saint-FlourVL. LectureNotesin Math.598 249-447. Springer,New York. ground [13] NEwMAN, C. M. and STEIN,D. L. (1994). Spinglass modelwithdimension-dependent statemultiplicity. Phys.Rev.Lett.72 2286-2289. Ann.Probab. [14] PEMANTLE, R. (1991). Choosinga spanningforthe integerlatticeuniformly. 19 1559-1574. An Introduction. [15] PREPARATA, F. P. and SHAMOS, M. I. (1985). ComputationalGeometry: Springer,New York. Bell Syst.Tech [16] PRIM,R. C. (1957). Shortestconnecting networksand somegeneralizations. J. 36 1389-1401. DEPARTMENT OF MATHEMATICSDRB 155 UNIVERSITY OF SOUTHERN CALIFORNIA Los ANGELES, CALIFORNIA90089-1113
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The Annals ofProbability 1995, Vol. 23, No. 1, 87-104
PERCOLATION AND MINIMAL SPANNING FORESTS IN INFINITE GRAPHS' BY KENNETH S. ALEXANDER Universityof Southern California The structureof a spanningforestthat generalizesthe minimal spanningtreeis consideredforinfinite graphswitha value f(b) attached to each bond b. Of particularinterestare stationaryrandomgraphs; values f(b) and theVoronoior examplesincludea latticewithiid uniform completegraphon thesitesofa Poissonprocess,withf(b) thelengthofb. The corresponding percolation modelsare Bernoullibondpercolation and the "lilypad" modelof continuumpercolation, It is shown respectively. that under a mild "simultaneousuniqueness"hypothesis, withat most one exception, each treein the foresthas one topologicalend,thatis, has no doublyinfinite paths.Ifthereis a treein theforest, necessarilyunique, withtwotopologicalends,it mustcontainall sitesofan infinite clusterat the criticalpointin the corresponding model.Treeswithzero, percolation or threeor more,topologicalends are not possible.Applicationsto invasion percolationare given.If all trees are one-ended,thereis a unique fromeach site. optimal(locallyminimaxforf) pathto infinity
1. Introduction. For a finiteset V c d a Euclidean minimalspanning tree(MST) of V is a tree withsite (that is, vertex)set V and minimaltotal lengthofall bonds(thatis, edges).More generally,givena finitegraphwith site set V and bond set A, and a labeling functionf: M
-*
[0, oo),a minimal
spanningtree of (V, M, f) is a tree in (V, M) spanningV with EbEf(b)
minimal among all such trees; (V, A, f) determines a labeled graph. It is
natural to ask whetherthere is a structureanalogous to the MST when (V, A, f) is an infinitelabeled graph,and if so to considerits properties, especiallyforrandomlabeled graphs.Two particularcases ofinterestare: the lattice/ uniformmodel: (V, M) is a latticein Rd and {f(b), b EA } are iid random variablesuniformin [0,1] and the Poisson/ Euclidean model: (V, M) is the complete graphon the set ofsites of a Poissonprocessand f is Euclideanlength. ReceivedDecember1993;revisedApril1994. DMS-92-06139. 'ResearchsupportedbyNSF Grant, 'AMS 1991 subjectclassifications. Primary60K35; secondary82B43,60D05, 05C80. invasion continuum Keywordsand phrases.Minimalspanningtree,percolation, percolation, percolation.
87
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In the latter case one can allow more general stationarypoint processes, / Euclidean model. yieldingthe stationary The natural way to definesuch an MST analog is to finda propertyof bondsin a labeled graphwhich(i) in finitegraphs,characterizesmembership then graphsas well.This property in theMST and (ii) makes sense in infinite of membershipin the MST analog. One such definibecomesthe definition in [16] of the MST in finite tion, based on Prim's inductiveconstruction graphs,was used by Aldous and Steele [1] for the stationary/Euclidean model;it yieldsa "minimalspanningforest"(MSF) in whicheverycomponent is equivadespiteappearingquite different, tree.Our definition, is an infinite lent to theirsin a large class of models,whichincludesthe lattice/uniform and Poisson/Euclidean. The main structureofan infinitetreeis givenbyits numberoftopological paths fromany fixed ends, which is the numberof infiniteself-avoiding vertex.Thus a zero-endedinfinitetreemustcontainsites ofinfinitedegree,a a two-endedtree one-endedtreeis like an infinitesystemofrivertributaries, consistsof a single doublyinfinitepath (the trunk)with finitebranches emanatingfromit and a treewiththreeor moreends mustcontainat least one branchpoint,thatis, a pointfromwhichthereare at least threedisjoint paths. Aldous and Steele [1] conjecturedthat forthe infiniteself-avoiding Poisson/Euclideanmodel,theirMSF consistsofa singleone-endedtree. to any infiniterandomlabeled graph X = (V, M, f), there Corresponding is a percolationmodel as follows.We say a bond b is occupiedat levelr if f(b) < r. Let X, x, y ED=d, with( x, y> and ( y, x > identified. Such a graph can include sites of infinitedegree,but a.s. has onlyfinitely manysites in each boundedregion;the asymptoticdensityof sites may be infinite. Multiplebondsbetweena fixedpair ofsites are notallowed,but one can obtainresultsforgraphswithsuch multiplebondsby deletingall bonds betweeneach fixedpair x and y except the one with the smallest label. Bonds ofform( x, x>, called loop bonds,are allowed.We let V denotethe set of sites, M the set of bonds and f the labelingfunction,so that X can be Rd maybe replacedthroughwiththe triple(V, A, f). Alternately, identified out by a lattice L, withtranslationallowedby elementsof L only;without furthermention,we use this formulationwhen appropriate,as in the model. lattice/uniform To ensurethatthe graphsG = (V, A, f) we deal withhave a well-defined (1) all bond unique MST or MSF, we will make twoassumptionsthroughout: labels are distinctand (2) foreverycomponentC and everyfiniteproper bondamongall bondsthat subset A of V n C, thereis a unique f-minimizing conditionfor(2) connectsitesin A to sitesin V \ A. Assuming(1), a sufficient
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is thateverysite has finitedegreein G, r foreveryr; thisis satisfieda.s. in modelprovided model and in the stationary/Euclidean the lattice/uniform the stationarypointprocessofsites is locallyfinite.We call G locallyfiniteif V n R is finitefor all bounded regions R and call a labeled graph that satisfies (1) and (2) ambiguity-free.
