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Submitted to the Annals of Applied Probability ASYMPTOTIC SHAPE FOR THE CONTACT PROCESS IN RANDOM ENVIRONMENT By Olivier Garet and Regine Marchand University of Nancy The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems gen- eralize known results for the classical contact process. In particular, if Ht denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set Ht/t almost surely converges to a compact set that only depends on the law of the envi- ronment. To this aim, we prove a new almost subadditive ergodic theorem. 1. Introduction. The aim of this paper is to obtain an asymptotic shape theorem for the contact process in random environment on Zd. The ordinary contact process is a famous interacting particle system modelling the spread of an infection on the sites of Zd. In the classical model, the evolution depends on a fixed parameter ? ? (0,+∞) and is as follows: at each moment, an infected site becomes healthy at rate 1 while a healthy site becomes infected at a rate equal to ? times the number of its infected neighbors. For the contact process in random environment, the single infection parameter ? is replaced by a collection (?e)e?Ed of random variables indexed by the set Ed of edges of the lattice Zd: the random variable ?e gives the infection rate between the extremities of edge e, while each site becomes healthy at rate 1.

- environment ?
- independence properties
- site becomes
- when ? ?
- random environment
- contact process
- translation operator
- infected before time

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The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems gen-eralize known results for the classical contact process. In particular, ifHt denotes the set of already occupied sites at timet, we show that for almost every environment, when the contact process survives, the setHt=talmost surely converges to a compact set that only depends on the law of the envi-ronment. To this aim, we prove a new almost subadditive ergodic theorem.

1.IcgedRhPgidc.The aim of this paper is to obtain an asymptotic shape theorem for the contact process in random environment onZB. The ordinary contact process is a famous interacting particle system modelling the spread of an infection on the sites ofZB. In the classical model, the evoluti n depends on a ﬁxed parameter o ζ2(0τ+∞) and is as follows: at each moment, an infected site becomes healthy at rate 1 while a healthy site becomes infected at a rate equal toζtimes the number of its infected neighbors. For the contact process in random environment, the single infection parameterζis replaced by a collection (ζE)Eαof random 2E variables indexed by the setEBof edges of the latticeZB: the random variableζE gives the infection rate between the extremities of edgeS, while each site becomes healthy at rate 1. We assume that the law of (ζE)E2Eαis stationary and ergodic. From the application point of view, allowing a random infection rate can be more realistic in modelizing real epidemics { note that in his book [15], Durrett already underlined the inadequacies of the classical contact process in the modelization of an infection among a racoon rabbits population, and proposed the contact process in random environment as an alternative. Our main result is the following: if we assume that the minimal value taken by the (ζE)E2Eαis aboveζA(ZB) { the critical parameter for the ordinary contact process oZB{ then there exists a normηonRBsuch that for almost every environment n ζ= (ζE)E2Eα, the setHbof points already infected before timeisatisﬁes: P(9T >0iT=)(1 ")iHb+⊂)= 1τ ε⊂˜ (1 +")iε

whereH˜b=Hb+ [0τ1]B,εis the unit ball forηandPis the law of the contact process in the environmentζ tact, conditioned to survive. The growth of the co n process in random environment conditioned to survive is thus asymptotically linear

1AC2(((bdLVPMcMXKbbUMKcU][b:Primary 60K35; secondary 82B43 9Phf]aNbK[NpTaKbPb:random growth, contact process, random environment, almost subad-ditive ergodic theorem, asymptotic shape theorem

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OLIVIER GARET AND R EGINE MARCHAND

