First-passage percolation on width-two stretches with exponential link weights PDF
Preview First-passage percolation on width-two stretches with exponential link weights
FIRST-PASSAGE PERCOLATION ON WIDTH-TWO STRETCHES WITH EXPONENTIAL LINK WEIGHTS ECKHARDSCHLEMM Abstract. Weconsiderthefirst-passagepercolationproblemoneffectivelyone-dimen- sionalgraphswithvertexset{1...,n}×{0,1}andtranslation-invariantedge-structure. For threeofsixnon-trivialcasesweobtainexactexpressionsfortheasymptoticpercolation rateχbysolvingcertainrecursivedistributionalequationsandinvokingresultsfromer- 2 godictheorytoidentifyχastheexpectedasymptoticone-stepgrowthofthefirst-passage 1 timefrom(0,0)to(n,0). 0 2 n a J 1 1. Introduction 2 LetG = (V,E)beagraphwithvertexsetV = V(G)andunorientededgesE = E(G) ⊂ ] V2. Twoverticesu,v ∈ V aresaidtobeadjacent, forwhichwewriteu ∼ vif(u,v) ∈ E R and the edge e = (u,v) is said to join the vertices u and v. Assume there is a weight- P function w : E → R. For any two vertices u,v ∈ V a path p joining u and v in G is . h a collection of vertices {u = p0,p1,...,pn−1,pn = v} such that pν and pν+1 are adjacent at for 0 ≤ ν < n; a path p is called simple if each vertex occurs in p at most once. To a m given path p we associate the set of its comprising edges pˆ = {(pν,pν+1) : 0 ≤ ν < n}. [ The weight w(p) of a path p is then defined as (cid:80)e∈pˆw(e). We define dG : V ×V → R by dG(u,v) = inf{w(p): papathjoininguandvinG}, andcall dG(u,v) the first-passage 1 timebetweenuandv. ApathpjoininguandvinG,suchthatw(p) = d (u,v)iscalleda v G shortestpath. ThroughoutthisworkweassumethatGisfiniteandconnectedandwewill 2 8 beinterestedinthecasethattheweightsw(e),e∈ E,arerandomvariables;thegoalisthen 4 to make probability statements about first-passage times or related quantities. In general 4 wenoteherethatashortestpath p = {u = p ,...,p = v}isalwayssimpleandthateach 0 n . 1 sub-path {pk,...,pl}, 0 ≤ k < l ≤ n, is also a shortest path. Moreover, for continuously 0 distributed,independentedgeweightstheshortestpathbetweenanytwoverticesisalmost 2 surelyunique(Kesten,1986). 1 The typical first-passage percolation problem is based on two-dimensional regular in- : v finite graphs G with vertex sets V(G) = Z2. The edge weights w(e), e ∈ E(G), are i.i.d Xi random variables in L1 with some common distribution P such that P-almost surely w(e) ispositive. Letl ,0≤m≤n,denotethefirst-passagetimefrom(m,0)to(n,0)subject r m→n a totheconditionthatthecontributingpathsconsistonlyofverticeswithfirstcoordinateν, m ≤ ν ≤ n and write l (cid:66) l . By Lemma 4.2, l ≤ l +l and the theory of n 0→n 0→n 0→m m→n sub-additiveprocesses(Proposition4.1)impliesthatlim 1l existsandisalmostsurely n→∞ n n constant. Thisnumber, whichdependsonlyonthegraphG andthedistributionP, isde- notedbyχ(G,P)andcalledthe(asymptotic)percolationrateortimeconstant.Theexplicit calculationofthepercolationrateevenforthesimplestregularinfinitegraphsanddistri- butions P is characterized in Graham et al. (1995, p. 1937) as a ”hopelessly intractable” problem. 