Law of large numbers for random walks on attractive spin-flip dynamics
aa r X i v : . [ m a t h . P R ] S e p Law of large numbers for random walkson attractive spin-flip dynamics
Stein Andreas Bethuelsen Markus Heydenreich September 28, 2018
Abstract
We prove a law of large numbers for certain random walks on certain attractive dynamicrandom environments when initialised from all sites equal to the same state. This result ap-plies to random walks on Z d with d ≥
1. We further provide sufficient mixing conditions underwhich the assumption on the initial state can be relaxed, and obtain estimates on the largedeviation behaviour of the random walk.As prime example we study the random walk on the contact process, for which we obtaina law of large numbers in arbitrary dimension. For this model, further properties about thespeed are derived.
MSC2010.
Primary 82C41; Secondary 82C22, 60K37, 60F10, 60F15, 39B62.
Key words and phrases.
Random walks, dynamic random environments, strong law of large num-bers, large deviation estimates, coupling, monotonicity, sub-additivity, contact process.
Random walks in random environment (RWRE) gained much interest throughout the last decades.Such models serve as natural extensions of the classical random walk model and have broad ap-plications in physics, chemistry and biology.RWRE models show significantly different behaviours than the simple random walk model.This was already observed in one of the first models studied, Solomon [25], where it was shownthat the random walk can behave sub-ballistically. Non-Gaussian scaling limits were establishedfor the same model in Kesten, Kozlov, and Spitzer [15] and Sina˘ı [24]. These characteristics are dueto trapping phenomena.RWRE models on Z are by now well understood in great generality whenever the environmentis static, i.e. it does not change with time. On the other hand, for RWRE models on Z d , d ≥ Technische Universität München, Fakultät für Mathematik, Boltzmannstr. 3, 85748 Garching,Germany. Email: [email protected] Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, Germany.Email: [email protected]
1n the last decade, much focus has been devoted to models where the random environmentevolves with time, i.e. random walks in dynamic random environments (RWDRE). It is believedthat the extent to which trapping phenomena occur for RWDRE models depends on the correlationstructure of the dynamics.At a rigorous level, it is known to great generality that RWDRE models scale diffusively whenthe environment is only weakly correlated in space-time; see for instance Redig and Völlering[22]. These results are not restricted to random walks on Z , but are valid in any dimension. Hereweakly correlated essentially means that the environment becomes approximately independent ofits starting configuration within a space-time cone, also known as cone mixing environment.Little is known at a general level when the environment has a non-uniform correlation struc-ture, though trapping phenomena are conjectured to occur for some specific models (Avena andThomann [1]). Avena, Hollander, and Redig [2] have shown rigorously that a random walk on theone-dimensional exclusion process exhibits trapping phenomena at the level of large deviationsunder drift assumptions. On the other hand, several other models with a non-uniform correlationstructure have been shown to possess diffusive scaling limits, for example Avena, Santos, and Völ-lering [4], Hilário, den Hollander, Sidoravicius, dos Santos, and Teixeira [11], den Hollander anddos Santos [12], Huveneers and Simenhaus [14] and Mountford and Vares [19].In this paper we present a strong law of large numbers (SLLN) for random walks on certainattractive (or monotone) interacting particle system (IPS). For this, restrictions on both the randomwalk and the IPS are required. In particular, we assume that the sites of the IPS take values 0 or 1and that the IPS has a graphical representation coupling which is monotone with respect to theinitial configuration. One class of IPS satisfying the latter assumption are additive and attractivespin-flip systems.The SLLN is obtained when the IPS is initialised at time 0 from a configuration where all siteshave the same value, assuming in addition that the jump transitions of the random walk only de-pend on the state of the IPS at the position of the random walk. Under certain mixing conditions,we are able to relax the restriction on the starting configuration.An important feature of the SLLN is that it does not rely directly on the correlation structureof the environment, but rather assumes monotonicity. In particular, the SLLN applies to a largeclass of models with non-uniform correlation structure not previously considered in the literature.Furthermore, the SLLN applies to random walks on Z d for any d ≥ Outline
The rest of the paper is organised as follows. In the next subsection we give a precise definition ofour model and in Subsection 1.3 we present our main results. Subsection 1.4 contains a discussionof related literature. Section 2 is devoted to a particular coupling construction of the environment2nd the random walk, yielding a monotonicity property, important for our results. In Section 3 wepresent the proofs of our main theorems for general attractive environments. Section 4 is devotedto the special case of a random walk on the contact process.