f) and a subgraphH, define Givena graphG = (V, AW, dH:= {b = (x, y> Em': x e H, y 4 H}.
we willcall f(b) the lengthof b and referto In a mildabuse ofterminology bonds as shorter,longest,and so forth;this should not cause confusion because we never make use of Euclidean lengthof bonds exceptwhen it coincideswiththe labelingby f. The MST in finitegraphscan be constructedby an inductive"invasion" procedureknownas Prim'salgorithm[16]; the same procedurewas used by Aldousand Steele [11to definetheirMSF forcertaininfinitegraphs.Specifif) be a labeledgraphand let v E V be a site.Let Jo(v) = {v} cally,let (V, AW,
and, given Jn(v), let Jn+ (v) be Jn(v) together with the shortest (that is, f-minimizing) bond Bn 1 in 9Jn(v), including the endpoint of BnI 1 in V \ Jn(v). Let J0,(v) U n 2 0Jn(V). We say a bond b is invaded from v if b E Joo(v). path in a graphand u, v sitesin y, let zag denotethe For y a self-avoiding
path y in a labeled graph segmentof y fromu to v. We say a self-avoiding
(V, IW,f) is locally f-minimax if for every pair u, v of sites in y and every path a fromu to v,
max(f(b): b E ZaJ< max( f(b): b Era). All the equivalences To defineour MSF, we willneed the nextproposition. are well knownin the case offinitegraphs;see [15]. For infinitegraphs,we essentiallyneed onlyverifythatexistingideas forfinitegraphscan establish equivalenceamong(2.1)-(2.5) withoutusingequivalenceto (2.6) or finiteness of the graph. The proofsof.this and all results in this sectionappear in Section3. Let G = (V, A, f) be a (finiteor infinite)ambiguity-free labeled graph and let b = <x, y> _Wwith x = y. The following are equivalent: PROPOSITION 2.1.
(2.1)
There exists a set A of sites such that b is the shortestbond fromAtoV\A.
(2.2) (2.2)
There exists no path from x to y with all bonds strictly shorterthan b.
(2.3)
b is a bond in some locally f-minimaxpath.
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The followingare equivalent: (2.4)
b is invaded eitherfromx or fromy.
(2.5)
x and y are in distinctstrictf(b)-clusters and these clusters are not both infinite.
If strictsimultaneous uniqueness holds for G, then (2.1)-(2.5) are all equivalent. If G is finite,then (2.1)-(2.5) are each equivalent to (2.6)
b is a bond of the MST of G.
Again,shortermeans havinga smallerlabel. Equivalenceof(2.4) and (2.6) the forfinitegraphs is the basis of Prim's algorithm[151 forconstructing MST and showsthatthe set ofinvadedbondsdoes notdependon the starting criterion, by analogyto a site in finitegraphs.We call (2.2) the creek-crossing hikertryingto crossa creekby steppingfromstoneto stone,avoidinggetting wet by nevertakingan available step if a path of all strictlyshortersteps exists. Aldous and Steele [1] defineda spanningforestconsistingof all bonds satisfying(2.4) and provedthat (2.4) and (2.5) are equivalent.We preferto that is, we definethe criterionas our definition, use the creek-crossing minimal spanning forest (or MSF) of an ambiguity-freelabeled graph G
(V, A, f) to be the graphwithsite set V and bondset
=
{b = (x, y> e A: thereis no pathfromx to y
consistingentirelyofbondse withf( e) < f( b)}. Because simultaneousuniquenessplays a majorrole in thiswork,we will conditionforit from[2]. For this we need the notionof give a sufficient positivefiniteenergy:fora full definitionin our context,see [2]; the idea on the appears in [6]. Loosely,positivefiniteenergymeans that conditioning about graph outside a finitebox, togetherwith certainpartial information bonds crossingthe box boundary,a.s. yields a nonzeroprobabilitythat all sites withinthat box are.connectedat a givenlevel r, at least forr large enoughthat percolationoccursat level r. LettingAt denote[- t, tVd, let us definethe site densityof a stationary random graph X = (V, A, f) to be
lim.IV n Ati/lAt i,
t X-.0
where [X]denotescardinalityforfinitesets and volumeforregionsin Stationarityensuresthatthislimitexistsa.s.
Rd.
PROPOSITION2.2 ([2], Theorem 1.8 and Remark 1.10). Suppose X is a stationary random labeled graph in Rd, with positive finiteenergyand finite site density a.s. Then both simultaneous uniqueness and strictsimultaneous uniqueness hold forX, a.s.