in time, and governed by a shape theorem as in the case of the classical contact process oZ. nB Until now, most of the work devoted to the study of the contact process in random environment focuses on determining conditions for its survival (Liggett [31], Andjel[3],NewmanandVolchan[33])oritsextinction(Klein[29]).Theyalso mainly deal with the case of dimensionR1. Concerning the speed of the growth= whenR= 1, Bramson, Durrett and Schonmann [6] show that a random environment can give birth to a sublinear growth. On the contrary, they conjecture that the growth should be of linear order forR2 as soon as the survival is possible, and that an asymptotic shape result should hold. For the classical contact process, the proof of the shape result mainly falls in tw o parts: The result is ﬁrst proved for large values of the infection rateζby Durrett and Griﬀeath [16] in 1982. They ﬁrst obtain, for largeζ, estimates essentially implying that the growth is of linear order, and then they get the shape result with superconvolutive techniques. Later, Bezuidenhout and Grimmett [4] show that a supercritical contact pro-cess conditioned to survive, when seen on a large scale, stochastically dom-inates a two-dimensional supercritical oriented percolation: this guarantees the at least linear growth of the contact process. They also indicate how their construction could be used to obtain a shape theorem. This last step essen-tially consists in proving that the estimates needed in [16] hold for the whole supercritical regime, and is done by Durrett [1 ] in 1989. 7 Similarly, in the case of a random environment, proving a shape theorem can also fall into two diﬀerent parts. The ﬁrst one, and undoubtedly the hardest one, would be to prove that the growth is of linear order, as soon as survival is possible: this correspondstotheBezuidenhout{Grimmettresultinrandomenvironment.The second one, which we tackle here, is to prove a shape theorem under conditions assuring that the growth is of linear order: this is the random environment analogous oftheDurrett{Griﬀeathwork.Wethuschosetoputconditionsontherandom environment that allow to obtain, with classical techniques, estimates similar to the ones needed in [16] and to focus on the proof of the shape result, which already presentsseriousadditionaldicultieswhencomparedtotheproofintheclassical case. The history of shape theorems for random growth models begins in 1961 with Eden [18] asking for a shape theorem for a tumor growth model. Richardson [35] proves then in 1973 a shape result for a class of models including Eden model, by using the technique of subadditive processes initiated in 1965 by Hammersley and Welsh [21] for ﬁrst-passage percolation. From then, asymptotic shape results for random growth models are usually proved with the theory of subadditive proce se , s s and more precisely with Kingman’s subadditive ergodic theorem [27] and its exten-sions. The most famous example is the shape result for ﬁrst passage-percolation oZB(see also diﬀerent variations of this model: Boivin [5], Garet and Marc-n hand[19],Vahidi-AslandWierman[39],HowardandNewmann[24],Howard[23], Deijfen [10]).

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CONTACT PROCESS IN RANDOM ENVIRONMENT

2

The random growth models can be classiﬁed in two families. The ﬁrst and most studied one is composed of the permanent models, in which the occupied set at timeiis non-decreasing and extinction is impossible. First of all are of course Richardson models [35]. More recently, we can cite the frog model, introduced in its continuous time version by Bramson and Durrett, and for which Ram´rez and Sidoravicius [34] obtained a shape theorem, and also the discrete time version, ﬁrst studied by Telcs and Wormald [38] and for which the shape theorem has been obtained by Alves et al. [1, 2]. We can also cite the branching random walks by CometsandPopov[9].Inthesemodels,themainpartoftheworkistoprovethat the growth is of linear order, and the whole convergence result is then obtained by subadditivity. The second family contains non-permanent models, in which extinction is possi-be.Inthiscase,weratherlookforashaperesultunderconditioningbythesurvival. l Hammersley [20] himself, from the beginning of the subadditive theory, underlined thedicultiesraisedbythepossibilityofextinction.Indeed,ifwewanttoprove that the hitting times (i()d2Zαare such thati(c)ωcconverges, Kingman’s the-ory requires subadditivity, stationarity and integrability properties for the collectio n i(). Of course, as soon as extinction is possible, the hitting times can be inﬁ ite. n Moreover, conditioning on the survival can break independence, stationarity and even subadditivity properties. The theory of superconvolutive distributions was de-velopped to treat cases where either the subadditivity or the stationarity property lacks: see the lemma proposed by Kesten in the discussion of Kingman’s paper [27], andslightlyimprovedbyHammersley[20](page674).Notethatrecently,Kesten and Sidoravicius [26] use the same kind of techniques as an ingredient to prove a shape theorem for a model of spread of an infection. Following Bramson and Griﬀeath [7, 8], it is on these “superconvolutive" tech-niques that Durrett and Griﬀeath [16] rely to prove the shape result for the classical contact process onZB{ see also Durrett [15], that corrects or clariﬁes some points of [16]. However, as noticed by Liggett in the introduction of [30], superconvolutive techniques require some kind of independence of the increments of the process that canlimititsapplication.Itisparticularlythecaseinarandomenvironmentsetting: for the hitting times, we have a subadditive property of type i((c e))≤i(c) +i˜ndλ(e) +g(cτ eτ )σ + Here, the exponent gives the environment,i˜ndλ(e) has the same law as the hitting time ofebut in the translated environmentcσζ, andg(cτ eτ ) is to be thought as a small error term. Following the superconvolutive road would require thati(c) andi˜ndλ(e) are independent and thati˜ndλ(e) has the same law asi(eoN.),w if we work with a given { quenched { environment, we lose all the spatial stationarity properties:i˜ndλ(e) has no reason to have the same law asi(e).Buweroitwfk under the annealed probability, we lose the markovianity of the contact process and the independence properties it oﬀers: we thus cannot use, at least directly, the superc nvolutive techniques. o Liggett’s extension [30] of the subadditive ergodic theorem provides an alternate approach when independence properties fail. However, it does not give the possi-bility to deal with an error term. Some works in the same decade (see for instance