We first introduce the environment. For this, let d ≥ Ω = {0, 1} Z d the configurationspace and by D Ω [0, ∞ ) the corresponding path space, that is, the set of càdlàg functions on [0, ∞ )taking values in Ω .As the environment we consider an IPS, ξ = ( ξ t ) t ≥ , such that ξ t = © ξ t ( x ) : x ∈ Z d ª is in Ω , t ∈ [0, ∞ ), and ξ ∈ D Ω [0, ∞ ). The process ξ starting from ξ = η is denoted by ξ η and its law is givenby P η . When ξ is drawn from µ ∈ P ( Ω ), the set of probability measures on Ω , we write ξ µ for thecorresponding process. Its law is denoted by P µ and is given by P µ ( · ) = Z Ω P η ( · ) µ ( d η ). (1.1)We assume throughout that ξ is translation invariant, that is, P η ( θ x ξ t ∈ · ) = P θ x η ( ξ t ∈ · ) (1.2)with θ x denoting the shift operator θ x η ( y ) = η ( y − x ), η ∈ Ω .Further, to the configuration space Ω we associate the partial ordering such that ξ ≤ η with ξ , η ∈ Ω if and only if ξ ( x ) ≤ η ( x ) for all x ∈ Z d . A function f : Ω → R is called increasing if ξ ≤ η implies f ( ξ ) ≤ f ( η ). For two measures µ , µ ∈ P ( Ω ), µ stochastically dominates µ , written µ ≤ µ , provided that Z Ω f d µ ≤ Z Ω f d µ (1.3)for all increasing continuous functions f on Ω . We denote by δ ¯0 , δ ¯1 ∈ P ( Ω ) the extremal measureswhich put all their weight on the configurations ¯1 and ¯0, respectively, where ¯ i ( x ) = i for all x ∈ Z d , i ∈ {0, 1}. Obviously, it holds that δ ¯0 ≤ δ ¯1 .For a fixed realisation of ( ξ t ) t ≥ , let ( W t ) t ≥ be the time-inhomogeneous Markov process on Z d that, given W t = x , jumps to x + z at rate α ( ξ t ( x ), z ) for some function α : {0, 1} × Z d → [0, ∞ ). Wecall this process the random walk . Further, we assume throughout that γ : = max i ∈ {0,1} ( α ( i , o ) + X z ∈ Z d k z k α ( i , z ) ) < ∞ , (1.4)where o ∈ Z d denotes the origin. Thus, the speed of the simple random walk seeing only occupiedsites ( i =
1) or only vacant sites ( i =
0) is given by the local drifts u i : = X z ∈ Z d α ( i , z ) z , i ∈ {0, 1} . (1.5)We say that ( W t ) is elliptic if, for the unit vectors { ± e j } j = d , we havemin i ∈ {0,1} min j ∈ {1,..., d } © α ( i , ± e j ) ª >
0. (1.6)We also say that ( W t ) has finite second moments ifmax i ∈ {0,1} ( X z ∈ Z d α ( i , z ) k z k ) < ∞ (1.7)3nd that it has finite exponential moments if there exist κ > i ∈ {0,1} ( X z ∈ Z d α ( i , z ) exp ( ǫ k z k ) ) < ∞ , for all 0 < ǫ < κ . (1.8)Lastly, for ξ ∈ D Ω [0, ∞ ) and x ∈ Z d , let P ξ x denote the law of ( W t ) starting from W = x in a fixedenvironment ξ , which is the quenched law of W . The annealed law of W is given by P µ , x ( · ) = Z D Ω [0, ∞ ) P ξ x ( · ) P µ ( d ξ ). (1.9) General IPS can formally be constructed by defining a generator, see Liggett [16, Chapter I.1-3].Alternatively, one can describe an IPS via a countable set of Poisson processes I , yielding a moreprobabilistic description, see Durrett [8]. The probabilistic construction has the advantage thatit yields a natural coupling, denoted by b P , of the dynamics starting from any configuration on ajoint probability space. For many interacting particle systems this coupling can be constructedexplicitly and is known as the graphical representation. Important to our approach is the existenceof such a coupling b P of the dynamic environment ξ which satisfies the following monotonicityproperty, b P ¡ ξ η t ≤ ξ ω t , ∀ t > η , ω ∈ Ω satisfying η ≤ ω ¢ =
1. (1.10)A coupling b P satisfying (1.10) is said to be an attractive graphical representation coupling . Theorem 1.1 (Strong law of large numbers) . Assume that ξ has an attractive graphical representa-tion coupling. Then, for each i ∈ {0, 1} , there exists ρ i ∈ [0, 1] such that lim t →∞ t W t = ρ i u + (1 − ρ i ) u , P δ ¯ i , o -a.s. and in L . (1.11)Note that Theorem 1.1 does not require ( W t ) to be elliptic nor set restrictions on finite rangejump transitions, technical assumptions often present in the literature. Further, Durrett [8, Theo-rem 2.5] yields a large class of IPS with spin-flip dynamics having an attractive graphical represen-tation coupling to which Theorem 1.1 applies.The proof of Theorem 1.1 makes use of a particular coupling construction of the random walktogether with the sub-additive ergodic theorem. The coupling construction (given in Section 2)enables us to transfer monotonicity properties of the environment to a functional of ( W t ). Infor-mally, this functional counts the number of occupied sites the random walk observes at its jumptimes, as a function of time. The monotonicity property of this functional together with the graph-ical representation of the environment naturally leads to a sub-additive structure which we use toobtain a SLLN by employing the sub-additive ergodic theorem (with limit equal to ρ i as in (1.11)).This is the content of Theorem 3.2 below. As a last step, the SLLN in Theorem 3.2 is transferred intoa SLLN for ( W t ), whose proof is given in Section 3.1.The restriction to the extremal starting configurations in Theorem 1.1 can in many cases berelaxed. For m >
0, let V m : = n ( x , t ) ∈ Z d × [0, ∞ ) : k x k < mt o (1.12)4e a cone of inclination m opening upwards in space-time. Let ( ξ t ) be an IPS with an attractivegraphical representation coupling and, for η ∈ Ω , i ∈ {0, 1} and T ∈ [0, ∞ ), define φ ( m ) i ( η , T ) : = b P ³ ∃ ( x , t ) ∈ V m ∩ Z d × [ T , ∞ ) : ξ η t ( x ) ξ ¯ it ( x ) ´ . (1.13)Note that ξ is cone mixing, as defined in [3, Definition 1.1], iflim T →∞ φ ( m )1 (¯0, T ) = ∀ m ∈ [0, ∞ ). (1.14)Further, denote by ¯ ν , ¯ ν ∈ P ( Ω ) the “lower” and “upper” invariant measures to which ( η t ) con-verges when initialised from ¯0 and ¯1 respectively. That is, we have ¯ ν = lim t →∞ P ¯0 ( ξ t ∈ · ) and¯ ν = lim t →∞ P ¯1 ( ξ t ∈ · ). These limits exist and are invariant under ( ξ t ) (see [16, Theorem III.2.3]).Lastly, recalling (1.5), we denote the convex hull of u and u by U ( u , u ) : = conv ( u , u ) . (1.15) Theorem 1.2.