1991MathematicsSubjectClassification. Primary:60K35;secondary:60J05. Keywordsandphrases. ergodicity,first-passagepercolation,Markovchains,percolationrate. 1 FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 2 2. Themodel In this work we first focus on the first-passage percolation problem as described in thelastparagraphforcertainfamiliesofregular,effectivelyone-dimensionalgraphswith independent,randomhypweights. LetV ={0,1,...,n}×{0,1}and n V ={((i,0),(i+1,1)):0≤i<n} n W ={((i,1),(i+1,0)):0≤i<n} n X ={((i,0),(i+1,0)):0≤i<n} n Y ={((i,1),(i+1,1)):0≤i<n} n Z ={((i,0),(i,1)):0≤i≤n}. n ToeachsubsetE⊂{V,W,X,Y,Z}correspondsafamilyofgraphs GE ={GEn}n∈N ={(Vn,En)}n∈N withvertexsetsV andedgesE =(cid:83) L .Foreachgraphtheedgeweightsareindepen- n n L∈E n dent exponentially distributed random variables which are labelled V,W,X,Y,Z in the i i i i i obviousway. Bytime-scalingitisnorestrictionofgeneralityifweassumetheparameter oftheedgeweightdistributionstobeunity.WedenotetheprobabilitymeasurebyP,thatis dP(x)=e−xdx.Wepointoutthatbyconstruction,foranyselectionofedgesE,GEisasub- n graphofGE ,anobservationwhichformsthebasisforourinductiveargumentdescribed n+1 below. The method is based upon Flaxman et al. (2011) where it has successfully been employedtocomputetheasymptoticpercolationrateonG{X,Y,Z} wheretheedgeweights aretakentobeindependentlyonewithprobability pandzerowithprobability1−p. They also consider continuous edge-weight distributions and show that one can replace those by suitably chosen discrete ones to obtain arbitrarily good, yet approximative values for the time constant. We do not adopt this method here but rather work explicitly with the continuousdistributions. 3. Results ItturnsoutthatforonlysixfamiliesofgraphsGE thefirst-passagepercolationisnon- trivial. For three of these six, namely for G{X,Y,Z}, G{V,W,X,Y} and G{W,X,Y,Z} we derive expressions for χ which seem not to have been known so far (see Table 1). We are con- fident that our method works for the three remaining families, G{V,W,X}, G{V,W,X,Z} and G{V,W,X,Y,Z},aswell.Increasedcomplexityinthecalculationsinvolved,however,prevents us from obtaining analytic results; the numerical values of χ for theses cases, which are alsorecordedinTable1,wereobtainedbymeansofsimulations. 4. Subadditivityandergodictheorems Inthissectionwepresentsometheoreticalresultsaboutfirst-passagetimes.Inparticular we show that the asymptotic percolation rate lim 1l on graphs with vertex set V (cid:66) n→∞ n n n {0,...,n}×{0,1}andtranslation-invariantedgestructureis,withprobabilityone,equalto adeterministicnumberandprovideaformulaforitscomputation(Theorem4.4). Proposition 4.1. Let G be the graph GE for some E ⊂ {V,W,X,Y,Z}. If the edge n weights w(e), e ∈ E , are independent, identically P-distributed, positive and integrable n randomvariablesthenalmostsurelylim 1l existsandisequaltoadeterministicnum- n→∞ n n berχ(G,P). Toprovethisweneedalemmaaboutapropertyoffirst-passagetimes,calledsubaddi- tivity. Lemma4.2. ForanyweightedgraphG,thefunctiond :V(G)2 →Rissubadditive,that G isd (u,w)≤d (u,v)+d (v,w)forallu,v,w∈V(G). IfinparticularV(G)=V ,thissays G G G n l ≤l +l holdsforany0≤m≤n. 0→n 0→m m→n FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 3 Table1. Width-twostretchesGE: Non-trivialsolvedandunsolvedcases E Pictograph χ(E) 3 J (2) {X,Y,Z} − 1 2 2J (2) 2 √ 3 J ( 2) {V,W,X,Y} − √0 √ 4 2 2J ( 2) 1 2tan1−2 {W,X,Y,Z} 2tan1−1 {V,W,X} ≈.51 {V,W,X,Z} ≈.45 {V,W,X,Y,Z} ≈.35 Proof. Subadditivity of d is a consequence of the simple observation that the set of all G paths joining u and w inG is a superset of the set of all paths joining u and w inG and containingvandthatonecancombineanytwopathspu→vjoininguandvandpv→wjoining vandwtogetapath pu→w joininguandwthatsatisfiesw(pu→w) = w(pu→v)+w(pv→w). The remark about the special case G = GE is clear. (Take u = (0,0), v = (m,0), w = n (n,0)) (cid:3) Toconcludefromthissubadditivitypropertythattheasymptoticpercolationrateisal- most surely a deterministic number we need the following subadditive ergodic theorem duetoLiggett,whogeneralizedaresultofKingman. (SeeDurrett(1991,theorem6.1)for aproof.) Theorem 4.3 (Liggett). Suppose a family of random variables X = {X : 0 ≤ m ≤ n} m,n satisfies (i) X ≤ X +X . 0,n 0,m m.n (ii) Foreachk∈N,(Xnk,(n+1)k)n≥0isastationarysequence. ((iiivi)) ETh(cid:104)eXd+is(cid:105)tr<ib∞utiaonndoffo(rXema,mc+hkn)k≥∈0Ndo,eEs(cid:2)nXotd(cid:3)e≥peγndhoonldms.withγ >−∞. 0,1 0,n 0 0 ThenX =lim X /nexistsalmostsurely. n→∞ 0,n Intheformulationof(iv)weused,aswewilloftendointhefollowing,thenotation x+ asashort-handformax{0,x}. Wecannowgiveaproofoftheassertionthattheasymptotic percolationratealmostsurelyequalsanon-randomnumber. ProofofProposition4.1. Wewillfirstarguethatthefamily{X =l :0≤m≤n}sat- m,n m→n isfiestheconditionsofTheorem4.3.InLemma4.2wehaveseenthat(i)holds.(ii)and(iii) followfromthetranslationalinvarianceofthegraphandthefactthat{w(e),e∈ E}isi.i.d. Thelastcondition,(iv),holdsbyassumptionandinparticularfortheexponentialdistribu- tion. Wecanthusconcludethatl /n→χalmostsurelyasn→∞. Toseethatχisindeed n constant,weenumeratetheedges{e∈ E}insomeway,saye ,e ,....Sinceχisdefinedas 1 2 lim l /nandeachl isasumofonlyafinitenumberofedge-weights,namelyofasub- n→∞ n n setofthosecontainedinGEn, χismeasurablewithrespectto(cid:84)n≥1σ(w(en),w(en+1),...), thetail-σ-algebraofthei.i.d.sequencew(e ),w(e ),....Kolmogorov’sZero-Onelaw(see 1 2 FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 4 forexampleBreiman(1968,Theorem3.12))thenimpliesthatforanyBorelsetBtheprob- abilitythatχtakesavaluein Biseither0or1whichmeansthatχisalmostsurelyequal toadeterministicnumber. (cid:3) Inviewofournexttheorem,whichprovidesanexplicitformulaforthepercolationrate χ, we could have done without Proposition 4.1 and without invoking Kingman’s general subadditivityresult. Wechosetoincludeit,however,inordertodistinguishthisuniversal aspectoffirst-passagepercolationtheoryfrompropertiesspecifictoourmodel. Theresult isthefollowing: Theorem 4.4. Let Λ (cid:66) l −l and assume there exists an S-valued ergodic Markov n n n−1 chain(M ) andameasurablefunction f :S →RsatisfyingΛ = f(M )forallpositive n n≥1 n n integersn. Then,almostsurely, (cid:90) l (1) χ= lim n = f(s)π(s), n→∞ n S