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and Poisson/Euclideanmodelsclearlyhave positive The lattice/uniform finiteenergyand finitesite density,so Proposition2.2 showsthat forthese (2.4) used byAldous (2.2) is equivalentto the definition models,ourdefinition and Steele [1]. The distinctionbetweenstrictand nonstrictsimultaneousuniquenessis by way we will explicatethe distinction importantin our results.Therefore, of the followingresult,thoughProposition2.2 makes it nonessentialto our main results. LEMMA 2.3. (i) Suppose G is an infinitelabeled graph with all labels distinct. If strictsimultaneous uniqueness holds for G, then so does simultaneous uniqueness. (ii) Suppose X is a stationary random labeled graph in Rd that a.s. has finitesite densityand all labels distinct. Then withprobability 1, simultaneous uniqueness holds forX if and only if strictsimultaneous uniqueness holds forX.
The assumptionofall labels distinctcannotbe eliminatedin Lemma2.3. It is easilyverifiedthatin the stationaryrandomlabeled graphofExample 1.9 of[2], strictsimultaneousuniquenessholdsa.s. butnotsimultaneousuniqueness. See also Example 5.2 below. (2.2) thatthelongestbond criterion It is immediatefromthecreek-crossing in any cyclein a labeled graphG is notin the MSF. Thus the MSF is acyclic. connectedlabeled Criterion(2.1) ensures that in an infiniteambiguity-free graph,everycomponentof the MSF is infinite.In particular,there are no meaningtheMSF spans thesite set V. This provesthe one-pointcomponents, whichjustifiesthe name "minimalspanningforest." following, LEMMA 2.4. In an ambiguity-freelabeled graph G = (V, A, f) with all componentsinfinite,the MSF is a forestthat spans V and consists of infinite trees.
way of obtaininga random For an infinitelattice,a completelydifferent spanningforestis consideredin [141. Let C0,(G,r) denotethe unionofall infiniter-clustersin the graphG and let C.t(G, r) be the union of all infinitestrictr-clusters.Here is our main result. THEOREM 2.5. Suppose X is a stationaryrandom labeled graph in Rd that a.s. is ambiguity-free,has finitesite densityand has all componentsinfinite. Then withprobability 1:
(i) The MSF contains no zero-ended treesor trees with threeor more ends. (ii) If simultaneous uniqueness holds forX, then the MSF includes at most one two-endedtree; all othertrees are one-ended. If a two-endedtree T exists, then thereis percolation at level r (X) in the correspondingpercolation model,
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T contains all sites of CQt(X, rc(X))
and the trunk of T is contained in
CQ'(X9 rc(X)).
If X is ergodic,thenthereis a nonrandomalmost-surevalue ofthe critical
point rj(X), which we denote rc.
Considera planar graph G, withoutloop bonds, embeddedin R2. The bonds, viewed as curves in R2, divide the plane into faces, which are connectedcomponents ofthe complement ofthe graph.We call G latticelikeif (i) G is locallyfinite,(ii) everysite in G has finitedegree,(iii) G is connected, (iv) G is planar,(v) G has no loopbondsand (vi) everyfaceis bounded.For a lattice-likegraph, a dual graph, also planar, is obtainedby selectingan arbitrary "facesite"in each faceand then,foreach bond b thatformspartof the boundarybetweentwodistinctfaces,puttinga dual bond b* betweenthe face sites in these faces.For G labeled,a dual bond b* is said to be strictly occupiedat level r ifand onlyif f(b) 2 r. THEOREM2.6.
Suppose G is an ambiguity-free lattice-likelabeled graph in
R2. If the MSF of G consists of more than one tree,then thereis percolation of
strictlyoccupied dual bonds at level rc(G) in the correspondingpercolation model.
EXAMPLE lattice,it 2.7. For the lattice/uniform modelon the hypercubic large d [91thatthere is knownfordimensionsd = 2 [11] and forsufficiently percolationmodel; is no percolationat the criticalpointin the corresponding large"[10]. Therefore, thebestresultat presentis that d 2 19 is "sufficiently all treesin the MSF are one-ended.For d = 2, wherethe latticeand its dual thereis also no percolationin the dual graphat level rc [111, are isomorphic, so the MSF consistsofa singleone-endedtree,as was provedin [4]. EXAMPLE 2.8. For the two-dimensional model,in stationary/Euclidean place ofthe completegraphon the site set V, one may considerthe Voronoi graph, definedas follows.Let d(, ) denote Euclidean distance.Given V, dividethe plane intothe polygonalcells Q(v) := y e R2: d(y, v) < d(y, x) forall xEVV},
vEV.