Assume that ξ has an attractive graphical representation coupling. Let i ∈ {0, 1} andassume that for some ǫ , m > we have U ( u , u ) × {1/ γ } ⊂ V m (1 − ǫ ) and lim T →∞ φ ( m ) i ( η , T ) = for ¯ ν i − a . e . η ∈ Ω . (1.16) Then, for all ν ∈ P ( Ω ) such that ν ≤ ¯ ν i (if i = ) or ¯ ν i ≤ ν (if i = ), lim t →∞ t W t = ρ i u + (1 − ρ i ) u P ν , o -a.s. and in L . (1.17)Theorem 1.2 relaxes the assumption on the starting configuration in Theorem 1.1. Note that(1.16) is weaker than cone mixing (in the sense of (1.14)) and applies to IPS with non-uniformcorrelation structure. The proof of Theorem 1.2 uses a different coupling construction than thatneeded for the proof of Theorem 1.1 and is given in Section 3.2.From the proof of Theorem 1.1, we also obtain certain large deviation estimates. These arepresented in Section 3.3. In particular, we have the following theorem. Theorem 1.3 (Large deviation estimates) . Assume that ξ has an attractive graphical representationcoupling. Further, assume ρ = ρ = : ρ (with ρ and ρ as in Theorem 1.1). Consequently, ( W t ) satisfies (1.11) P µ , o -a.s with limiting speed v : = ρ u + (1 − ρ ) u , irrespectively of µ ∈ P ( Ω ) . If, inaddition, ( W t ) has finite exponential moments, then for any ǫ > there exists C ( ǫ ) > such that P µ , o ¡ k W t − t v k > ǫ t ¢ ≤ exp( − C ( ǫ ) t ) , for all µ ∈ P ( Ω ). (1.18)The proof of Theorem 1.3 is given in Section 3.3, where we also prove some additional largedeviation properties. The key observation for the proof is that the sub-additive structure obtainedfor the functional used in the proof of Theorem 1.1 yields one-sided large deviations estimates forthis functional. The assumption that ρ = ρ implies two-sided large deviation estimates, however,for the same functional. An additional argument is needed in order to conclude large deviationestimates for ( W t ). For this, we use the assumption that ( W t ) has finite exponential moments.5 andom walk on the contact process One classical IPS having an attractive graphical representation coupling is the contact process ξ = ( ξ t ) t ≥ . Given λ ∈ (0, ∞ ), the contact process on Z d with “infection rate” λ is defined via its localtransition rates, which are given by η → η x with rate ½
1, if η ( x ) = λ P x ∼ y η ( y ), if η ( x ) = η x is defined by η x ( y ) : = η ( y ) for y x , and η x ( x ) : = − η ( x ), and P x ∼ y denotes the sum overnearest neighbours.Much is known about the contact process; see [17, Chapter 1] for a thorough introduction. Inparticular, the empty configuration ¯0 is an absorbing state for the contact process. On the otherhand, starting from the full configuration ¯1, the contact process evolves towards an equilibriummeasure ¯ ν λ , called the “upper invariant measure”, which is stationary and ergodic with respect to( ξ t ). Further, there is a critical threshold λ c ( d ) ∈ (0, ∞ ), depending on the dimension d , such that¯ ν λ = δ ¯0 for λ ∈ (0, λ c ( d )] and, for all λ ∈ ( λ c ( d ), ∞ ), we have ¯ ν λ ( η ( o ) = >
0. In particular, for λ > λ c ( d ), the contact process does not satisfy (1.14). Theorem 1.4.
Consider the contact process on Z d with d ≥ and infection rate λ ∈ ( λ c ( d ), ∞ ) . a) There exists ρ = ρ ( λ ) ∈ [0, 1] such that for all ν ∈ P ( Ω ) with ¯ ν λ ≤ ν , lim t →∞ t W t = u ρ + (1 − ρ ) u P ν , o -a.s. and in L . (1.19) b) The function ρ : ( λ c ( d ), ∞ ) → [0, 1] , λ ρ ( λ ) , is non-decreasing and right-continuous in λ .Moreover, if ( W t ) has finite second moments, then ρ ( λ ) ∈ (0, 1) and lim λ →∞ ρ ( λ ) =
1. (1.20)Theorem 1.4 extends the law of large numbers of [12], obtained for the nearest neighbour ran-dom walk on the supercritical contact process on Z , to higher dimensions and beyond the nearestneighbour assumption.Concerning the proof of Theorem 1.4, note that a) follows immediately from Theorem 1.1 andthe graphical representation coupling of the contact process when started from ¯1. To extend thisto any measure stochastically dominated by ¯ ν λ , we prove that the contact process satisfies (1.16).The function ρ ( · ) in Theorem 1.4b) is the same functional as considered in the proof of Theo-rem 1.1. That this is non-decreasing and right-continuous is not difficult to show and follows bymonotonicity considerations. Most of the proof of Theorem 1.4b) goes about showing that (1.20)holds. Note that this result implies that the SLLN in (1.19) is non-trivial in the sense that the speedof ( W t ) is neither u nor u . For the proof of (1.20), we treat the two cases d = d ≥ d = d ≥ ρ ( · ). Section 4 is dedicated to the proof of Theorem 1.4.
