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SYNCHRONIZATIONOFFINITE-STATEPULSE-COUPLEDOSCILLATORSON(cid:90) HanbaekLyu1andDavidSivakoff2 Abstract 7 1 We study class of cellular automata called κ-color firefly cellular automata, which were introduced 0 recentlybythefirstauthor. Ateachdiscretetimet,eachvertexinagraphhasastatein{0,...,κ−1}, 2 andaspecialstateb(κ)=(cid:98)κ−1(cid:99)isdesignatedasthe‘blinking’state. Atstept,simultaneouslyforall 2 n vertices, thestateofavertexincrementsfromk tok+1 modκunlessk >b(κ)andatleastoneof a J itsneighborsisinthestateb(κ). Westudythissystemon(cid:90)wheretheinitialconfigurationisdrawn 2 fromauniformproductmeasureon{0,...,κ−1}(cid:90). Weshowthatthesystemclusterswithhighprob- ability for all κ≥3, and give upper and lower bounds on the rate. In the special case of κ=3, we ] R obtainsharpresultsusingacombinationoffunctionalcentrallimittheoremforMarkovchainsand P generatingfunctionmethod. Ourproofreliesonaclassicideaofrelatingthelimitingbehaviorwith . anaccompanyinginteractingparticlesystem,aswellasanothermonotonecomparisonprocessthat h t weintroduceinthispaper. a m Keywords: [ cellularautomaton,excitablemedia,interactingparticlesystems,coupledoscillators, 1 synchronization v 9 1 3 1. Introduction 0 0 1. Anexcitablemediumisanetworkofcoupleddynamicunitswhosestatesgetexciteduponapartic- 0 ularlocalevent. Ithasthecapacitytopropagatewavesofexcitation,whichoftenself-organizeinto 7 spiralpatterns. Examplesofsuchsystemsinnatureincludeneuralnetworks,Belousov-Zhabotinsky 1 : reaction,aswellascoupledoscillatorssuchasfirefliesandpacemakercells. Inadiscretesetting,ex- v citablemediacanbemodeledusingtheframeworkofgeneralizedcellularautomaton(GCA).Given i X asimpleconnectedgraphG=(V,E)andafixedintegerκ≥2,themicrostateofthesystematagiven ar discrete time t ≥0 is given by a κ-coloring of vertices Xt :V →(cid:90)κ =(cid:90)/κ(cid:90). A given initial coloring X0evolvesindiscretetimeviaiteratingafixeddeterministictransitionmapτ:Xt (cid:55)→Xt+1,whichde- pendsonlyonlocalinformationateachtimestep. Thatis,foreachv ∈V, Xt+1(v)isdeterminedby X restrictedonN(v)∪{v},whereN(v)isthesetofneighborsofv inG. Thisgeneratesatrajectory t (Xt)t≥0,anditslimitingbehaviorinrelationtothetopologyofG andstructureofτisofourinterest. 1DepartmentofMathematics,TheOhioStateUniversity,Columbus,OH43210,[email protected] 2DepartmentsofStatisticsandMathematics,TheOhioStateUniversity,Columbus,OH43210, [email protected] Greenberg-Hastings Model (GHM) and Cyclic cellular automaton (CCA) are two particular discrete excitable media which have been studied extensively since the 90s. GHM was introduced by GreenbergandHastings[11]tocapturephenomenologicalessenceofneuralnetworksinadiscrete setting,whereasCCAwasintroducedbyBramsonandGriffeath[2]asadiscretetimeanalogueofthe cyclicparticlesystems.InGHM,thinkofeachvertexofagivengraphasaκ-stateneuron.Anexcited neuron(i.e.,oneinstate1)excitesneighboringneuronsatrest(instate0)andthenneedstowaitfor arefractoryperiodoftime(modeledbytheremainingκ−2states)tobecomerestedagain. InCCA, eachvertexofthegraphisinhabitedbyoneofκdifferentspeciesinacyclicfoodchain. Speciesof coloriareeaten(andthusreplaced)byspeciesofcolor(i+1)modκintheirneighborhoodateach timestep.Moreprecisely,thetransitionmapsforκ-colorGHMandCCAaregivenbelow:  1 ifGHMt(v)=0and∃u∈N(v)s.t.GHMt(u)=1  GHMt+1(v)= 0 ifGHMt(v)=0and(cid:216)u∈N(v)s.t.GHMt(u)=1 (1)  GHM (v)+1(modκ) otherwise t (cid:40) CCA (v)+1(modκ) if∃u∈N(v)s.t.CCA (u)=CCA (v)+1(modκ) CCAt+1(v)= t t t (2) CCA (v) otherwise t