The Voronoigraph has site set V and bondset
{ (u, v:
u, v E V, Q(u) and Q(v) have an edgein common),
labeled by Euclideanlength.It is easily checkedthatonlyVoronoibondscan satisfy(2.1) or (2.2), so the MSF is the same fortheVoronoigraphas forthe completegraph.This is well knownforfinitegraphs;see [15]. The Voronoi so Theorem2.6 can be applied.Percolationat level r graphis a.s. lattice-like, in the graphdual to the Voronoigraphis equivalentto percolationofvacant blobmodelat level r (radius r/2). In the Poisson space in the corresponding case, it is provedin [3] thatthereis no percolationofvacantor occupiedspace
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at level rc. Therefore,the MSF consists of a single one-endedtree, as conjecturedbyAldousand Steele [1]. 3. Proofs of the main results. PROOF OF PROPOSITION2.1. We will show (2.1) < (2.2), (2.2) < (2.3) and, understrictsimultaneousuniqueness,(2.2) < (2.5). The equivalence(2.4) < (2.5) is provedin [1]. The equivalence(2.1) < (2.6) forfinitegraphscan be foundin [15]. Suppose firstthat(2.1) holds and x E A, y E V \A. If y is a path fromx to y, theny includesa bond e fromA to V\A. By (2.1), f(e) 2 f(b). Since y is arbitrary,(2.2) follows.Converselysuppose (2.2) holds and let A be the strict f(b)-clustercontainingx. By (2.2), y E V\A and by definitionof "strictf(b)-cluster," thereis no bond shorterthan b fromA to V\A. Thus (2.1) holdsand we have (2.1) (2.2). Next suppose (2.3) holds. Suppose a is a path fromx to y. Then by definitionof "locallyf-minimax," max{f(e): e Eca) 2 f(b). Since a is arbitrary,(2.2) holds.Conversely,(2.2) says that b by itselfconstitutesa locally f-minimax path.Thus (2.3) < (2.2). Now (2.2) says that x and y are in distinctstrictf(b)-clusters.Under strictsimultaneousuniqueness,these clustersare necessarilynot bothinfinite,so (2.2) and (2.5) are equivalent. [1
For a graphG in Rd and x E Rd., let OG denotethe translationof G by -x; thusforeverysite v of G0 QvGhas a site at 0. The proofof Theorem2.5 is based principallyon the idea that certain and possiblestructuresin labeled graphsare prohibiteda.s. by stationarity finitesite density.We beginwithfourpropositionson this theme.The first says,loosely,thatanythingthathappensonlyfinitely manytimesperinfinite clusteractuallyneverhappens. PROPOSITION 3.1 ([2]). Suppose X = (V, ~, f) is a stationary random labeled graph in Rd withfinitesite density,and A is a set oflabeled graphs G in which the origin is a site in an infinitecomponentof G. Suppose that with probability 1, thereare onlyfinitelymany sites v in each infinitecomponentof X for which QvX e A. Then e V] =0. P[QvXeAforsomev
PROOFOF LEMMA2.3. (i) Suppose G includestwo disjointinfiniter-clusters forsome r. Since thereis at mostone bond in G withlabel r, each of these two r-clusterscontainsan infinitestrictr-cluster,so strictsimultaneous uniqueness fails.
(ii) We may assume the randomgraphis ergodic,so rc(X) = rc a.s. Suppose that in the graph X, simultaneousuniqueness holds. We may
assume all labels are distinct. For each r > rc, M(r) := U 8< rCoo(X,s) is an infinitestrictr-cluster;we call any otherinfinitestrictr-clusterextraneous.
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For r > rc we call an infinitestrictr-clusterhollowifit containsno infinite s-cluster,for every s < r. Thus every extraneousinfinitestrictclusteris hollow.For each site v of X let r.(v) = infIr> rc: v E CO(X,r)}. If v is a site ofan extraneousinfinitestrictr-clusterforsome r > rc,thenv t CO(X,s) for any s < r, so r (v) = r; in particular,thereis at mostone such r foreach site v ofX. Therefore, distinctextraneousinfinitestrictclustersare disjoint,even ifthe corresponding values of r are distinct. If C is an extraneousinfinitestrictr-clusterforsome r > rc,then since C M(X, theremustbe a bond b = (x, y) E dC withx E C and r) is connected, = f(b) r. Since labels are distinct,thereis at mostone suchchoiceofb, x and y; we call b the attachmentbond and x the attachment site of C. To detach all extraneousclustersfromeach other,we wish to replace the attachment bond <x, y) withthe loop bond <x, x). We call <x, x) the alteredattachment bond of C and giveit the same label r as the attachmentbond.Let Y be the stationaryrandom labeled graph whose componentsare the extraneous infinitestrictclustersin X togetherwiththeiralteredattachmentbonds.In each componentof Y, the altered attachmentbond is the unique longest bond. Let A be the set of labeled graphsin which0 is an endpointof the unique longestbond in its connectedcomponent.If v is a site of Y, then OvYe A if and onlyif v is the alteredattachmentsite of some extraneous infinitestrictclusterin X. Hence O Y E A forexactlyone site v in each infiniteclusterin Y. By Proposition3.1, we may assume thereare no sites v with OvYE A; this means Y is empty,so there are no extraneousinfinite strictclustersin X. It remainsto establishuniquenessofthe strictrc-cluster. AnotherapplicationofProposition3.1 and the distinct-labels propertyshowsthat thereare a.s. no bonds b in X withf(b) = rc.However,thismeans everyinfinitestrict is also an infiniterc-cluster, and thereis at mostone ofthe latter. rc-cluster El
If F is a finiteset of sites in a singlecomponentof a graph G, we write C(G, F) forthis componentand let G \ F denotethe subgraphof G obtained by deleting F and all bonds emanating from F. We write C(G, v) for C(G, {v}). We call such a finiteF a core if thereare infinitely manyfinite componentsin G \F that are containedin C(G, F); that is, removingF splitsoffinfinitely manynew finiteclusters.An exampleis furnishedby the "infinite-spoked bicyclewheel,"in whicha countablenumberof "rim sites" are locatedon somecircle,and thereare twoother"hub-end"sitesnoton this circle;thereis a bondbetweeneach hub-endsite and each rimsite.The two hub-endsites thenforma core.Neitherhub-endsite is a coreby itself. PROPOSITION 3.2 ([2]). SupposeX is a stationary randomlabeledgraph in Rd withfinitesitedensity.Then withprobability1, X containsno core.
The easy proofofthe following lemmais containedin the proofofLemma 2.3 of[2].
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LEMMA3.3. Suppose G is an infiniteconnectedgraph that containsno path core, and v is a site of G. Then G containsan infiniteself-avoiding startingat v. PROPOSITION3.4. Suppose T is an infinitetreethat has at mostfinitely manytopologicalends and containsno core. Then all sites of T have finite degree.