1. The SLLN for random walks on a 2-state IPS has been proven earlier by Avena, Hollander,and Redig [3] under strong mixing assumptions on the environment, known as cone mixing.6his has been extended to more general IPS by Redig and Völlering [22], however, still undera uniform mixing assumption similar to cone mixing.Theorem 1.1 in this paper yields an extension of the SLLN in [3] to random walks on IPSwhich are not cone mixing, but satisfy a monotonicity property. Indeed, instead of conemixing, we assume that the IPS has an attractive graphical representation coupling and isstarted from a configuration where all sites are equal. Theorem 1.2 present sufficient mixingconditions for relaxing the restriction on the starting configuration.Contrary to [3] and [22], it is essential to the proof of Theorem 1.1 that the random walkonly has two transition kernels. That is, at jump times, the random walk chooses one amongtwo transition kernels, depending on the environment. It is not clear how to extend ourargument to random walks having more than two transition kernels. In fact, there are exam-ples showing that the monotonicity property crucial to our proof (see Lemma 2.1) does notalways hold for such systems already when the random walk depends on three states; seeHolmes and Salisbury [13].2. Important to the proof of Theorem 1.1 is the fact that a certain functional of the environment ξ and ( W t ) is monotone in ξ . This functional counts, as a function of time, the number ofoccupied sites the random walk observes at the jump times of the random walk (see Section2.1 for a definition). The monotonicity property of this functional is proven in Lemma 2.1.We note that this property has earlier been exploited by Holmes and Salisbury [13] to studymonotonicity properties of random walks on i.i.d. static Z , e.g. Hilário, den Hollander, Sido-ravicius, dos Santos, and Teixeira [11] and Huveneers and Simenhaus [14], monotonicityproperties of the random walk have played an important role. Lemma 2.1 seems useful inorder to extend their results to random walks on Z d with more general transition kernels.3. In Peres, Popov, and Sousi [20], sufficient conditions for general RWRE models to be transientwere proven. In particular, [20, Proposition 1.4] implies that ( W t ), as studied in this paper, istransient if it is elliptic and d ≥
5. If u = u =
0, and under weak moment assumptions, ([20,Theorem 1.2]) yields that ( W t ) is transient when d ≥ ν and ¯ ν , see the remarkat the end of Section 3.2. In particular, the statement of Theorem 1.2 holds if ( W t ) is el-liptic and the dynamic environment is initialised from a measure µ ∈ P ( Ω ) which satisfies(1.16) (with µ replacing ¯ ν i ). As an example, the contact process started from any measure µ stochastically dominating a non-trivial Bernoulli product measure satisfies (1.16) with i = Z . Note that, Theorem1.3 extends the estimates in [2, Proposition 2.5] to hold throughout the cone mixing regimeand for random walks on Z d with d ≥ d ≥
2, our proof7eems to transfer to this case using that such processes also survives in (tilted) space-timeslabs, as shown in Bezuidenhout and Gray [5].Non-triviality of the speed for RWDRE as in Theorem 1.4b) has previously been proven bydos Santos [23] for a random walk on the exclusion process, by employing multi-scale argu-ments. This argument can perhaps be adapted to yield a different proof of ρ ( λ ) ∈ (0, 1) forthe random walk on the contact process. In this section, we describe a particular coupling construction of the random walk. This construc-tion is at the heart of the argument for the proof of Theorem 1.1, as it yields an important mono-tonicity property; see Lemma 2.1 below.To construct the evolution of the random walk, let ( N t ) be a Poisson jump process with jumprate γ ∈ (0, ∞ ) and with inverse process ( J k ) k ≥ . We call these times the jump times of the ran-dom walk. Essential to our approach and for the proof of Theorem 1.1 is the introduction of twoindependent sequences of i.i.d. UNIF [0,1] random variables, O = ( O j ) j ≥ and V = ( V j ) j ≥ . Here O stands for occupied, whereas V stands for vacant.Given α = ( α ( i , z )) i ∈ {0,1}, z ∈ Z d as introduced in (1.4), enumerate Z d = { z , z , . . .} and define( p ( m )) ∞ m = and ( p ( n )) ∞ n = by setting p (0) = p (0) = p ( m ) = γ m X j = α (1, z j ) and p ( n ) = γ n X j = α (0, z j ), n , m ∈ N . (2.1)For convenience, we shall assume that the maximum in (1.4) is attained by both α (1, · ) and α (0, · )by adapting the values for α (0, o ) and α (1, o ) appropriately. Note that this does not affect the be-haviour of the random walk. Moreover, we can arrange thatlim m →∞ p ( m ) = lim n →∞ p ( n ) =
1. (2.2)Given a fixed environment ( ξ t ) t ≥ we next define the discrete-time random walk S = ( S k ) k ∈ N . Forthis, we also introduce the functional ( ρ ( k , ξ )) k ∈ N , taking values in N . Let S = o and ρ (0, ξ ) = S k and ρ ( k , ξ ), define S k + and ρ ( k +
1) iteratively by ρ ( k + ξ ) = ρ ( k , ξ ) + ξ J k ( S k ) (2.3)and S k + = S k + (1 − ξ J k ( S k )) ∞ X n = [ p ( n − p ( n )) ( V k + − ρ ( k , ξ ) ) z n + ξ J k ( S k ) ∞ X m = [ p ( m − p ( m )) ( O ρ ( k , ξ ) + ) z m . (2.4)Note that, in (2.4), since ( ξ t ) is càdlàg and independent of ( J k ) k ≥ , we have that ξ J − k ( S k ) is a.s. equalto ξ J k ( S k ) for all k . Further, we have that ρ ( k , ξ ) = k − X i = ξ J i ( S i ), k ∈ N , (2.5)8ounts the number of occupied sites the (discrete-time) random walk ( S k ) has observed at the first k jump times. The (continuous-time) random walk ( W t ) with transition kernels α ( i , · ), i ∈ {0, 1}, isobtained by setting W t = S N t , as follows by the construction, using that ( ξ t ) is right-continuous.The continuous version of the (rescaled) ρ ( n , ξ ) is given by ρ t ( ξ ) : = γ ρ ( N t , ξ ), t ∈ [0, ∞ ). (2.6) In the proof of Theorem 1.4b) in Section 4.3 we carry out a domination argument. For this purpose,we consider a generalisation of the construction in Section 2.1. For any t ∈ [0, ∞ ), denote by ξ [0, t ] the space-time environment from time 0 to time t . Furthermore, consider a family of Booleanfunctions ( f k ), measurable with respect to the σ -algebra σ ( ξ [0, J k ] , N , O , V ). The construction inSection 2.1 can then be generalised in the same manner by setting S = ρ ( d ) (0, ξ ) =
0, anditeratively, ρ ( d ) ( k + ξ ) = f k + + ρ ( d ) ( k , ξ ) S k + = S k + (1 − f k ) ∞ X n = [ p ( n − p ( n )) ( V k + − ρ ( d ) ( k , ξ ) ) z n + f k ∞ X m = [ p ( m − p ( m )) ( O ρ ( d ) ( k , ξ ) + ) z m . (2.7)Thus, in this more general setup, ρ ( d ) ( k , ξ ) = P k − i = f i , k ∈ N . Note that, we recover (2.4) when f k = ξ J k ( S k ). In Section 4.3, we consider cases where f k = R k ξ J k ( S k ) for some event R k ∈ σ ( ξ [0, J k ] , N , O , V ),for which we readily see that ρ ( d ) ( k , ξ ) ≤ ρ ( k , ξ ). The construction in the previous two subsections provides us with a coupling that keeps track ofthe number of occupied sites the random walk has observed at any given time. The key propertyof the coupling construction in Subsection 2.1 is a monotonicity property, which we state next. Forthis, denote the elements of D Ω [0, ∞ ) by ( η t ) t ≥ and write ( η t ) t ≥ ≤ ( ω t ) t ≥ if η t ( x ) ≤ ω t ( x ) for all x ∈ Z d and t ∈ [0, ∞ ). Lemma 2.1 (Monotonicity of particle density) . For any η = ( η t ) t ≥ and ω = ( ω t ) t ≥ contained inD Ω [0, ∞ ) satisfying ( η t ) t ≥ ≤ ( ω t ) t ≥ ; ρ s ( η ) ≤ ρ s ( ω ) ∀ s ∈ [0, ∞ ). (2.8) Proof.