In[13],thefirstauthorproposedadiscretemodelforcoupledoscillators,andstudiedcriteriafor synchronizationonsome classesoffinite graphs. The basic setupisthe sameas CCAor GHM. Fix anintegerκ≥3andlet(cid:90)κ=(cid:90)/κ(cid:90)withlinearordering0<1<2<···<κ−1bethecolorspace. Let b(κ)=(cid:98)κ−1(cid:99)bethedesignatedblinkingstate. Considereachvertexasaκ-statefirefly,whichblinks 2 wheneverithascolorb(κ).Duringeachiteration,post-blinkingfireflies(whosecoloris>b(κ))witha blinkingneighbor(aneighborofcolorb(κ))stayatthesamecolor,andalltheothersincrementtheir colorsby1moduloκ.Moreprecisely,thetimeevolutionfromagiveninitialκ-coloringX isgivenby 0 (cid:40) X (v) ifX (v)>b(κ)and|{u∈N(v): X (u)=b(κ)}|≥1 (FCA) Xt+1(v)= Xt(v)+1(modκ) othetrwise t (3) t for t ≥0. The discrete dynamical system (Xt)t≥0 generated by the iteration of the above transition mapiscalledtheκ-colorfireflycellularautomaton(FCA)onG.Wecallaunitoftimea“second”. Thecentralnotioninthedynamicsoftheabovethreediscretemodelsforexcitablemediaisexci- tation. Wesayasitex isexcitedattimet ifitsinternaldynamicsareaffectedbyitsneighborsattime t.Thatis,ifCCAt+1(x)=CCAt(x)+1 modκ,ifGHMt(x)=0andGHMt+1(x)=1,orifXt+1(x)=Xt(x) for FCA. Note that excitation always comes from local disagreements, and in all three models sites excite their neighbors to remedy the current disagreement. It is the non-linear aggregation of this mutualefforttosynchronizewithneighborsthatmakesstudyingtheglobaldynamicsinteresting. Both CCA and GHM dynamics have been extensively studied on integer lattices G =(cid:90)d using probabilisticmethods,whereonetakestheinitialconfiguration X asarandomκ-coloringonsites 0 accordingtotheuniformproductprobabilitymeasure(cid:80)on((cid:90)κ)(cid:90)d. Weintroducesometerminology forFCAtodescribeitsbehavior,whichmaybedefinedforCCAandGHMsimilarly. Wesaythat X t fixatesifeverysiteisexcitedonlyfinitelymanytimes(cid:80)-a.s.,andsynchronizesifforeverytwovertices x,y ∈V, there exists N = N(x,y)∈(cid:78) such that X (x)= X (y) for all t ≥ N (cid:80)-a.s.. It is not hard to t t 2 seethatfixationandsynchronizationareequivalentnotionsifandonlyifanyinitialcoloringonthe completegraphwithtwoverticessynchronize,whichisthecaseforGHMandFCAforallκ≥3and CCAonlyforκ=3:CCAforκ≥4hasapairofdistinctbutnon-interactingcolors,resultinginfixation withoutsynchronization. We say X fluctuates if it does not fixate, in which case different limiting behaviors can arise, t dependingonthescalingofthenumberofexcitationsbytimet, t−1 ne (x):=(cid:88)1(xisexcitedattimes). (4) t s=0 Wesay Xt synchronizesweakly iflimt→∞net(x)/t =0forall x ∈(cid:90)d (cid:80)-a.s.. Notethatweaksynchro- nization does not imply synchronization, as the time evolution may be dominated by increasingly rarewavesofperturbations,whichcanpreventanysitefromrestingatafinal“phase”. Forinstance, letX (x)=(cid:80)∞ (−n modκ)1(κn≤x<κn+1),whereallsiteson(−∞,0)havecolor0andtotheright 0 n=0 colors decrement by 1 after every geometrically increasing number of steps. In this example, it is easy to see that the color boundaries move to the left with speed 1/κ, that is, Xκt =(cid:80)∞n=−t(−n+t modκ)1(κn−t ≤x <κn+1−t). Hence,inthesedynamics,eachsiteexcitesinfinitelyoftenbutwith frequencytendingtozero,so X synchronizesonlyweakly. Asimilarbutslightlydifferentnotionto t weak synchronization is clustering. We say X clusters if it is overwhelmingly likely to see a single t coloronanyfinitefixedregionin(cid:90)d afteralongtime,thatis, lim (cid:80)(X (x)(cid:54)=X (y))=0 (5) t→∞ t t foranytwositesxandyin(cid:90)d.Sinceexcitationoriginatesfromlocaldisagreement,clusteringimplies