PROOF. Suppose T is an infinitetreethathas k topologicalends (O < k
0, k ? 1 and v a branch pointin a componentC of X, let C(')(t, k), i = 1,..., nv(t,k), be a listingof ofC \ {v) thathave at least k sitesin thetranslatev + At. thosecomponents large; let ofbranchpoint,nv(t,k) 2 3 if t is sufficiently Fromthe definition A(t, k) be the set of graphsin which0 is a branchpointwith no(t, k) 2 3. Then fors > 0, the hypothesesofLemma 2 of [6] are satisfiedforV n At+s (in place of S), VA(tk) n As (in place of R) and C ()(t, k) (in place of C()) yielding IVA(tk)
n AsJ? k'IV n At+sJ.
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PERCOLATION AND FORESTS IN GRAPHS
Dividing by IA81and letting s -3
00, we
97
see that
P(VA(tk) X) < kjp(V, X) a.s. (3.2) Now A(t, k) increases to A as t -) o0, and p(V., X) is a.s. a measure, so lettingt -0oo in (3.2), we obtain p(VA,X) < k-p(V, X) a.s. Since k is arbitrary,this shows P(VA,X) = 0, so by (3.1) VA = O a.s. E1
in the lemmais relatedto the criterion(2.3) formembership The following MSF. labeledgraph withMSF F. Then LEMMA 3.6. Let G be an ambiguity-free path in F is locallyf-minimax. everyself-avoiding path in F, let u, v be sitesin y and let a PROOF. Let y be a self-avoiding be anotherpath fromu to v in G. Followingyuvfromu to v, thenfollowing a backwardfromv to u producesa circuit.If the longestbond b in this circuitappears in onlyone of yu and a, then eitherb is a loop bond or b b is criterion (2.2), so b is nota bondofF. Therefore, failsthe creek-crossing a bondof a and the lemmafollows.E1 LEMMA3.7. Suppose T is a componentof theMSF of an ambiguity-free labeledgraph G and b = (x, y> E dT, withx E T. Thenf(b) 2 r,(G). infinite self-avoiding If all sitesof T have finitedegreein T, thenthereis an infinite < e bonds in y. path y in T startingat x such thatf(e) f(b) forall PROOF. Let bo = b, x0 = x and yo = y, and let F be the MSF of G. Then bo 0 F, so thereexists a path yo in G fromx0 to yo consistingentirelyof bonds e withf(e) < f(b0). Let b1 = (x1, y1>be the firstbondin yo thatis in dT, withx1 E T; such a bondnecessarilyexistssince x0 E T and yo 0 T. By path fromx0 to xl lies in T and Lemma 3.6, the unique locallyf-minimax also consistsofbondse withf(e) < f(bo), so we mayassume the sectionofyo fromx0 to x1 is preciselythis f-minimax path;in particularthismeans this sectionof y0 lies in T. Similarly,thereexistsa path y1 in G fromx1 to y1 consistingentirelyofbondse withf(e) in y1 that is in dT, with the section of y1 fromx1 to x2 lyingin T; Let y be the path in T thisprocesscan be continuedindefinitely. inductively, that followsyo from x0 to x1, then yj from x1 to x2 and so on. Let S = y U {b1,b2,... }. Now thebonds bi are distinct,since f(b0) > f(b1) > so S is infiniteand connected,and all bonds e in S have f(e) < f(b). Therefore, f(b) ? rc(G). If all sitesin T have finitedegree,thensinceall bi are distinct,theremust be infinitely y containsan infiniteself-avoidmanysites xi in y. Therefore, ing path in T startingat x, consistingofbonds e withf(e) < f(b). :1
Suppose T is a two-endedtree.Then foreach site v in T thereis a unique trunksite, denotedz(v), such that everyinfinitepath in T startingfromv firstmeetsthe trunkat z(v).
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PROOFOF THEOREM2.5. Let F denotethe MSF of X. We may assume X is ergodic,so r,(X) = r, a.s. (i) The absenceofzero-endedtreesfollowsfromProposition3.2, appliedto F, and Lemma3.3. The absenceoftreeswiththreeor moreends followsfrom Proposition3.5 appliedto F. (ii) Suppose T is a two-endedtreeofthe MSF withtrunky. Let the bonds ofy be labeled as { ... ,e_1,eo, ej, ... } in the orderin whichtheyappear when followingy in an arbitrarily chosendirection, and let l+(T)
= limsupf(en), n-
+-o
I-(T)
=
limsupf(en). -X n
o
By Proposition3.1, foreach rational q thereis a.s. no firstor last bond en with f(en) > q, so we musthave l+(T) = I-(T); thus we denotethe common value by l(T). If f(en) > 1(T) forsome n, thenthereis an f-maximizing bond in y. However,by Proposition3.1 again, using the factthat all labels are thereis a.s. no f-maximizing distinct, bondin y and nobondwithf(en) = 1(T). < for all Therefore, 1(T) n; thus y is partofa infinitestrictl(T)-cluster f(en) in X and l(T) ? rc. By strictsimultaneousuniqueness,the infinitestrictl(T)-clusterin X is unique; we claim that T containsall sites ofthis cluster.If T = F, thereis nothingto prove,so suppose T # F and let b e dT. By Propositions3.2 (applied to F) and 3.4, all sites of T have finitedegreein T. Therefore, by Lemma 3.7 thereis an infiniteself-avoiding path in T consistingofbonds e withf(e) < f(b). An infiniteself-avoiding pathin T mustinclude{en: n ? m} or {en: n < m} forsome m, so it followsthat l(T) < f(b). Since b E dT is arbitrary,the infinitestrict l(T)-cluster cannot cross dT and the claim follows. Let us show that l(T) = rc. Suppose not and fix l(T) > r > rc. Since T containsall sites of C.t(X, l(T)), T also containsall sites of C.t(X, r). By Propositions3.2, 3.4 and 3.5, z-'(v) is finiteforeach trunksite v, so there must exist x, y E C.t(X, r) with z(x) # z(y). By uniqueness,C.t(X, r) is connected,so forany such x, y thereis a path f3in C.t(X, r) fromx to y. Thereis also a self-avoiding path a in T thatgoes fromx to z(x), thenvia y to z(y), thento y. By Lemma3.6, a is f-minimax. Since all bondsin 13have label less than r, the same mustbe trueforall bondsin a, so a is contained in C.t(X, r). It followsthat y n C.t(X, r) is a nonemptyconnectedsubsetof y. By Proposition3.1, this connectedsubset a.s. has no firstor last bond,so must be all of y. However,this would mean l(T) < r, contraryto our assumption.Hence no such r exists,thatis, l(T) = rc,so T containsall sites of an infinitestrictrc-cluster. Since thereis at most one such cluster,the theoremfollows.El PROOF OF THEOREM2.6. Suppose T is a tree of the MSF F of G, with T 0 F. Let d*T be the set ofdual bonds{b*: b E daT}.Since G is lattice-like, dT is infiniteand every componentof d*T is infinite.By Lemma 3.7,