Consider two (discrete-time) random walkers ( S n ) n ≥ and ( S m ) m ≥ , both constructed as inSubsection 2.1 using the same realisation of (( N t ), ( O k ), ( V k )) and having identical transition ker-nels, seeing environment ( η t ) and ( ω t ), respectively. We claim that ρ ( n , η ) ≤ ρ ( n , ω ) for all n ≥
0. (2.9)To see this, we argue by induction. First note that, by definition, we have that ρ (0, η ) = ρ (0, ω ). Asthe induction hypothesis, we assume that ρ ( k , η ) ≤ ρ ( k , ω ) for some k ≥ ρ ( k , η ) < ρ ( k , ω ), we have ρ ( k + η ) ≤ ρ ( k , η ) + ≤ ρ ( k , ω ) ≤ ρ ( k + ω ). Thus, theinduction step holds in this case. In the other case, where ρ ( k , η ) = ρ ( k , ω ), we have by constructionthat S k = ρ ( k , η ) X i = £ ∞ X m = [ p ( m − p ( m )) ( O i ) z m ¤ (2.10) + k − ρ ( k , ω ) X j = £ ∞ X n = [ p ( n − p ( n )) ( V j ) z n ¤ = S k . (2.11)Since η ≤ ω , it holds that η J k ( S k ) ≤ ω J k ( S k ), which in particular implies that ρ ( k + η ) ≤ ρ ( k + ω ).Hence, also the second case satisfy the induction step and consequently (2.9) holds. Finally, byreplacing n in (2.9) by N t and multiplying by γ − , this proves (2.8). Remark.
For any attractive IPS ξ , Lemma 2.1 transfers to an almost sure statement with respect tothe annealed measure (and thus also the quenched measure). In this case, for every µ , ν ∈ P ( Ω )with µ ≤ ν there exists a coupling b P of P µ , o , P ν , o and ξ such that b P ( ρ t ( ξ µ ) ≤ ρ t ( ξ ν )) =
1. The exis-tence of such a coupling follows by [16, Theorem II.2.4] and the construction above.
Remark.
The coupling construction in Section 2.1 for random walks on a dynamic random envi-ronment is to our knowledge new. Apparently the same coupling construction has previously beenused to study monotonicity properties for certain specific random walks in static random environ-ment by Holmes and Salisbury [13]. Lemma 2.1 can be seen as an immediate extension of [13,Theorem 4.1i)].
Remark.
Lemma 2.1 can be extended to hold in certain cases under the general construction con-sidered in Section 2.2. For this, the functions ( f k ) need to be monotone in the sense that (for η ≤ ξ as above), if ρ ( d ) ( k , η ) = ρ ( d ) ( k , ξ ), then ρ ( d ) ( k + η ) ≤ ρ ( d ) ( k + ξ ). In this subsection we first present the proof of Theorem 1.1 for the case when the environment isstarted from all sites occupied. Essentially the same proof can be applied to the case where theenvironment is started from all sites vacant. We comment at the end of this subsection on whichchanges to the proof are necessary for this case.The main idea is to show that ρ t ( ξ ) is sub-additive, by using that ξ has an attractive graphicalrepresentation coupling and Lemma 2.1. Subsequently, the subadditive ergodic theorem appliedto ρ t ( ξ ) yields that t − ρ t ( ξ ) converges towards a deterministic constant. This, in turn, identifiesthe limiting speed.Let ξ be an IPS with an attractive graphical representation coupling, b P , where by I we denotethe corresponding collection of Poisson point processes. In order to formulate the proof, we haveto be more specific about I and write I as a countable set of Poisson point processes indexedby the lattice Z d , I = ³ ( X y ) y ∈ Z d , ( X y ) y ∈ Z d , . . . ´ , where every X iy is an (independent) Poisson pointprocess on [0, ∞ ). Further, for x ∈ Z d and t ∈ [0, ∞ ) let Θ x , t be the space-time shift operator on therealisations of I : Θ x , t ³ ( X s , y ), ( X s , y ), . . . ´ s ∈ [0, ∞ ), y ∈ Z d = ³ ( X s + t , y + x ), ( X s + t , y + x ), . . . ´ s ∈ [0, ∞ ), y ∈ Z d .10or the contact process, the set I as considered in Section 4, consists of the Poisson processes © H x , I x , e : x , e ∈ Z d , | e | = ª , and Θ x , t shifts crosses and arrows in space by x and in time by t . Fur-ther, in [8, Theorem 2.5], I is the set of birth and death events (which in [8] are denoted by n T y , in : n ≥ y ∈ Z d , i ∈ {0, 1} o ).To emphasise the graphical representation, we write ρ t ( ξ ) = ρ t ( η , I , N , O , V ) for ξ = η and let b P denote the joint law of the graphical construction coupling and N , O and V . Note that, by Lemma2.1 and (1.10), for any η ∈ Ω , ρ t ( η , I , N , O , V ) ≤ ρ t (¯1, I , N , O , V ), b P − a . s . (3.1)Moreover, let N ( s ) = ( N ( s ) t ) t ≥ : = ( N t + s − N s ) t ≥ , O ( s ) = ( O ( s ) n ) n ≥ : = ( O n + γρ s ( ξ ) ) n ≥ , V ( s ) = ( V ( s ) n ) n ≥ : = ( V n + N s − γρ s ( ξ ) ) n ≥ .Similar to Θ x , t we introduce the