thatthedensityofsitesthatgetexcitedattimet tendstozero.TheconverseistruefortheGHMand FCA,butnotnecessarilyfortheCCA.Toseethis,supposetwoadjacentsitesx and y arenotexcited insomelongenoughtimewindow.Thequestioniswhethertheycouldstayindisagreementwithout eitheronebeingexcited.ThisispossibleinCCAforκ≥4,butnotintheothertwomodels. Next, we briefly summarize known results for the CCA and GHM in 1-dimension, and some of themainprooftechniques.Mostresultson1-dimensionalmodelsrelyonaparticlesystemsanalogy whereweplace“edgeparticles"ontheboundariesbetweendistinctlycoloredregionsandconsider their time evolution. By counting “live edge” particles against “blockade” particles and with some carefularguments, Fisch[7]showedthatκ-colorCCAon(cid:90)fixatesifandonlyifκ≥5, andalso[8] thatthe3-colorCCAon(cid:90)clusters,withanexactasymptotic (cid:114) (cid:80)(X (x)(cid:54)=X (x+1))∼ 2 t−1/2. (6) t t 3π The embedded particle system description of the 3-color CCA is as follows. At time 0, place a r or lparticle oneachedge independently withprobability 1/3. rparticles move to rightwithconstant unit speed and l particles behave similarly; if opposing particles ever collide or have to occupy the same edge, they annihilate each other and disappear. Now if there is a r particle on the edge (0,1) attime t, thisparticlemusthavebeenontheedge(−t,−t+1)attime0andmusttraveldistance t withoutbeingannihilatedbyanopposingparticle. Thiseventisdeterminedbythenetcountsofr 3 andlparticlesattime0startingfromtheedge(−t,−t+1)andmovingrightward.Namely,theexcess numberofrparticlesonsuccessiveintervals[−t,−t+s], 1≤s ≤2t+1, formarandomwalk. Ther particle moves as long as this random walk survives (stays at positive height), which is an event of probabilityΘ(t−1/2). AsimilartechniquewasincorporatedbyDurrettandSteifin[5]toshowsimilar clusteringresultsforGHMon(cid:90)forκ=3, andlateritwasextendedtoarbitraryκ≥3byFischand Gravnerin[9]. Noclusteringresultsareknownfor4-colorCCAon(cid:90), butsimulationindicatesthat themeanclustersizeofsuchsystemgrowsataratedifferentfromt1/2[8]. TheFCAsharesasimilarembeddedannihilatingparticlesystemstructure, butwithadditional queuingandflippingphenomena. Theinitialsitecoloringattime0inducesacanonicalassignment ofedgeparticles. IntheFCAdynamics,thesystemtakesafiniteamountof“burn-in”period,during which particles may flip their directions and thereafter they stabilize and move in only one direc- tionwithannihilationuponcollision. Thisfiniteburn-inperiodintroducesdependenciesbetween edge particles, so the associated random walk has correlated increments. For example, consider a 3-configuration···120··· on(cid:90), whichcorrespondstotwoconsecutiverightparticles. Applyingthe 3-colorFCArule,itevolvesto···221···,whichhasoneleftparticlebetween2and1,asiftherparticle between1to2flipstherparticletoitsrightbetween2and0. Asimilarphenomenonoccursforall κ≥3, so an associated random walk has correlated increments. We study survival probabilities of randomwalkswithcorrelatedincrementsusingacombinationofdifferentmethods,andobtainour firstmainresult,whichstatesthattheFCAon(cid:90)forarbitraryκ≥3clusters: Theorem2. LetG=(cid:90)betheintegerlatticewithnearest-neighboredges,fixκ≥3,andletX bedrawn 0 fromtheuniformproductmeasure(cid:80)on(cid:90)(cid:90)κ. (i) Foranyκ≥3andx∈(cid:90),wehave (cid:80)(X (x)(cid:54)=X (x+1))=O(t−1/2). (7) t t (ii) Forκ=3andanyx∈(cid:90),wehave (cid:80)(X (x)(cid:54)=X (x+1))∼ 17(cid:112)6 t−1/2. (8) t t 467 2π Whilethefirstpartoftheabovetheoremgivesanupperboundontherateatwhichexcitations ateach time stepdisappear, asimilar