whereπistheuniqueinvariantdistributionof(M ) . Putdifferently, χ = E[Λ],where n n≥1 Λistheweaklimitofthesequence(Λ ) ,i.e. Λ →D Λ. n n≥1 n Remark 4.5. ItwillbeestablishedinLemma5.6thatthereindeedexistsaMarkovchain (M ) andafunction f satisfyingtheconditionsofthetheorem. n n≥1 Proof. As an instantaneous function of the ergodic Markov chain (M ) the sequence n n≥1 (Λ ) isergodicaswell. Itfollowsthatthetimeaverage 1(cid:80)n Λ convergestoE[Λ]. Sinncen≥c1learlyl =(cid:80)n Λ theclaimfollows. n ν=1 ν (cid:3) n ν=1 n 5. CalculationsforG{X,Y,Z} Inthissectionweproveourresultsaboutthetimeconstantsforfirst-passagepercolation on width-two stretches (Table 1) in the case of E = {X,Y,Z}. The calculations for the othertwocasesmentionedarecompletelyanalogousandcanbefoundinSchlemm(2008). Wedenotethelengthoftheshortestpathfrom(0,0)to(n,0)byl ,thelengthoftheshortest n pathfrom(0,0)to(n,1)byl(cid:48) andwelet∆ =l(cid:48) −l . Ourfirstgoalistofindarecurrence n n n n relationbetweenthedistributionsof∆ and∆ . n n−1 Proposition5.1. Thesequenceofrandomvariables(∆ ) isareal-valuedMarkovchain n n≥0 withinitialdistributionP(∆ ≤d)=1−e−d andtransitionkernel 0 (2) K(δ,d)=e−|d|1eee1−−−|((δdδ|−−dδ)) ddddd ><=><00000,∧∧∧∧δδδδ≤≤>>dddd.,,, Proof. Itisclearthatwemusttrytoexpressρ ,theprobabilitydensityfunctionof∆ ,in n n termsofρ ,thedensityfunctionof∆ . Toachievethiswedefineeventsonwhich∆ n−1 n−1 n isadeterministicfunctionofX ,Y ,Z and∆ ,whichamountstoseparatelytakinginto n n n n−1 accountthedifferentpossiblebehavioursofthelaststepoftheshortestpath. Considerfirst theshortestpath p0→1 from(0,0)to(n,1),thelastedgeofwhichmustbeeitherY orZ . n n Similarly,thelastedgeof p0→0,theshortestpathfrom(0,0)to(n,0),mustbeeitherX or n Z . Wenotethatalmostsurely,theedgeZ isnotpartofboth p0→1 and p0→0 becausethis n n would contradict the uniqueness property mentioned in the introduction, so that we have FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 5 toconsiderthreecases: (cid:110) (cid:111) (3a) = Y +l(cid:48) ≤Z +X +l ∧X +l ≤Z +Y +l(cid:48) n n−1 n n n−1 n n−1 n n n−1 ={Z ≥|X −Y −∆ |} n n n n−1 (cid:110) (cid:111) (3b) = Z +X +l ≤Y +l(cid:48) ∧X +l ≤Z +Y +l(cid:48) n n n−1 n n−1 n n−1 n n n−1 ={Y ≥ X +Z −∆ } n n n n−1 (cid:110) (cid:111) (3c) = Y +l(cid:48) ≤Z +X +l ∧Z +Y +l(cid:48) ≤ X +l n n−1 n n n−1 n n n−1 n n−1 ={X ≥Y +Z +∆ } n n n n−1 Inparticularwehavetherelation (4) ∆ =min{∆ +Y ,X +Z }−min{X ,∆ +Y +Z }. n n−1 n n n n n−1 n n Thereasonwhythepictographsabovehavebeenchosentorepresentthedifferentevents isthefollowing: solidlinescorrespondtoedgesbeingusedbytheshortestpathtoeither (n,0) or (n,1), double solid lines represent edges being part of both these paths, while dashedlinesstandforunusededges. Usingtheseeventswenowcomputethecumulative distributionfunctionof∆ as n (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (5) P(∆ ≤d)=P {∆ ≤d}∩ +P {∆ ≤d}∩ +P {∆ ≤d}∩ . n n n n Ontheevent ,∆ isequalto∆ +Y −X ,sothefirsttermis n n−1 n n (cid:90) (cid:90) (cid:16) (cid:17) P ∆n ≤d∩ = dδρn−1(δ) dP3(x,y,z)I{z≥|x−y−δ|}I{δ+y−x≤d}. R R3 Rewritingtheindicatorfunctionsasintegrationboundsweobtain (cid:90) (cid:90) ∞ (cid:90) ∞ (cid:90) ∞ dδρ (δ) dP(y) dP(x) dP(z) n−1 R 0 (δ+y−d)+ |x−y−δ| andafterdoingthetediousbuteasyx-,y-andz-integralswearriveat (cid:90) (cid:16) (cid:17) P ∆ ≤d∩ = dδρ (δ)G (δ,d) n n−1 1 R with G1(δ,d)= 14eeeeeδ2δ−−(((δδd12((−12δ+−)++2222δ(dδδ−)−−eδ−e)2−)d2)(d−δ)) ddddd <<≥≥≥00000∧∧∧∧∧δδδ0δ≤≤>><dd0dδ,,,.