space-time shift θ x , t on Ω [0, ∞ ) by( θ x , t ξ s ) s ≥ = ( θ x ξ s + t ) s ≥ with space-shift θ x introduced in (1.2). Next, define the continuous-time process ( X t , s ) ≤ t ≤ s by X t , s : = ρ s − t (¯1, Θ W t , t I , N ( t ) , O ( t ) , V ( t ) ), for 0 ≤ t ≤ s . (3.2)Note that, if ξ is such that ξ = ¯1, then X s = ρ s (¯1, I , N , O , V ) = ρ s ( ξ ), s ∈ [0, ∞ ). (3.3)In the next statement and in the proceedings, for µ ∈ P ( Ω ) and x ∈ Z d , we write b P µ , x to emphasisethe starting configuration of both ξ and ( W t ). Lemma 3.1 (Sub-additivity) . The process ( X t , s ) ≤ t ≤ s has the following properties. i) X = and for all t , s ∈ [0, ∞ ) : X t + s ≤ X t + X t , t + s . ii) For all t ∈ (0, ∞ ) , ( X t , k + t ) k ≥ has the same distribution as ( X k ) k ≥ . iii) For all t ∈ [0, ∞ ) , ( X ( k − t , kt ) k ≥ is a sequence of i.i.d. random variables. iv) For all t ∈ [0, ∞ ) , the expectation b E δ ¯1 , o [ X t ] is finite and X t ≥ .Proof. Fix t , s ∈ [0, ∞ ) and recall (3.1). By the Markov property of the Poisson point process I , wehave that X t + s = ρ t + s (¯1, I , N , O , V ) = ρ t (¯1, I , N , V , O ) + ρ ( s , θ W t , t ξ , Θ W t , t I , N ( t ) , O ( t ) , V ( t ) ) ≤ ρ t (¯1, I , N , O , V ) + ρ ( s , ¯1, Θ W t , t I , N ( t ) , O ( t ) , V ( t ) ) = X t + X t , t + s .Properties i) and ii) follow from the equality X =
0, the translation invariance in (1.2) and theequality in distribution X s = X t , t + s . Moreover, iii) follows by the Markov property of ξ and thegraphical representation. Lastly, property iv) holds trivially, since X t is non-negative by definitionand since X t ≤ N t . 11emma 3.1 enables us to prove the SLLN for the process ρ t ( ξ ) when ξ is initialised at time 0 by¯1, by applying the subadditive ergodic theorem. Theorem 3.2 (Law of large numbers for ρ t ( ξ )) . Assume that ξ has an attractive graphical represen-tation coupling. There exists ρ ∈ [0, 1] such that lim t →∞ t ρ t ( ξ ) = ρ b P δ ¯1 , o -a.s. and in L . (3.4) Moreover, ρ = inf t ≥ t − b E δ ¯1 , o ( ρ t ( ξ )) .Proof. By Lemma 3.1 we know that X satisfies property a)-d) of [16, Theorem VII.2.6]. In particu-lar, by the independence property in Lemma 3.1iii), the process is stationary and ergodic. Hence,the conclusion of Theorem 3.2 holds when t takes integer values. This can easily be extended tocontinuous t . Indeed, for any t ∈ (0, ∞ ) we have that X ⌊ t ⌋ ≤ X t ≤ X ⌈ t ⌉ . (3.5)In particular, by dividing by t in (3.5) and taking t → ∞ (as in (3.4)), we conclude the proof.We are now in position to present the proof of Theorem 1.1. Proof of Theorem 1.1.
By the construction in Section 2.1, W t can be written as W t = ρ ( N t , ξ ) X i = µ ∞ X m = [ p ( m − p ( m )) ( O i ) z m ¶ + N t − ρ ( N t , ξ ) X j = µ ∞ X n = [ p ( n − p ( n )) ( V j ) z n ¶ . (3.6)Dividing by t > W t t = ρ ( N t , ξ ) t ρ ( N t , ξ ) ρ ( N t , ξ ) X i = µ ∞ X m = [ p ( m − p ( m )) ( O i ) z m ¶ (3.7) + N t − ρ ( N t , ξ ) t N t − ρ ( N t , ξ ) N t − ρ ( N t , ξ ) X j = µ ∞ X n = [ p ( n − p ( n )) ( V j ) z n ¶ . (3.8)Taking the limit as t → ∞ and applying Theorem 3.2 we obtainlim t →∞ t W t = ρ u + (1 − ρ ) u P δ ¯1 , o -a.s. and in L , (3.9)where ρ is as in Theorem 3.2 and u , u ∈ R d are as in (1.5). This proves Theorem 1.1 for the casewhen the environment is started from all sites equal to 1.We next comment on the changes necessary in the argument for proving Theorem 1.1 whenstarted from all sites equal to 0. For this case we can define the process ( Y t , s ) ≤ t ≤ s given by Y t , s : = ρ ( s − t , ¯0, Θ W t , t I , N ( t ) , O ( t ) , V ( t ) ) for 0 ≤ t ≤ s . (3.10)By the same arguments as in Lemma 3.1 we can prove that − Y is a sub-additive process satisfy-ing property ii) and iii) as in Lemma 3.1. Moreover, since Y t is dominated by N t it follows that b E δ ¯0 , o [ Y t ] ≤ b E δ ¯0 , o [ N t ] = t . This is sufficient in order to apply [16, Theorem VII.2.6]. By a literaladaptation of the proof under b P δ ¯1 , o above we obtainlim t →∞ t W t = ρ u + (1 − ρ ) u P δ ¯0, o -a.s. and in L , (3.11)where ρ is the limit in Theorem 3.2 when b P δ ¯1 , o is replaced by b P δ ¯0 , o . This completes the proof ofTheorem 1.1. 12 .2 Proof of Theorem 1.2 In this subsection we present the proof of Theorem 1.2. The presentation is inspired by the proofof [12, Proposition 3.3] (see also Remark 3.4 therein), and the proof of Theorem 1.2 is an extensionof their proof to higher dimensions. We only provide the proof when i =