argumentdoes not easily give a corresponding lower bound. Instead,wetakeanovelapproachofconstructingamonotonecomparisonprocess,whichwasfirst developedbytheauthorsandGravnerinarecentwork[10]tostudy3-colorCCAandGHMonarbi- trarygraphs.WedevelopasimilartechniqueforFCAon(cid:90),bywhichweareablerelatethemaximum ofanassociatedrandomwalktothenumberofexcitations. Applyingafunctionalcentrallimittheo- remforrandomwalkswithcorrelatedincrements,weobtainoursecondmainresult: Theorem3. LetG=(cid:90)betheintegerlatticewithnearest-neighboredges,fixκ≥3,andletX bedrawn 0 fromtheuniformproductmeasure(cid:80)on(cid:90)(cid:90)κ. Let Z ∼N(0,1)beastandardnormalrandomvariable. Thenwehavethefollowings: (i) Foreachκ≥3,thereexistsaconstantσ=σ(κ)>0suchthatforanyx∈(cid:90)andr ≥0,wehave (cid:112) liminf (cid:80)(cid:161)ne (x)≥r tσ(cid:162)≥4(cid:80)(Z ≥r)(cid:80)(Z ≤r). (9) t→∞ t 4 (ii) Forκ=3,wehave (cid:112) (cid:179) (cid:180) lim (cid:80) ne (x)≥r 8t/81 =4(cid:80)(Z ≥r)(cid:80)(Z ≤r). (10) t→∞ t foranyx∈(cid:90)andr ≥0. (cid:112) Inwords,ittellsusthatallsitesmustexciteatleastabout t tim(cid:112)esinthefirstt seconds,whichagrees withtheprobabilityofdisagreementattimet decayinglike1/ t. Thispaperisorganizedasfollows. InSection2wecarefullydefinetheembeddedparticlessys- tem for FCA, and establish combinatorial lemmas that relate the survival of random walks arising fromcountinginitialparticlestotheeventofhavinglocaldisagreementonaparticularedgeatapar- ticulartime. InSection3wecoarsegrainthecorrelatedincrementsofassociatedrandomwalkinto i.i.d. chunksandprovetheTheorem2(i),usingaclassictheoremofSparreAndersonaboutsurvival of random walks with i.i.d. increments. To obtain the complimentary result in Theorem 3, in Sec- tion4,weintroduceanewnotionoftournamentexpansionontopoftheembeddedparticlesystem whichwewillhavedevelopedinSection2. ByusingafunctionalCLTforMarkovianincrements,we thenproveTheorem3inSection5. InSection6,weproveTheorem2(ii)byusingageneratingfunc- tionmethodtocomputethesurvivalprobabilityofawalkwithMarkovianincrements. Thedetailed computationsforgeneratingfunctionsaregivenintheappendix. 2. ParticlesystemexpansionofFCAon(cid:90) LetG=(V,E)bea(notnecessarilyfinite)simplegraphandfixκ≥3andletm=(cid:98)κ/2(cid:99).LetE betheset oftheorderedpairsofadjacentnodes,i.e.,E={(u,v),(v,u)|uv∈E}.Foreachκ-coloringX :V →(cid:90)κ, wedefineanassociated1-formdX :E→[−m,m]by  m ifκ=2m,|X(v)−X(u)|=m,andX(u)∈[0,b(n)]   dX(u,v)= −m ifκ=2m,|X(v)−X(u)|=m,andX(v)∈[0,b(n)] (11)  X(v)−X(u) otherwise wherethesubtractionistakenmoduloκintheinterval[−m,m].NotethatdX isanti-symmetric,i.e., dX(u,v)=−dX(v,u)foreach(u,v)∈E. ConsideranFCAtrajectory(Xt)t≥0 startingfromaκ-coloringX0 onG. Thisinducesasequence ofassociated1-forms(dXt)t≥0. WevieweachdXt asanedgeconfigurationwhereforeachadjacent pair(u,v)∈E withdX (u,v)=a>0,weimagineastackofa unitparticlesontheedgeuv directed t fromutov.Notethat(cid:107)dXt(cid:107)∞≤κ/2,sotheheightsofthestacksontheedgesisuniformlybounded. With this interpretation, we may view (dXt)t≥0 as an embedded edge-particle system. This “dual” systemadmitsasimpledescriptionwhentheunderlyinggraphis(cid:90). Fixanintegerκ≥3andfixaκ-colorFCAtrajectory(Xt)t≥0on(cid:90). Weidentifyeachorientededge (x,x+1)withitsleftsitex. Wedefinetheparticlesystemexpansionof(Xt)t≥0 attimet =a,denoted by(ξt)t≥a=(ξt;a)t≥a,asfollows. (i)Foreacht≥a,ξ isanedgeparticleconfiguration,thatis,ξ :(cid:90)×{l,r}→((cid:90)×(cid:78))∪{∞}suchthatξ t t t isinjectiveonξ−1[(cid:90)×(cid:78)].Theinterpretationofξ ((cid:96),r)=(x,h)isthatthereisanr(rightmoving) t t 5 particlelabeled(cid:96)attheedge(x,x+1)anditsqueueheightish.Thestate∞isagraveyardstate toaccountforparticlesthatgetannihilated. (ii)Attimet=a,weendoweachedge(x,x+1)withdX (x,x+1)=k>0(resp.