≤d, Onecanalsoconfirmthiscomputationandsimilaroneswhichfollowbyuseofacomputer algebrasystem. Infact,weusedMathematicatoverifyourresults. Ontheevent ,∆ is n givenbyZ ,soforthesecondtermin(5)weobtain: n (cid:90) (cid:90) (cid:16) (cid:17) P ∆n ≤d∩ = dδρn−1(δ) dP3(x,y,z)I{y≥x+z−δ}I{z≤d} R R3 (cid:90) (cid:90) ∞ (cid:90) d+ (cid:90) ∞ = dδρ (δ) dP(x) dP(z) dP(y) n−1 R 0 0 (x+z−δ)+ (cid:90) (cid:67) dδρ (δ)G (δ,d), n−1 2 R FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 6 wherethefunctionG isgivenby 2 G (δ,d)= 1I 4eδ−(1e−−δe(cid:16)−32d+)e−2(d−δ)+2δ(cid:17) δ0≤<0δ,≤d, 2 4 {d≥0}4−4e−d−2de−δ δ>d. Finally,ontheevent ,∆ isequalto−Z ,sowecomputeforthethirdtermin(5) n n (cid:90) (cid:90) (cid:16) (cid:17) P ∆n ≤d∩ = dδρn−1(δ) dP3(x,y,z)I{x≥y+z+δ}I{z≤−d} R R3 (cid:90) (cid:90) ∞ (cid:90) ∞ (cid:90) ∞ = dδρ n(δ) dP(y) dP(z) dP(x) −1 R 0 (−d)+ (y+z+δ)+ (cid:90) (cid:67) dδρ (δ)G (δ,d) n−1 3 R and G (δ,d)= 1e4−e2δ(δ+−de)δ(−2(d−δ)−3) dd <<00∧∧δδ≤>dd,, 3 4e4−+δ eδ(2δ−3) dd ≥≥00∧∧δδ≤>00., It is interesting to note that, due to the symmetry of the events (3b) and (3c), the sum of G (δ,d)andG (−δ,−d)doesnotdependond;explicitlyitholdsthat 2 3 G (δ,d)+G (−δ,−d)= 1e−|δ|1 δ≤0. 2 3 4 4eδ−2δ−3 δ>0 Puttingeverythingtogetheritfollowsthat (cid:90) P(∆ ≤d)= dδρ (δ)[G (δ,d)+G (δ,d)+G (δ,d)]. n n−1 1 2 3 R Differentiating this equation with respect to d and interchanging differentiation and inte- grationontherighthandsideweobtain (cid:90) ρ (δ)= dδρ (δ)∂ [G [δ,d]+G (δ,d)+G (δ,d)], n n−1 d 1 2 3 R sothetransitionkernelK :R→RisgivenbyK(δ,d)=∂ [G [δ,d]+G (δ,d)+G (δ,d)] d 1 2 3 andtheclaimfollowsfrombasiccomputations. (cid:3) FromtheexplicitdescriptionofthetransitionkernelK(δ,d)wecandeduceusefulprop- ertiesof(∆ ) : n n≥0 Lemma 5.2. The Markov chain (∆ ) is ergodic. In particular, as n → ∞, (∆ ) n n≥0 n n≥0 convergesindistributiontoanon-degeneratelimitingrandomvariable∆withprobability densityfunctionρ =lim ρ . ∞ n→∞ n Proof. It is enough to note that the kernel function K from Proposition 5.1 is strictly positive and continuous. The claim therefore follows from an extension of the Perron- Frobeniustheoremtocontinuoustransferoperators(seeJentzsch,1912). (cid:3) Corollary5.3. Letρ =lim ρ . Then ∞ n→∞ n (i) Thedensityρ ofthestationarydistributionof(∆ ) satisfiestheintegralequation ∞ n n≥0 (cid:90) d (cid:90) ∞ (6a) ρ (d)=ed dδρ (δ)+e2d dδρ (δ)e−δ d <0 ∞ ∞ ∞ −∞ d (cid:90) d (cid:90) ∞ (6b) ρ (d)=e−2d dδρ (δ)eδ+e−d dδρ (δ) d ≥0. ∞ ∞ ∞ −∞ d FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 7 (ii) Thedensityρ isanevenfunction,thatisρ (d)=ρ (−d)foralld ∈R. ∞ ∞ ∞ Proof. For the first claim we observe that ρ satisfies the integral equation ρ (d) = (cid:82) ∞ ∞ dδρ (δ)K(δ,d)withthekernel K givenin(2). Thesecondassertionfollowsfromthe R ∞ symmetryK(δ,d)= K(−δ,−d). (cid:3) Inordertosolvethisintegralequationwetransformitintoadifferentialequation. Lemma5.4. Thedensityρ ofthestationarydistributionof(∆ ) satisfiesthedifferen- ∞ n n≥0 tialequation (cid:104) (cid:105) (7a) ρ(cid:48)(cid:48)(d)=− 2+ed ρ (d)+3ρ(cid:48) (d) d <0, ∞ ∞ ∞ (cid:104) (cid:105) (7b) ρ(cid:48)(cid:48)(d)=− 2+e−d ρ (d)−3ρ(cid:48) (d) d ≥0. ∞ ∞ ∞ Proof. Weonlyprovetheclaimford < 0. Thecased ≥ 0canbeshowninthesameway or one uses (ii) of Corollary 5.3. In the case d < 0 we know that ρ is a solution to the ∞ integralequation(6a). Differentiatingthisequationwithrespecttodandusing(6a)again weobtain (cid:90) d ρ(cid:48) (d)=edρ (d)+ed dδρ (δ) ∞ ∞ ∞ −∞ (cid:90) ∞ (8) −edρ (d)+2e2d dδρ (δ)e−δ ∞ ∞ d (cid:90) d =2ρ (d)−ed dδρ (δ). ∞ ∞ −∞ Differentiatingagainandeliminatingtheremainingintegralvia(8)yields (cid:90) d ρ(cid:48)(cid:48)(d)=2ρ(cid:48) (d)−edρ (d)−ed dδρ (δ) ∞ ∞ ∞ ∞ −∞ (cid:104) (cid:105) =3ρ(cid:48) (d)− 2+ed ρ (d). ∞ ∞ (cid:3) Proposition5.5. Thedensityρ ofthestationarydistributionoftheMarkovchain(∆ ) ∞ n n≥0 is 1 (cid:16) (cid:17) (9) ρ∞(d)= e−32|d|J1 2e−12|d| , 2J (2) 2 whereJ areBesselfunctionsofthefirstkind. ν (cid:16) (cid:17) Proof. For d < 0, we write ρ∞ as ρ∞(d) = e32dρ˜∞ 2e21d with a function ρ˜∞ which is to bedetermined. Ifthisansatzisinsertedinto(7a)andifwereplacedby2log z weobtain 2 z2d2ρ˜∞(z)+zdρ˜∞(z)+(z2 −1)ρ˜ (z) = 0. This is the differential equation the solution to dz2 dz ∞ which are by definition the Bessel functions of the first and second kind, denoted by J ν andY ,respectively(seeAbramowitzandStegun,1965,equation9.1.1). Inthisparticular ν caseν=1. Onecanapplyananalogousmethodforthecased >0tofindthatthegeneral solutionofequation(7)isgivenby (cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105) (10) ρ∞(d)=e−32|d| c1J1 2e−12|d| +c2Y1 2e−12|d| . In order to determine the two constants c and c we insert this general solution into the 1 2 integralequation(6)andtakethelimitd → 0(itdoesnotmakeadifferencewhetherwe taked <0ord ≥0). Wechoosed <0andobtain (cid:90) 0 (cid:90) ∞ ρ (0)= dδρ (δ)+ dδρ (δ)e−δ, ∞ ∞ ∞ −∞ 0 FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 8 inwhichwereplaceρ (δ)bytheexpressiongivenin(10)toget ∞ (cid:34)(cid:90) 0 (cid:16) (cid:17) (cid:90) ∞ (cid:16) (cid:17) (cid:35) c1J1(2)+c2Y1(2)=c1 dδe23dJ1 2e12d + dδe−23dJ1 2e−12d e−δ −∞ 0 (cid:34)(cid:90) 0 (cid:16) (cid:17) (cid:90) ∞ (cid:16) (cid:17) (cid:35) +c2 dδe32dY1 2e12d + dδe−23dY1 2e−21d e−δ −∞ 0 (cid:34)(cid:32) (cid:33) (cid:32) (cid:33)(cid:35) 1 2 =c [J (2)+J (2)]+c +Y (2) + − +Y (2) , 1 2 0 2 π 2 π 0 SinceJ (2)=J (2)+J (2),Y (2)=Y (2)+Y (2)and−1/π(cid:44)0itfollowsthatc =0. 