Fluctuations of a surface submitted to a random av

ElectronicJournalofProbabilityVol.3(1998)Paperno.6,pages1-34.JournalURL:~ejpecp/PaperURL:~ejpecp/EjpVol3/paper6.abs.htmlAuthors’URL’s:~pablo,~lrenatoFluctuationsofasurfacesubmittedtoarandomaverageprocessP.A.FerrariL.R.G.FontesUniversidadedeS~[e-mail],[e-mail]-USP,CxPostal66281,05315-970,S~aoPaulo,BrazilAbstract:Weconsiderahypersurfaceofdimensiondimbeddedinad+1dimensionalspace.Foreachx2Zd,lett(x)2Rbetheheightofthesurfaceatsitexattimet.Atrate1thex-thheightisupdatedtoarandomconvexcombinationoftheheightsofthe‘neighbors’ofx.Thedistributionoftheconvexcombinationistranslationinvariantanddoesnotdependontheheights.Thismotion,namedtherandomaverageprocess(RAP),isoneofthelinearprocessesintroducedbyLiggett(1985).SpecialcasesofRAPareatypeofsmoothingprocess(whentheconvexcombinationisdeterministic)andthevotermodel(whentheconvexcombinationconcentratesononesitechosenatrandom).Westarttheheightslocatedonahyperplanepassingthroughtheoriginbutdierentfromthetrivialone(x)0.Weshowthat,whentheconvexcombinationisneitherdeterministicnorconcentratingononesite,thevarianceoftheheightattheoriginattimetisproportionaltothenumberofreturnstotheoriginofasymmetricrandomwalkofdimensiond.Undermildconditionsonthedistributionoftherandomconvexcombination,thisgivesvarianceoftheorderoft1=2indimensiond=1,logtindimensiond=2andboundedintindimensionsd3.Wealsoshowthatforeachinitialhyperplanetheprocessasseenfromtheheightattheoriginconvergestoaninvariantmeasureonthehypersurfacesconservingtheinitialasymptoticslope.Theheightattheoriginsatisesacentrallimittheorem.Toobtaintheresultsweuseacorrespondingprobabilisticcellularautomatonforwhichsimilarresultsarederived.Thisautomatoncorrespondstotheproductof(innitelydimensional)independentrandommatriceswhoserowsareindependent.Keywords:randomaverageprocess,randomsurfaces,productofrandommatrices,linearprocess,votermodel,smoothingprocess.AMSsubjectclassication:60K35,82CSubmittedtoEJPonApril10,1997.FinalversionacceptedonMay15,1998.11IntroductionWeconsiderastochasticprocesstinRZd.Toeachsitei2Zdateachtimetcorrespondsaheightt(i).Theseheightsevolveaccordingtothefollowingrule.Foreachi2Zdletu(i;)bearandomprobabilitydistributiononZd.EachheighthasanindependentPoissonclockofparameter1.Whentheclockringsforsiteiattimet,arealizationofuischosen,independentofeverything,andthentheheightatsiteimovestothepositionXj2Zdu(i;j)t(j):Inotherwords,atrateoneeachheightisreplacedbyarandomconvexcombinationofthecurrentheights.Theweightsofthisconvexcombinationarechosenindependentlyateachtime.Wecallthisprocesstherandomaverageprocess(RAP).Themotioniswelldenedundersuitableconditionsonthedistributionsofuand0.TheformalgeneratorisgivenbyLf()=XiZd(u)[f(ui)f()](1.1)whereisthedistributionofthematrixuanduiisdenedastheoperator:(ui)(k)=((k)ifk6=iPju(i;j)(j)ifk=i.(1.2)Thatis,uireplacestheheightatibyaconvexcombinationoftheheightsatZd.TheRAPisaspecialcaseofthelinearprocessesofchapterIXofLiggett(1985).ParticularexamplesoftheRAParethenoiselesssmoothingprocessandthevotermodel.Inthenoiselesssmoothingprocessthedistributionconcentratesallitsmassonacon-stantmatrix;thatis,thereexistsamatrixasuchthat(u=a)=1.LiggettandSpitzer(1981),Andjel(1985)andLiggett(1985)studiedthe(noisy)smoothingprocess:ateacheventofthePoissonprocesses,the(deterministic)randomconvexcombinationismultipliedbyanindependentpositiverandomvariableWofmean1|thenoise.Theabovepapersstudiedquestionsaboutexistenceandergodicityoftheprocesswhentheheightsarerestrictedtobenonnegative.Itwouldbeinterestingtostudytheergodicityquestionsforthecaseofgeneralinitialconditions.Inthevotermodelconcentratesmassonthesetofmatricesuwiththefollowingproperty:foreachithereisexactlyonejwithu(i;j)dierentfromzero|andhenceequalto1.Inotherwords,whentheclockringsforheighti,itisreplacedbytheheightjforwhichu(i;j)=1.Intheusualvotermodeltheheightsassumeonlytwovalues.SeeDurrett(1996)forarecentreviewonthevotermodel.Inthispaperwewillconcentrateontheothercases,butsomemarginalresultswillconcernthesmoothingprocessandthevotermodel.Thelattermodelwillbediscussedbrieyalsointhenalsectionofthepaper.Formostofourresultstheinitialcongurationisahyperplanepassingthroughtheorigin:givenavector2Rd,foreachi2Zdweset0(i)=i;(1.3)2the(matrix)productofiandthetransposeof(i.e.theinnerproduct).Noticethatif0,thent0becausethisinitialcongurationisinvariantfortheprocess.Weassumethatthedistributionofu(i;j)istranslationinvariant:u(i;j)andu(i+k;j+k)havethesamedistribution.Ourmainresultistoshowthatwhentheinitialcongurationisahyperplane,thevarianceoftheheightattheoriginattimetisproportionaltotheexpectedtimespentintheoriginbyasymmetricrandomwalkperturbedattheorigin.DenotingVasthevariance,weshowthatif0(i)=iforalli,thenVt(0)=(2+2)Zt0P(Ds=0jD0=0)ds(1.4)where=Xjju(0;j);2=24Xjju(0;j)235(1.5)andDtisarandomwalk(symmetric,perturbedattheorigin)withtransitionratesq(0;k)=Xj(u(0;j)u(0;j+k))q(‘;‘+k)=u(0;k)