<0)withkrparticles a (resp. −k lparticles)thatarestackedinaverticalfirst-infirst-outqueue. Defineξ tobeany a labelingofsuchaparticleconfiguration,i.e.,  {((cid:96),r)} forsome(cid:96)=(cid:96)(x,h)∈(cid:90)ifdXa(x,x+1)≥h  ξ−1(x,h)= {((cid:96),l)} forsome(cid:96)=(cid:96)(x,h)∈(cid:90)ifdX (x,x+1)≤−h a a  (cid:59) otherwise andξ−1(∞)=(cid:59). a (iii)Inwords,foreacht ≥a,thetransitionmapξt (cid:55)→ξt+1 isasfollows. Foreacht ≥a,callanedge (x,x+1)activeattimet ifdX (x,x+1)>0(or,resp. dX (x,x+1)<0)and X (x)=b(κ)(resp. t t t X (x+1)=b(κ))andinertotherwise.Thenthedynamicsproceedasfollows. t (release)Thebottomr(resp. l)particleoneachactiveedge(x,x+1)isreleasedattimet and headedtowardtheedge(x+1,x+2)(resp.(x−1,x)); (annihilation)Anypairofbottomrandlparticlesannihilateeachotherifatleastoneofthem isreleasedandtheyhavetooccupythesameedgeorcrossattimet+1; (queuing)Allotherparticlesreleasedattimet gotothetopofthequeueatthetargetedgeat timet+1. Moreprecisely,fixalabeledrparticle((cid:96),r). Ifξt((cid:96),r)=∞,thenξt+1((cid:96),r)=∞. Ifξt((cid:96),r)=(x,h) forsome(x,h)withh≥2,then (cid:40) (x,h) if(X (x)(cid:54)=b(κ))and[(cid:216)(i,l)s.t.ξ (i,l)=(x+1,1)orX (x+2)(cid:54)=b(κ)] ξt+1((cid:96),r)= (x,h−1) othetrwise. t t Ifξ ((cid:96),r)=(x,1),thenwedefine t ∞ if∃(i,l)s.t.ξ (i,l)=(x+1,1)and[X (x+2)=b(κ)orX (x)=b(κ)]  t t t   ∞ if∃(i,l)s.t.ξ (i,l)=(x+2,1)andX (x)=X (x+3)=b(κ) ξt+1((cid:96),r)=(x,1) ifXt(x)(cid:54)=b(κt)and[(cid:54)∃(i,l)s.t.ξt(i,lt)=(x+1t,1)orXt(x+2)(cid:54)=b(κ)]   (x+1,k+1) otherwise, wherekisthenumberofrparticles(i,r)suchthatξt(i,r)=(x+1,h)forsomehandξt+1(i,r)(cid:54)= ∞. Foralabeledlparticle((cid:96),l), weinterchangerandlintheabovedefinitionsandflipall+ signsrightafterx’sto−signs. Tomotivatetheaboveconstructionoftheparticlesystemexpansion,wewalkthroughsomeex- ampleswhenκ=6. Recallthatb(6)=2istheblinkingcolorandtherearethreepost-blinkingcolors 3,4,and5whoseupdatetothenextcolorisinhibitedwhenincontactwithcolor2.SupposeX (x)=2 t and X (x+1)=4 so that there are two r particle on the edge (x,x+1) and site x blinks at time t. t 6 Supposetheedge(x+1,x+2)isvacantandsite x+3doesnotblinkattime t; so X (x+2)=4and t X (x+3)=∗(cid:54)=2(seeFigure2.2(a)). Thenattime t+1sites x,x+1,and x+2havecolors3,4,and t 5 respectively, so there is a single r particle on each of the edges (x,x+1) and (x+1,x+2) at time t+1. Weviewthisasthebottomrparticleontheedge(x,x+1)havingmovedontothevacantedge (x+1,x+2). IfX (x+3)=2,thenboththebottomparticlesonedges(x,x+1)and(x+2,x+3)tryto t moveintothevacantedge(x+1,x+2),resultingintheirannihilationattimet+1(seeFigure2.2(b)). Similarannihilationoccurswhentheseparticlesareclosertoeachothe1r, i.e., theedge(x+1,x+2) alsohasanlparticle(seeFigure2.2(c)). → → ← → → (cid:2870) (cid:2870) (cid:2872) (cid:2870) (cid:2870) time t →(cid:2869) →(cid:2869) ←(cid:2875) →(cid:2869) ←(cid:2872) ←(cid:2875) 1 →(cid:2869) →(cid:2877) 2 4 4 * 2 4 4 2 2 4 3 2 2 4 5 * → (cid:2869) → → → ← → ← → → time t 1 (cid:2870) (cid:2869)→ (cid:2870) (cid:2872)→ ← (cid:2870) (cid:2875) → (cid:2870) (cid:2877) → 3 4 5→(cid:2870) 3 4 4 →3(cid:2870) ←(cid:2872) 3 4 3 3 →(cid:2870) ← ←3 4 0 →(cid:2870) → time t (cid:2869) (cid:2869) (cid:2875) (cid:2869) (cid:2872) (cid:2875) (cid:2869) (cid:2877) (cid:4666)a(cid:4667) 2 4 4 * (cid:4666)b(cid:4667) 2 4 4 2 (cid:4666)c(cid:4667) 2 4 3 2(cid:4666)d(cid:4667) 2 4 5 * → Figure1: Queuin gandannihi→latio →nru lesfortheparticlesyst→em expan←sion ofthe6-colorFC→Ao n←(cid:90). Wedenoteparticle → →(cid:2869) (i,r)byt→imie antds1imilarlyforlpar(cid:2870)ticles(cid:2869).