1 0 2 1 0 2 2 Therequirementthatρ integratetooneimplies ∞ (cid:90) (cid:16) (cid:17) 1(cid:90) 2 c−11 = Rdδe−32|δ|J1 2e−21|δ| = 2 0 dyy2J1(y)=2J2(2). (cid:3) Lemma5.6. ThereexistsanR×R3-valuedergodicMarkovchain(M ) andafunction + n n≥1 f :R×R3 →RsuchthatΛ = f(M )foralln≥1. + n n Proof. Set M = (∆ ,X ,Y ,Z )and f(δ,x,y,z) = min{x,δ+y+z}. Itisthenclearthat n n−1 n n n f(M ) = Λ (see(12))and(M ) isMarkovbecause∆ canbewrittenasmin{∆ + n n n n≥1 n−1 n−2 Y ,X +Z }−min{X ,∆ +Y +Z }. (cf. equation(4)intheproofofPropo- n−1 n−1 n−1 n−1 n−2 n−1 n−1 sition5.1.) ErgodicityisadirectconsequenceofLemma5.2. (cid:3) WewillnowusethisresulttocomputethedistributionofΛ(cid:66)lim Λ ,theexpected n→∞ n valueofwhichis,byTheorem4.4,thepercolationratewearelookingfor. Lemma5.7. Foreachnaturalnumbern,thedensityη ofthedistributionofΛ isη (l)= (cid:82) n n n dδρ (δ)Q(δ,l)withQ:R2 →Rgivenby R n−1 (11) Q(δ,l)=e−lee−δ((ll−−δ)δ(1)+2(l−δ)) ll≥<00∧∧δδ≤≤ll,, 10 lot≥he0rw∧isδe>. l, (cid:82) Inparticular,η (l)= dδρ (δ)Q(δ,l). ∞ R ∞ Proof. Asbeforewedefineeventsforeachpossiblebehaviourofthelaststepoftheshort- estpathfrom(0,0)to(0,n): (cid:110) (cid:111) = X +l ≤Z +Y +l(cid:48) ={X ≤∆ +Y +Z } n n−1 n n n−1 n n−1 n n (cid:110) (cid:111) = X +l ≥Z +Y +l(cid:48) ={X ≥∆ +Y +Z }. n n−1 n n n−1 n n−1 n n Clearly, (12) Λ =min{X ,∆ +Y +Z }, n n n−1 n n i.e. on the first of the two events, Λ is given by X while on the second it equals Y + n n n Z +∆ . NowtheprocedurecontinuesverysimilartotheproofofProposition5.1. We n n−1 computethecumulativedistributionfunctionofΛas (cid:16) (cid:17) (cid:16) (cid:17) (13) P(Λ ≤l)=P {Λ ≤l}∩ +P {Λ ≤l}∩ , n n n FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 9 becauseoneachofthesetwoeventswehaveanexplicitexpressionforΛ . Thefirstterm n leadsto (cid:90) (cid:90) (cid:16) (cid:17) P {Λn ≤l}∩ = dδρn−1(δ) dP3(x,y,z)I{x≤δ+y+z}I{x≤l} R R3 (cid:90) (cid:90) ∞ (cid:90) ∞ (cid:90) min{δ+y+z,l}+ = dδρ (δ) dP(z) dP(y) dP(x) n−1 R 0 0 0 (cid:90) (cid:67) dδρ (δ)P (δ,l), n−1 1 R wherethefunctionP istheresultofthex-,y-andz-integrationandgivenexplicitlyby 1 (cid:104) (cid:105) P1(δ,l)=I{l≥0}11144e−(cid:104)−42e−l−+lδe−eδ2l−(3e−−2l2+δδ)(2−(l3−−δ2)(+l−3)δ(cid:105)) δδ0≤><0lδ.,≤l, Verysimilarlyweobtainforthesecondtermin(13) (cid:90) (cid:90) (cid:16) (cid:17) P {Λn ≤l}∩ = dδρn−1(δ) dP3(x,y,z)I{x≥δ+y+z}I{δ+y+z≤l} R R3 (cid:90) (cid:90) ∞ (cid:90) (l−z−δ)+ (cid:90) ∞ = dδρ (δ) dP(z) dP(y) dP(x) n−1 R 0 0 (δ+y+z)+ (cid:90) (cid:67) dδρ (δ)P (δ,l), n−1 2 R withP givenby 2 P2(δ,l)=11014e−−−2e14ll−e−δδδ(cid:104)((cid:104)e132−l−−δ2eδ+2+δl()1e−+2l2(1(l+−2δ()l)−(cid:105) δ))(cid:105) lllot<≥≥he000rw∧∧∧isδδ0e≤≤≤. l0δ,,≤l, (cid:82) Puttingthingstogether,itfollowsthatP(Λ ≤l)= dδρ (δ)[P (δ,l)+P (δ,l)]. Taking n R n1 1 2 the derivative with respect to l and interchanging differentiation and integration on the righthandsideweobtainQ(δ,l)=∂ [P (δ,l)+P (δ,l)]. Thestatementabouttherelation l 1 2 betweenthestationarydensitiesη andρ isclear. (cid:3) ∞ ∞ (cid:104) (cid:105) Theorem5.8. ThepercolationrateχonG{X,Y,Z}is 3 − J1(2) ≈0.68.... 2 2J2(2) Proof. We have already argued that the time constant is the expectation of Λ, that is (cid:82) χ = dllη (l). From Lemma 5.7 we know the explicit form of η so we obtain χ = (cid:82) R (cid:82)∞ ∞ dδρ (δ) dllQ(δ,l), where the change of the order of integration is easily justified R ∞ R usingFubini’stheorem. Adirectcalculationshows (cid:90) dllQ(δ,l)= 18+4δ−eδ(5−2δ) δ<0, R 44−e−δ δ≥0 andthepercolationrateis (cid:34)(cid:90) 1 (cid:16) (cid:17)(cid:104) (cid:105) χ= dδe23δJ1 2e12δ 8+4δ−eδ(5−2δ) 2J2(2) R− (cid:90) (cid:35) (cid:16) (cid:17)(cid:104) (cid:105) + dδe−32δdJ1 2e−12δ 4−e−δ R + (cid:34) (cid:35) 1 4J (2)−7J (2) 4J (2)−J (2) 3 J (2) = 1 0 + 2 0 = − 1 . 