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f(b) 2 r,(G) forall b E daT,so everycomponentof d*T is part ofan infinite clusterofoccupieddual bondsat level r,(G). EJ 4. Invasion percolation and optimal paths to infinity. The following is immediatefromProposition2.1, Lemma 3.6 and Theorem2.5. PROPOSITION4.1. For X as in Theorem2.5 with strictsimultaneous uniquenessholdinga.s., withprobability1 foreach site v thereexisteither If one or two self-avoiding locallyf-minimax paths in X fromv to infinity. percolation thereis no percolationat the criticalpoint in the corresponding model,thereis a uniquesuchpath foreach v.
path in X For X as in Theorem2.5, let yx. denotethe locallyf-minimax fromx to infinity whenthereis onlyone, and the unionofbothwhenthere makes yo, or are two.In the lattice/uniform model,the f-minimax property the treecontaining0, candidatesto be called an incipientinfinitecluster,but we have not investigatedhow their propertiesrelate to those of other candidatesthat have been put forthin the literature;see [7], Section7.4. In the Poisson/Euclideanmodel,the same holdswith0 replacedby the closest site to 0. Invasion percolation,introducedin the mathematicalliteraturein [5], is definedas followsin an ambiguity-free labeled graph. Let IO(x) = {x for somesite x. GivenIn(x), let An(x) be the set ofbondsnotin In(x) butwithat bond in An(x), least one endpointin In(x), let En+ 1(x) be the f-minimizing let In,1(x) = In(x) U {En+,(x)) and let I(x) = Un In(x). (Here bonds are viewedas containingthe sitesthatare theirendpoints.)Notethatthisdiffers fromPrim'salgorithm,the invasionproceduredescribedin Section2. More precisely,we call En+1(x) a backfillbond (withrespectto x) if it has both endpointsin In(x), and a breakoutbond ifit has onlyone endpointin In(x). The breakout bonds in I(x) are preciselythe bonds invaded in Prim's thatis, the bondsin J0(x). algorithm, THEOREM 4.2. Let X be as in Theorem2.5, with strictsimultaneous uniquenessholdinga.s. and letF denotetheMSF ofX. Thenwithprobability one:
(i) An invaded bond is a breakoutbond if and onlyif it is a bond ofF. (ii) For everyx, I=(x) containsylxo. I=(x)&IA,(y) is finiteifand onlyifx and y are (iii) The symmetric difference in thesame treeofF. For the square lattice, (iii) reproducesTheoremA.1 of [5], where,by Example 2.7, F consists of a single tree. Nonrigorousargumentsin [13] suggestthat forthe integerlattice in dimensiongreaterthan 8 thereis a forx # y that I4O(x)and I4(y) are disjoint,muchas the positiveprobability
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paths ofindependentrandomwalks startedat x and y can be disjoint.By (iii), thiswouldimplythatthe MSF is notconnectedin highdimensions. For A c G, let A = {b E G: b has an endpointin A}. PROOF OF THEOREM 4.2. We may assume that X is ergodicand, by Propositions3.2 and 3.4, that all sites have finitedegreein F. (2.1) that b is If b = is a breakoutbond,it followsfromthecriterion a bond of F. Conversely,if b is a backfillbond, then there is a path of breakoutbondsfromu to v. Since this path lies in F and F is acyclic,b is not a bondof F. This proves(i). If the treeT ofF thatcontainsx is one-endedand b is a bondofyz, then onlyfinitely manybondsof F can be reachedfromx via paths in F without passing throughb. Hence by (i) onlyfinitelymanybreakoutbonds can be b E Ij,(x) and (ii) followsforone-endinvadedwithoutinvadingb. Therefore, ed T. By Theorem2.5 and Proposition3.4, to prove(ii) in generalit remainsto considertwo-endedT, withall sites offinitedegreein T. The trunkconsists Let ao = z(x) and oftwo disjointpaths,say a and A, fromz(x) to infinity. let a,, a2, ... be the bonds of a and bl, b2, ... the bonds of 13,each listed startingfromz(x). From(i), I.(x) mustcontainat least one ofthese paths, say 13,and In(x) n (a U 18) is a singleintervalof a U 13foreach n. Suppose ao,..., ak1 are in I4x), with ak 1 invadedat some time T> 0 and I,(x) n (a U /3) = {akl, ... , a, ao, bl, ..., b1}forsome j. By Theorem 2.5, f(ak) < r, so thereis a.s. no infinitef(ak)-cluster;in particular,f(bm) > f(ak) forsome m > j. However,this means ak mustbe invadedbeforebi. Thus by induction,ak n I.(x) forall k. Hence I.(x) containsa U /8and (ii) followswhenT is a two-endedtreeas well. treesof F, thenit followsfrom Turningto (iii), if x and y are in different (i) that I.(x) n I.(y) = 4. Thus suppose x and y are in the same treeT of F. We considertwocases. Case 1: T is one-ended.Thereexists a site sxy,where x-> firstmeets yy., startingfromx or y. If sxyis neitherx nor y, then T\ {sxyl includesfinite Let Tx(or TY)be 4 if components Tx and Tycontainingx and y,respectively. x (or y) is sxy.Let 8n(x) be the nthbondinvaded,startingfromx, whichis bondin notin TxU Ty.Then 13n(x)is the f-minimizing b e G: b
OTXU TyU ({1(X),,...,
1(n-X)}
b has an endpointin Isxy, 131(x), **,
/3n-1( X) )}
if pi(x) = pi(y) forall 1 < i < n - 1, then 13n(x) = Pn(y). It folTherefore, lows that 13n(x)= 13n(Y)forall n and (4.1)
I4(X) t& 4(y) C TxU Ty,
Note that,althoughTx is finite,dTx and thus Tx may be infiniteif Tx has sites ofinfinitedegree.