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Derriennic[11],DerriennicandHachem[12],andSchurger[36,37])proposealmost subadditive ergodic theorems that do not require independence, but stationarity as-sumptions on the extra term are too strong to be used here. Thus we establish, with techniques inspired from Liggett, a general subadditive ergodic theorem allowing an error term that matches our situation. Infact,wedonotapplythisalmostsubadditiveergodictheoremdirectlyto the collection of hitting timesi(cudoehtenauqytit,butweratherintr)ξ(), that can be seen as a regeneration time, and that represents a time when siteis occupied and has inﬁnitely many descendants. Thisξhas stationarity and almost subadditive properties thatilacks and thus ﬁts the requirements of our almost subadditive ergodic theorem. Finally, by showing that the gap betweeniandξis ot too large, we transpose toithe shape result obtained forξ. n

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2.1.WN.fadUW3WhifelonIht,wngwiloebotenedykσk-andkσk∗the norms onRBrespectively deﬁned bykk-=PKB=-|K|andkk |. The ∗=-mKaxB K| notationkσkwill be used for an unspeciﬁed norm. We ﬁxζA(ZB)ψ ζ@in≤ζ@a[ψ+∞, whereζA(ZB) stands for the critical param-eter for the classical contact process inZBestrictourstudyfehtollogniwrew,In. E to random environmentsζ= (ζE)E2Eαtaking their value in = [ζ@niτ ζ@[]α. An a . environment is thus a collectionζ= (ζE)E2Eα2 Letζ2 be ﬁxed. The contact process (λb)b+in environmentζis a homo-geneous Markov process taking its values in the setP(ZB) of subsets ofZB. For z2ZBwe also use the random variableλb(z) =1⊗z2. Ifλb(z) = 1, we say thatz g t is occupied or infected, while ifλb(z) = 0, we say thatzis empty or healthy. The evolution of the process is as follows: an occupied site becomes empty at rate 1, an empty sitezbecomes occupied at rate∑λb(z−)ζ⊗zµz0gτ kz z0k)=-eachoftheseevolutionsbeingindependentfromtheothers.Inthefollowing,we denote byDml`agfunctionsfrohtsetefo`cdaR+toP(ZB): it is the set of trajectories for Markov processes with state spaceP(ZB.) To deﬁne the contact process in environmentζ2con-aHehsirrew,tesu struction[22].Itallowstocouplecontactprocessesstartingfromdistinctinitial conﬁgurations by building them from a single collection of Poisson measures onR+.

2.2.CdgLfaWefNRfOaWaiUWaeeiaPe.dNegNGWe endowR+with the Borelξ-+unting measures algebraB(Rwdna,)yotebedenBthe set of locally ﬁnite co b=P+K=+∗bi. We endow this set with theξ-algebraMgenerated by the maps b!7b(.), where.describes the set of Borel sets in R. + Wethendeﬁnethemeasurablespace(ΩτF) by setting α α α α E Z Ω =BEBZand MF
σ =M

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