∗∈(cid:90)6\{2}. (cid:2870) (cid:2872) (cid:2870) (cid:2875) (cid:2870) (cid:2877) →3 4 5 → 3 4 4 3 3 4 3 3 3 4 0 time t → → → → Lastly, co nsidert2hec3ase0w(cid:4666)hae(cid:4667)n thereareri2ght3par5ticl(cid:4666)e*bs(cid:4667) ontwoconsecutiveedg(cid:4666)ec(cid:4667)s , namely, (x,x+1) (cid:4666)d(cid:4667) and(x+1,x+2 )andsupposesitexblinksattimet.IfdX (x+1,x+2)=1,thenthebottomrparticle t → ontheed ge(x,x+1)m →igratestothetopoftheq →ueueontheedge(x+1,x+2)(seeFigure2.2(d)). → → time t 1 3 3 1 3 3 0 ← → → ← tim e t (cid:4666)a(cid:4667) → → (cid:4666)b(cid:4667) → → → ← x2 2 3 0 x2 x2 2 3 5 * x2 2 3 0  ← tim e t 1 a x0 ←← b x0 c x←←0 d x0 ←← x 3 3 x1 x3 3 0 x 3 3 1 1 1 1 1 (a)(cid:4666)a(cid:4667) (cid:4666)b(cid:4667) (b) (cid:4666)c(cid:4667) x x x x Figure2: Particle flippingfor2(a)κ=5and(b)κ=6,w2hicharenotincor2poratedintheparticlesyste2mexpansion(ξt)t≥a. ∗∈(cid:90)6\{2}.(c)BydefinitionofdX,whenκiseven,edgeswithmaximalnumberκ/2ofparticlesdonotflip. XH0ow ever,*t h*e0Ip0a0rt0ica0le0s0ysx0t0e0m0ξ0t0d0efi0n*eb*daIbovx*e0*m0i0ss0e0st0h0e0cfo0ll0ow0xi00ng0I0ph0e*n*omednon,xw0hichoccurs indXt. Ifκ= 6, Xit1(xx1)=2, dXt(x,x+1)x1=1, diXt(x+1,x+2)x=1 2and Xit1(x+3)=x1∗(cid:54)=2, theninthe next step dXt(x+1,x+2)=−3((as)ee Figure 2 (b)); in the particle syste(bm), this would correspond to theXri6ght-movi ngparticl*e*sa0t(0x*+*1,x+2)“flipping”tobeco*m*e0le0ft-*m*oving.Asimilarparticleflipping scenario forκ=5isillustratedinFigure2(a). Formally,wesaythatanedge(x,x+1)(orparticleson I I I hita)pflpipesnas tXitfi0amned to≥nil0y1*iiff*deXi0tht0(exr0,x0+010)0and0d0X0t0+10(x0i,x0+*1*)have*th*e0o0pp0o0si0te0isi10gn0.0It0is0ea0s0yt0o*se*ethatthis I I I i1 dX (x,x+1)≥b(κ)andxisiexcitedattimet i1 t 7 X6  **0 0**   **0 0 **  I I I i1 i i1 or dX (x,x+1)≤−b(κ)andx+1isexcitedattimet. t SeeFigure2foranillustration.Infact,wewillshowthatparticleflippingisatransientphenomenon; itoccursonlyduringafinite“burn-in”period,afterwhichtheparticlesfollowonlytheannihilation andqueuingrulesdefiningthedynamicsofξ .Thisisgivenbythefollowingproposition,whoseproof t wedelaytotheendofthissection: Proposition2.1. LetG=(V,E)beasimpleconnectedgraphwithmaximumdegree≤2.Fixκ≥3,and let(Xt)t≥0beaκ-colorFCAtrajectoryonG startingfromanarbitrarydeterministiccoloringX0. Then thefollowinghold: (i) Forκodd,noedgeflipsatanytimet≥(cid:98)κ/2(cid:99); (ii) Forκeven,noedgeflipsatanytimet≥5κ. Fromnowoninthissection,κdenotesaninteger≥3,(Xt)t≥0denotesaκ-colorFCAtrajectoryon (cid:90)startingfromanarbitrarydeterministicinitialcoloringX .Proposition2.1guaranteestheexistence 0 ofaninteger0≤t0≤5κafterwhichnoparticleflipsinthedynamics. (ξt)t≥t0=(ξt;t0)t≥t0 denotesthe particlesystemexpansioninitializedattimet=t .Inthefollowinglemma,weshowthattheparticle 0 systemexpansioniscompatiblewiththeoriginaldynamics. Lemma2.2. Let(Xt)t≥0,0≤t0≤5κ,and(ξt)t≥t0 beasbefore. Thenforeacht ≥t0andx∈(cid:90),theedge (x,x+1)doesnothavebothrandlparticlesatthesametimeand dX (x,x+1)=#{rparticleson(x,x+1)attimet}−#{lparticleson(x,x+1)attimet}. (12) t Proof. Thefirstpartoftheassertionfollowsfromtheconstructionofinitialparticleconfigurationξ t0 and the annihilation rule. We show the second assertion by an induction on t ≥t . The base case 0 t=t isclearfromtheconstruction,soassumetheassertionholdsfort≥t .Fixanedge(x,x+1).By 0 0 symmetrywemayassumedX (x,x+1)=k≥0. FirstsupposeX (x)=b(κ). Ifk=0,thenX (x+1)= t t t b(κ) and Xt+1(x)= Xt+1(x+1)=b(κ)+1 so