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Fix b = (u, v) E dTX, with u e T,. Suppose b E I.(x); we claim that thereis a b E Ic,(y). By (i), since b t F, b mustbe a backfillbond.Therefore, path A fromu to v consistingentirelyofbreakoutbonds(withrespectto x); criterion (2.2) that b by(i), Ais a path in T. It followsfromthe creek-crossing is the longestbond in A U {b). Further,since v 0 Tx, A must pass through sxysand thereforeb is invadedaftersy. Letting 80(x) := {sxy},we let n 2 0 be such that b is invaded(startingfromx) afterpn(x) is invadedbut before 8n+1(x) is invaded.Then (4.2) f( fn+1(Ax))> f( b) > f(e) forall bondse in A. Now startingfromy, sy is invaded beforepn+ (y) = n+ 1(x) is invaded. Since sxyis a site of A,it followsfrom(4.2) thatstartingfromy, all of A U {b} will be invadedbeforein+ 1(Y),and our claimthat b E Ij(y) follows. Conversely,forthe same b E dTx, if b E Ic0(y),then virtuallythe same we have proofshows b E Ij(x). Therefore, (I(x)A I4(y)) n (dTxu dTy)= 4. With(4.1) this establishes(iii) forone-endedtrees. Case 2: T is a two-endedtree with trunk-y.Let AV denote the (finite) componentofa site v in T \ {z(v)), togetherwill all bondsof X whichhave bothendpointsin this component(so A, = (Aif v E y). We claimthat
(4.3) Qs'(X, rj) c I.(v) c CQt(Xj rj) U;AV. Let o- be the least n such that In(v) containsz(v); such an n always exists by (i). Since, by Theorem2.5 z(v) e C~t(X,rc), at all futuretimes i 2 or, Ai(v) includesa bond of C~t(X, rc). Thus everybond b outside A, invaded aftertime oahas f(b) < r, so in factall suchbondsare in C~t(X, r,), and the secondinclusionin (4.3) follows.For the firstinclusion,supposethereexistsa bond b E C~t(X, rc)\ I4o(v).Since by uniqueness Ct(X, r,) is connected,we can findsuch a b withat least one endpointin Ij(v), so b E An(v)forsome n. However,then f(e) < f(b) forall e E I(v) \ In(v). The graph consistingof such e has onlyfinitely manyconnectedcomponentsso it includesan infinite f(b)-cluster.Since f(b) < r, thereis no such infiniteclustera.s. It follows that CQt(X,rd)\ 1(v) = 4 a.s. This proves (4.3), which in turn shows oo(X)A 4(y)
is finite. El
COROLLARY 4.3. In the lattice/uniform model,foreveryr > rc and every sitex, at mostfinitely manybondsof yx. are outsidethe infiniter-clusterof thecorresponding percolationmodel,a.s. PROOF. The analogous fact for I4o(x)was proved in [5], so the result followsfromTheorem4.1(ii).In [5], rc is replacedby a percolationthreshhold fromrc,but it was provedin [8] thatthetwothresholdsare possiblydifferent the same. El
5. Stability of the MSF under local changes. We considernext the extentto whichlocal changesin a graphcan produceglobal changesin the
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labeled graphG in Rd, let us define MSF. Specifically, foran ambiguity-free the functionSG on Rd X Rd by ( x y) 1 if x and y are sitesin thesame treeoftheMSF ofG,
{0:otherwise.
Local changesin G maycause bondsto be added to or deletedfromtheMSF, and as a resultthe function8G maychange. To begin, we define some modificationsof the graph G = (V, a, f ). For F C Rd let Dr(G) = {x E V nl: (x, y > E- forsomey E FC}. Definethe restriction GIr of G to F by = { <x, y> EG: x, y e F), O~r Glr =(V n rF, r, f) off to Sire To Here,in a slightabuse ofnotation,f is actuallytherestriction of G to F U DrC(G). add in bondscrossingdF, let Gj denotethe restriction We say twolabeled graphsG1 and G2 agree outside F if Gljc = G2 rc. THEOREM 5.1. (i) Suppose G1 and G2 are ambiguity-freelocally finite labeled graphs in Rd that agree outside At for some t > 0, and let Fi be the MSF of Gi. Then the symmetricdifferenceF1 A F2 is a finiteset of bonds. (ii) Let X be as in Theorem 2.5, with strict simultaneous uniqueness holding a.s. There exists a set A of labeled graphs such that (a) P(X e A) = 1 and (b) if t > 0 and Gi = (Vi, i, fi) are labeled graphs in A that agree outside At, then there exists s > t such that 8Gl(X, y) = 8G2(X,y) for all x, y e As
Roughlyspeaking,(ii) says that changesin a finitebox can onlyshifta treesoftheMSF. Finitechangescannot,for finitenumberofsitesto different treeintotwoone-endedtreesorgluetwoone-ended example,splita two-ended treesintoa two-endedtree,exceptpossiblyby creatinga labeled graphofa typethata.s. does notoccur,thatis, one notin A. PROOFOF THEOREM5.1.