1. The proof for the case i = Proof of Theorem 1.2.
We start with the construction of the random walk. Let U : = ( U k ) k ∈ N be ani.i.d. sequence of UNIF [0, 1] random variables, independent of the jump process N = ( N t ) t ≥ . Set S ( U )0 : = k ≥ S ( U ) k + : = S ( U ) k + (1 − ξ J k ( S ( U ) k )) ∞ X n = [ p ( n − p ( n )) ( U k ) z n + ξ J k ( S ( U ) k ) ∞ X m = [ p ( m − p ( m )) ( U k ) z m ,and let W ( U ) t : = S ( U ) N t and ρ ( U ) t = P N t k = ξ J k ( S k ). Clearly W ( U ) t and W t (as constructed in Section 2.1)are equal in distribution, and similarly for ρ ( U ) t and ρ t , and hence onwards we do not distinguishthem and write W t and ρ t for both processes.Let N (¯ ν ) , U (¯ ν ) and N ( δ ¯1 ) , U ( δ ¯1 ) be independent copies of U and N and denote by b P the jointlaw of b P , N ( a ) , U ( a ) , a ∈ © ¯ ν , δ ¯1 ª . Then W (¯ ν ) : = W ( ξ (¯ ν ) , N (¯ ν ) , U (¯ ν ) ) and ρ (¯ ν ) : = ρ ( ξ (¯ ν ) , N (¯ ν ) , U (¯ ν ) )under b P have the same law as W and ρ under b P ¯ ν , o . Similarly, W ( δ ¯1 ) : = W ( ξ ( δ ¯1 ) , N ( δ ¯1 ) , U ( δ ¯1 ) ) and ρ ( δ ¯1 ) : = ρ ( ξ ( δ ¯1 ) , N ( δ ¯1 ) , U ( δ ¯1 ) ) have the same law as W and ρ under P δ ¯1 , o . Further, for T > N = ( ˆ N s ) s ≥ and ˆ U = ( ˆ U n ) n ∈ N be defined byˆ U n : = ( U ( δ ¯1 ) n if n ≤ N ( δ ¯1 ) T ; U (¯ ν ) n otherwise, (3.12)ˆ N s : = ( N ( δ ¯1 ) s if s ≤ T ; N ( δ ¯1 ) T + N (¯ ν ) s − N (¯ ν ) T otherwise. (3.13)It is clear that ˆ W : = W ( ξ ( δ ¯1 ) , ˆ N , ˆ U ) and ˆ ρ : = ρ ( ξ ( δ ¯1 ) , ˆ N , ˆ U ) have the same laws as W ( δ ¯1 ) and ρ ( δ ¯1 ) .Furthermore, ˆ N and N (¯ ν ) are independent up to time T , and thus the jump times of W (¯ ν ) and ˆ W are independent in the time interval [0, T ]. By construction, for times later than T , the jumpingtimes of W (¯ ν ) and ˆ W are the same.Next, let ǫ , m > U ( u , u ) × {1/ γ } ⊂ V m (1 − ǫ ) andlim T →∞ φ ( m )1 ( η , T ) =
0, for ¯ ν -a.e. η ∈ Ω . (3.14)For T ≥
0, define the event D T : = n ( W (¯ ν ) t , t ) ∈ V m (1 − ǫ ) , ∀ t ≥ T o (3.15)and let Γ T : = D T ∩ n ξ (¯ ν ) s ( x ) = ξ ( δ ¯1 ) s ( x ), ∀ ( x , s ) ∈ V m ∩ Z d × [ T , ∞ ) o . (3.16)Note that, since b P ( Γ ) ≥ − ³b P ( D cT ) + R φ ( m )1 ( η , T )¯ ν ( d η ) ´ , by (3.14) and since U ( u , u ) × {1/ γ } ⊂ V m (1 − ǫ ) , it holds that lim T →∞ b P ( Γ T ) =
1. Furthermore, by stationarity under ¯ ν , and since ˆ N T is13ndependent of ( W (¯ ν ) t ) and ( ξ t ), we have that b P ³ lim t →∞ t − ρ (¯ ν ) t = ρ ´ = b P ³ lim t →∞ t − ρ (¯ ν ) t = ρ | N (¯ ν ) T = ˆ N T = ´ ≥ b P ³n lim t →∞ t − ρ (¯ ν ) t = ρ o ∩ Γ T | N (¯ ν ) T = ˆ N T = ´ = b P ³n lim t →∞ t − ρ ( δ ¯1 ) t = ρ o ∩ Γ T | N (¯ ν ) T = ˆ N T = ´ = b P ³ Γ T | N (¯ ν ) T = ˆ N T = ´ = b P ( Γ T ) T →∞ −−−−→ b P ³ lim t →∞ t − ρ (¯ ν ) t = ρ ´ =
1. From this and the monotonicity property obtained in Lemma2.1, and by arguing as in the proof of Theorem 1.1, we conclude the proof.
Remark.
The last paragraph of the proof of Theorem 1.2 is based on [12, Remark 3.4]. Followingthe proof of [12, Proposition 3.3], assuming that ( W t ) is elliptic, this part can be extended to holdfor any measure µ ∈ P ( Ω ) for which (1.16) holds, with ¯ ν replaced by µ . For this, since W ( µ ) T ∈ [ − mT , mT ] d on Γ T , we have that b P ³ lim t →∞ t − W ( µ ) t = v | Γ T ´ (3.17) = X x ∈ [ − mT , mT ] d b P ³ lim t →∞ t − W ( µ ) t = v | Γ T , W ( µ ) T = x ´ b P ³ W ( µ ) T = x | Γ T ´ . (3.18)In particular, it is sufficient to show that b P ³ lim t →∞ t − W ( µ ) t = v | Γ T , W ( µ ) T = x ´ = ∀ x ∈ [ − mT , mT ] d . (3.19)To prove the latter equation, use the ellipticity assumption to construct events A x , x ∈ [ − mT , mT ] d ,having the following properties: A x ⊂ { ˆ W T = x } and A x is independent of ( ξ t ) and ( W ( µ ) t ). Conclude(3.19) by first conditioning on A x and then noting that, under Γ T ∩ { W ( µ ) T = x } ∩ A x , we can replace{lim t →∞ t − W ( µ ) t } by {lim t →∞ t − ˆ W t }. We constructed in Lemma 3.1 an independent sub-additive process ( X t , s ) ≤ t ≤ s . Such processes arewell known to satisfy large deviation properties, see e.g. Grimmett [9]. In particular, we have thefollowing large deviation estimates for ρ t ( ξ ). Theorem 3.3 (Large deviation estimates for ρ t ( ξ )) . Assume that ξ has an attractive graphical rep-resentation coupling. Then, for any ǫ > , there exists R i ( ǫ ) > , i ∈ {0, 1} , such that b P δ ¯1 , o ¡ ρ t ( ξ ) > t ( ρ + ǫ ) ¢ ≤ exp ( − t R ( ǫ )) , for all t > b P δ ¯0 , o ¡ ρ t ( ξ ) < t ( ρ − ǫ ) ¢ ≤ exp ( − t R ( ǫ )) , for all t >
0. (3.20)
Proof of Theorem 3.3.