dXt+1(x,x+1)=0. On the other hand, there was no particle on the edge (x,x+1) and also there is no incoming particle into the edge (x,x+1) since X (x)=X (x+1)=b(κ). Thusthereisnoparticleontheedge(x,x+1)attimet+1asdesired. Now t t suppose k >0. In this case x+1 is excited by x so that dXt+1(x,x+1)=k−1≥0. In the particle system, the bottom r particle on the edge (x,x+1) is released at time t and there is no incoming r particlefromtheedge(x−1,x)into(x,x+1). Hencewhetherthereleasedrparticleisannihilatedor not,theedge(x,x+1)willhavek−1rparticlesattimet+1,asdesired. SupposeX (x)(cid:54)=b(κ)sothattheedge(x,x+1)isinertattimet.Ifneitherx norx+1areexcited t attimet,thenbothadvancetheirstatessothatdXt+1(x,x+1)=dXt(x,x+1)=k;inparallel,there isnoincomingparticleintotheedge(x,x+1)andalsonoparticleisreleased. Ifbothx andx+1are excitedattimet,thentheykeepthesamestatessothatdXt+1(x,x+1)=dXt(x,x+1)=k,whichis inagreementwiththefactthatanrparticlecomingfromtheedge(x−1,x)movestothetopofthe queueontheedge(x,x+1),whilethebottomrparticleisannihilatedbythelparticlecomingfrom theedge(x+1,x+2). Therearetworemainingcases. Supposeonlyx+1isexcitedattimet. Then dXt+1(x,x+1)=k−1, and in the particle system the l particle coming from the edge (x+1,x+2) 8 annihilates the bottom r particle on the edge (x,x+1), while there is no incoming r particle from left. Lastly, supposeonly x isexcitedattime t. Thisistheonlycaseinwhichparticlesontheedge (x,x+1)couldflip,butProposition2.1rulesthisoutfort ≥t . Therefore,inthislastcaseweknow 0 thatk<(cid:98)(κ−1)/2(cid:99),sodXt+1(x,x+1)=k+1. Intheparticlesystem,theedge(x,x+1)receivesanr particle,whichmovestothetopofthequeue,sotherearek+1rparticlesattimet+1. Thisproves theassertion. The following lemma translates the temporal event of seeing a particle on a particular edge at timetintothespatialeventofhavingpositivepartialsumsatpriortimes;thisisanalogoustoaduality relationinthegraphicalconstructionofaninteractingparticlesystem,inwhichonetrackstheorigin ofaparticlebackwardsintime.ItwillbecrucialinSections3and5. Lemma2.3. Let(Xt)t≥0,0≤t0≤5κ,and(ξt)t≥t0 beasbefore.Thenwehavethefollowings: (i) Everyparticlemovesinitsprescribeddirectionuntilannihilationwithspeedbetween1/(cid:98)κ/2(cid:99)(κ+1) and1/(b(κ)+2).Ifκ=3,everyparticlemovesonceinevery3timesteps,sothespeedis1/3. (ii) Suppose the particle labeled (i,r) is on the edge (0,1) at time t = a+τ. Then (i,r) was on edge (x,x+1)attimet=aforsomex∈[−τ/(b(κ)+2),−τ/(cid:98)κ/2(cid:99)(k+1)]and x+k (cid:88) dX (i,i+1)≥−(cid:98)κ/2(cid:99)+1 forall1≤k≤τ/(cid:98)κ/2(cid:99)(k+1). (13) a i=x+1 (iii) Supposeκ=3. Then(i,r)isontheedge(0,1)attime t =a+3τifandonlyif(i,r)wasonedge (−τ,−τ+1)attimet=aand −τ+k (cid:88) dX (i,i+1)≥0 forall1≤k≤2τ. (14) a i=−τ+1 Proof. Toshow(i),suppose(i,r)isreleasedfromanedge(y,y+1)attimet≥aandeventuallyreleased againfromtheedge(y+1,y+2)withoutbeingannihilated. Notethat(y+1,y+2)musthaveeither noparticlesorsomerparticlesattimet. Supposetheformer,whichisalwaysthecasewhenκ=3. Wehave X (y)=b(κ)and X (y+1)=X (y+2)∈(b(κ),κ−1],sothetimeofthenextblinkof y+1is t t t betweent+b(κ)+2andt+κ.Inparticular,thisshowstheassertionforκ=3.Ontheotherhand,since weareassumingthatnoparticleflipsaftertimet =a,therecouldbeatmost(cid:98)κ/2(cid:99)−1rparticleson theedge(y+1,y+2)attimet.Notethaty+1canbeexcitedonlybyyuntilallrparticleson(y+1,y+2) havebeenreleased,so y+1blinksatleastonceineveryκ+1secondsuntilthen. Henceittakesat most(cid:98)κ/2(cid:99)(κ+1)secondsfor(y+1,y+2)torelease(i,r).Thisshows(i). Thefirstpartof(ii)followsfrom(i). Toseethesecondpart,observethatif(i,r)wasatheight0 (firstinitsqueue)attimet =a,inorderforittoreachtheedge(0,1),thenumberofrparticlesmust exceedthenumberoflparticlesoneveryinterval[x,x+i]⊂[x,0]. ThisobservationandLemma2.2 imply thepartialsumsin(ii)arenonnegative. Ingeneral, (i,r)couldbeatheightatmost(cid:98)κ/2(cid:99), in whichcase(i,r)startswithatmost(cid:98)κ/2(cid:99)−1rparticlesaheadofit,whichdefineditagainstlparticles. Hencethepartialsumsmustbeboundedbelowbyatleast−(cid:98)κ/2(cid:99)+1. 