(i) For u, v distinct sites in DAt(Gi),define
:=inffr> 0: u is connectedto v at level r in GilAtK} Note DA(Gi) and GiLct do not dependon i. Suppose b = <x, y> is a bond of criterion (2.2), thereis GiIct thatis in F1 but not F2. Fromthe creek-crossing a path in G2 fromx to y consistingofbonds e withf(e) < f(b), but no such this path in G2 path existsin G1 and hence none existsin Gilt. Therefore, musthave a bondin G21At, so forsome pair u, v ofdistinctsites in DAI(Gi), thereexistdisjointpaths in G I+c fromx to u and fromy to v at somelevel less than f(b). Thus rUV< f(b). However,therecan be no pathfromu to v at a level less than f(b) in GiI ct,forotherwisewe wouldalso have such a path fromx to y. Thus ruv= f(b); since labels are distinctthereis at mostone such b foreach pair u, v. Since G1 and G2 are locallyfinite,(i) follows.
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(ii) We may assume X is ergodic,so by Theorem2.5 there exists an n E {0, 1} such thatthe MSF of X has exactlyn two-endedtrees,a.s. Let A be the set of graphsforwhichthe MSF has exactlyn two-endedtrees and in Theorem2.5(ii). If n = 0, thenforeach bond b in satisfiesthe description F1 \ F2, removingb fromF1 splitsthe tree of F1 containingb into a finite and an infinitepiece,and similarlyforb E F2 \ F1. If x and y are sites of X and neither x nor y is in one of these finitepieces, then 8G1(X,y) = 8G2(X,y). Thus forn = 0, (b) followsfrom(i). If n = 1, it followsfrom(i) thatto prove (b) we need onlyeliminatethe possibilitythatforsome site z in a one-ended treeT ofF1,thepath y,,.in T is containedin thetrunkofthetwo-endedtree of F2. However,thiswouldimplythatall bonds e in y%.have f(e) < rc,so T wouldmeetthe infinitestrictrc-cluster, so in G1 the infinitestrictrc-cluster wouldnotbe containedin the two-endedtree,meaningG1 0 A. El Withoutstrictsimultaneousuniqueness,Theorem5.1(ii) is false.The MSF of X maythenincludemultipletwo-endedtrees.Throughchangesin a finite box,it maybe possibleto in effect cutin halfthetrunksoftwosuchtreesand thenglue the fourresultingrays back togetherin a different pairing,as the following example,similarto Example 1.9 of[2], shows. EXAMPLE 5.2. Considera randomlabeled graphX in R2 withsite set Z2. Verticalbondsare nearestneighbor;horizontalbondsare longrange.Let us call the subgraphin the verticalline at m, columnm. Let {Um: m GEZ} be iid uniform[0,1] randomvariablesand let {Tl: j E ZZ}be the arrivaltimesofa timesnotidentically1. Let stationaryrenewalprocessin Z, withinterarrival us referto [?Tj, n Z as block on {Um: m E Z} and {T.: 1) j. Conditionally Tj, j E Z}, we label bonds as follows:For m in block j, verticalbonds ((m, k), in [0, Uj); form in blocki (m,k + 1)> in columnm getlabels Wnkiid uniform and n in blockj, each horizontalbond ((m, k), (n, k)> independently gets a label uniformin (max(Ui,Uj),1]. The discussionin Example 1.9 of[2] establishes that X does notsatisfysimultaneousuniqueness,strictor not,though forfixedr > 0 thereis a.s. a unique infiniteclusterat level r. Uniqueness failspreciselyat the levels Uj. It is easilyverifiedthatthe MSF of X consists manytwo-ended preciselyof all verticalbonds,so is composedof infinitely trees.We nowintroducelocal modifications a crossover to X. We can construct in a square [m, m + 1] x [k, k + 1] by deletingthe verticalbonds ((m, k), (m, k + 1)>, (m + 1, k + 1)> and adding diagonal bonds ((m, k), (m + 1, k + 1)> withlabel Wmkand ((m, k + 1), (m + 1,k)> withlabel Wm+ 1,k. Let {Zmk: m,k E ZZ}be idd takingvalues 0 and 1 withprobability1/2 each. Starting with the graph X, in each square [m, m + 1] x [k, k + 1] we constructa crossoverif m and m + 1 are in the same block,and Z- 1,k = 0, Zmk = 1 and Zm+ 1,k = 0. We denotethe resultinggraphY. Thus no twohorizontally adjacent cubes both have crossovers.The subgraphof Y consistingof premanydistinct ciselythe verticaland diagonalbondsis made up ofinfinitely infinitelines; let us call these lines strands.Note that a horizontalbond emanatingfroma givenstrandis longerthanany ofthebondsofthatstrand.
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K S. ALEXANDER
Usingthecreek-crossing criterion (2.2) it followseasilythattheMSF ofY a.s. consistspreciselyofthe infinitecollectionofstrands.ComparingY itselfto Y witha singleadded crossover,we see that Theorem5.1 (ii)(b) does not hold. [1
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