2J (2) 4 4 2 2J (2) 2 2 (cid:3) FIRST-PASSAGEPERCOLATIONONWIDTH-TWOSTRETCHES 10 6. Discussion The differences between the omitted computations for the cases {V,W,X,Y} and {W,X,Y,Z}andtheonespresentedareonlyamatterofdegree,notofkind;forinstance, therearenotonlythreedifferentcasestoconsiderwhenprovingtheanaloguesofPropo- sition 5.1, but rather eight and six, respectively. Moreover, for the case {W,X,Y,Z} the stationary density ρ of (∆ ) (cf. Proposition 5.5) is not symmetric. It is also ∞ n n≥1 the determination of this stationary distribution where things get more difficult with the unsolved cases: the characterizing differential equation (cf. Lemma 5.4) then becomes non-local, more specifically it involves both ρ (d) and ρ (−d), as well as their deriva- ∞ ∞ tives, at the same time and so one can not solve it separately for d > 0 and d < 0 as we did. Itmightbepossibletoremedythisbycomputingρ notasaneigenfunctiontothe ∞ transition kernel K itself but rather as an eigen function to its second convolution power (cid:82) K(2)(δ,d)= dσK(δ,σ)K(σ,d).Asthemethodusedinthispaperisverysimilartothatin R itsantecedentFlaxmanetal.(2011)itsrangeofapplicabilityisalsoessentiallythesame.It wouldbeanaturalgeneralizationtoconsidergraphswithvertexsets{1,...,n}×{0,...,k} for some integer k ≥ 1 and for the directed first-passage percolation problem, similar techniquesasusedhereapplyequallywelltothismoregeneralset-up. However,thecom- binatorial difficulties arising from the need to explicitly keep track of the shortest paths onGn+1\Gn seem very hard to overcome and for undirected percolation a similarly easy recursiveargumentasinthecasek=2isnotanoption. ManythanksgotoBa´lintVira´gforveryhelpfuladvice. References M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions, with formulas,graphs,andmathematicaltables,volume55ofNationalBureauofStandards AppliedMathematicsSeries. SuperintendentofDocuments,U.S.GovernmentPrinting Office,Washington,D.C.,1965. L.Breiman. Probability. Addison-WesleyPublishingCompany,Reading,Mass.,1968. R.Durrett. Probability: theoryandexamples. Wadsworth&Brooks/Cole,Monterey,CA, 1991. A.Flaxman,D.Gamarnik,andG.B.Sorkin. First-passagepercolationonaladdergraph, and the path cost in a VCG auction. Random Structures Algorithms, 38(3):350–364, 2011. ISSN1042-9832. R.L.Graham,M.Gro¨tschel,andL.Lova´sz,editors. Handbookofcombinatorics.Vol.2. ElsevierScienceB.V.,Amsterdam,1995. R.Jentzsch. U¨berIntegralgleichungenmitpositivemKern. J.ReineAngew.Math.,pages 235–244,1912. H. Kesten. Aspects of first passage percolation. In E´cole d’e´te´ de probabilite´s de Saint- Flour, XIV—1984, volume 1180 of Lecture Notes in Math., pages 125–264. Springer, Berlin,1986. E.Schlemm. First-passagepercolationonwidth-twostretches,2008. Diplomarbeit,Freie Universita¨tBerlin. WolfsonCollege,CambridgeUniversity E-mailaddress:[email protected]
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