9 For(iii),firstsupposethat(i,r)isontheedge(0,1)attimet=a+3τ.By(i),(i,r)wasontheedge (−τ,−τ+1)attimet=a.Ifforsomek∈[1,2τ], −τ+k (cid:88) dX (i,i+1)<0, (15) a i=−τ+1 then Lemma 2.2 implies that there are more l particles than r particles in the interval [−τ+1,−τ+ k+1]. Oneoftheseextralparticleswillannihilatewith(i,r)bytimea+3(cid:100)k/2(cid:101)≤a+3τ,whichisa contradiction,sowemusthaveallpartialsumsnonnegative. Conversely,suppose(i,r)startsonthe edge (−τ,−τ+1) at time t =a and is annihilated at time t =a+3r +s, where 0≤r ≤τ, 0≤s ≤2, and3r +s ≤3τ. Thentheremustbethesamenumberoflparticlesasrparticlesinatleastoneof theintervals:[−τ,−τ+2r],[−τ,−τ+2r+1]or[−τ,−τ+2r+2]attimet=a,dependingontheinitial position of the l particle that annihilates (i,r), the timing of when the particles jump (observe that thesetwoparticlesmustjumpatthesametime,sincetheyultimatelyannihilate),andwhetherornot s>0(ifs=0,thenthelastintervalcanbeexcluded). ByLemma2.2,thisimpliesthatthepartialsum in(iii)isnegativeforsomek∈{2r−1,2r,2r+1}(ifs=0,k∈{2r−1,2r}),whichprovestheassertion. Proof of Proposition 2.1. Suppose κ=2m+1 for some m ∈(cid:78). Note that b(κ)=m. To show (i), suppose for the sake of contradiction that an edge (x,x+1) flips at time t ≥ m. We may assume withoutlossofgeneralitythatdX (x ,x )=mandx isexcitedattimet.ThisrequiresX (x )=m+k t 1 2 1 t 1 for some k ∈[1,m] and x has another neighbor x ∈V such that X (x )=m. This yields X (x )≡ 1 0 t 0 t 2 2m+k≡k−1∈[0,m−1]moduloκ=2m+1.Sincex cannotbeexcitedwhenitscoloris<b(κ)=m, 2 it follows that Xt−k+1(x2) = 0 and Xt−k = 2m. On the other hand, note that x1 has no neighbors besidesx andx ,neitherofwhichblinksduringthetimeinterval[t−k,t−1]. Henceitfollowsthat 0 2 Xt−k(x1)=m.Butthenx1excitesx2attimet−k,sowemusthaveXt−k+1(x2)=2m,acontradiction. Nowsupposeκ=2mforsomeintegerm≥2.Weuseasimilarbacktrackingargumenttoruleout flippingparticlesafteraconstanttimedependingonm.Recallthatb(2m)=m−1istheblinkingcolor here.Suppose,forthesakeofcontradiction,wehaveaflippingparticleonedgex x attimet=N for 1 2 someintegerN >10m.Wefirstclaimthat{X (x ),X (x )}={m−1,2m−1}forsomet ≥9m.Without t1 1 t1 2 1 lossofgenerality,wemayassumedX (x ,x )=m−1andx isexcitedbyablinkingneighborx (cid:54)=x N 1 2 1 0 2 attimet=N.SoX (x )=m−1,X (x )=m−1+kforsomek∈[1,m],andX (x )≡2m−2+k≡k−2 N 0 N 1 N 2 moduloκ=2m.Asbefore,noneofx andx blinkduringtimeinterval(N−k,N],sobacktrackingk−2 0 2 iterationsgivesXN−k−2(x1)=m+1andXN−k−2(x2)=0andonemoreiterationgivesXN−k−1(x1)=m andXN−k−1(x2)=2m−1.ThisyieldsXN−k(x1)=m−1andXN−k−2(x2)=2m−1sincex1excitesx2at timet :=N−k.Thisshowstheclaim. 1 Next,weshowthataswefurtherbacktrackthedynamics,thesame“opposite”localconfiguration {X (x ),X (x )}={m−1,2m−1}appearswithperiodm+1. Sincet ≥9m≥6(m+1),thisyieldsthat t 1 t 2 1 thesamepatternappearsatleastseventimesduring[0,t ]. Wewillthenobtainacontradiction. For 1 simplicity,weworkwithκ=6casebutgeneralizingtoanyevenκisimmediate. In the following backtracking tables, time increases from right to left and each column gives a localconfigurationonx ,x andtheirneighbors.Thefirstcolumnshowstheoppositelocalconfigu- 1 2 10