We follow the proof of Theorem 3.2 in [10] and give the proof with respect to δ ¯1 only. The proof with respect to δ ¯0 follows analogously. Let ǫ > T > g T : = T b E δ ¯1 , o £ X T ¤ ≤ ρ + ǫ . (3.21)14e first consider the case when t = r T for some r ∈ N . Using the properties from Lemma 3.1,we have b P δ ¯1 , o ¡ X t ≥ t ( ρ + ǫ ) ¢ ≤ b P δ ¯1 , o ¡ Q + · · · + Q r ≥ t ( ρ + ǫ ) ¢ ,where Q i = X ( i − T , iT . Moreover, the Q i ’s are i.i.d., and, since Q is dominated by a Poisson randomvariable, b E δ ¯1 , o £ e zQ ¤ < ∞ for all z ∈ R .Next, let Z i = Q i − b E δ ¯1 , o ( Q i ), and note that (by (3.21)) b P δ ¯1 , o ¡ Q + · · · + Q r ≥ t ( ρ + ǫ ) ¢ ≤ b P δ ¯1 , o ( Z + · · · + Z r ≥ r T ǫ ) .Further, applying the exponential Chebyshev inequality implies that for each y ≥ b P δ ¯1 , o ( Z + · · · + Z r ≥ r T ǫ ) ≤ e − r T ǫ y b E δ ¯1 , o £ e y Z ¤ r .Since b E δ ¯1 , o £ e y Z ¤ < ∞ for all y ≤ b E δ ¯1 , o £ Z ¤ =
0, there exists a constant c = c (1) > b E δ ¯1 , o £ e y Z ¤ ≤ + c y for y ∈ [0, 1]. Hence, by setting y = ǫ c , for r large, b P δ ¯1 , o ( Z + · · · + Z r ≥ r T ǫ ) ≤ exp £ − r T ǫ y + r log(1 + c y ) ¤ ≤ exp £ − r T ǫ y + r c y ¤ = exp £ − t ǫ T c ¤ .This completes the proof for the case when t is a multiple of T .For general values of t , write t = r T + s , where 0 ≤ s < T , and note that X t ≤ X r T + X r T , t ,where the last two variables are independent. Further, notice that we can bound b P δ ¯1 , o ¡ X t ≥ t ( ρ + ǫ ) ¢ ≤ b P δ ¯1 , o ¡ X r T ≥ t ( ρ + ǫ /2) ¢ + b P δ ¯1 , o ¡ X s ≥ t ǫ /2 ¢ .By using that X s ≤ P ⌈ s ⌉ k = X ( k − k and Markov’s inequality, we obtain that b P δ ¯1 , o ( X s ≥ t ǫ ) ≤ e − t ǫ b E δ ¯1 , o ( e X ) ⌈ s ⌉ ,which completes the proof since b E δ ¯1 , o ( e X ) ⌈ s ⌉ ≤ b E δ ¯1 , o ( e X ) ⌈ T ⌉ < ∞ .We continue with the proof of Theorem 1.3. Proof of Theorem 1.3.
First note that, by the construction in Section 2, we have that W t = ρ ( N t , ξ ) X i = ˜ O i + N t − ρ ( N t , ξ ) X j = ˜ V j , (3.22)where ˜ O i = P ∞ m = [ p ( m − p ( m )) ( O i ) z m and ˜ V j = P ∞ n = [ p ( n − p ( n )) ( V j ) z n . Let v = ρ u + (1 − ρ ) u ,where ρ : = ρ ( = ρ by assumption). Then, for ǫ > µ ∈ P ( Ω ), we have that P µ , o ¡ k W t − t v k ≥ t ǫ ¢ = b P µ , o ð°°°° ρ ( N t , ξ ) X i = ˜ O i + N t − ρ ( N t , ξ ) X j = ˜ V j − t v °°°°° ≥ t ǫ ! ≤ b P µ , o ð°°°° ρ ( N t , ξ ) X i = ˜ O i − t ρ u °°°°° ≥ t ǫ ! + b P µ , o ð°°°° N t − ρ ( N t , ξ ) X j = ˜ V j − t (1 − ρ ) u °°°°° ≥ t ǫ ! .15o conclude the proof it is thus sufficient to show that both the latter terms decay exponentially in t . Since our argument is almost identical for both terms, we only provide the detailed proof for thefirst one. By Theorem 3.3, for any δ > b P µ , o ð°°°° ρ ( N t , ξ ) X i = ˜ O i − t ρ u °°°°° ≥ t ǫ ! (3.23) ≤ ⌈ t γ ( ρ + δ ) ⌉ X n =⌊ t γ ( ρ − δ ) ⌋ b P µ , o ð°°°° n X i = ˜ O i − t ρ u °°°°° ≥ t ǫ ! + exp( − t R ( δ )), (3.24)where R ( δ ) : = min{ R ( γδ ), R ( γδ )} >
0. Indeed, the estimates in Theorem 3.3 applies to the pro-cess started from µ , since by Lemma 2.1, for any µ ∈ P ( Ω ) and t , s >
0, we have b P δ ¯0 , o ¡ ρ t ( ξ ) ≥ s ¢ ≤ b P µ , o ¡ ρ t ( ξ ) ≥ s ¢ ≤ b P δ ¯1 , o ¡ ρ t ( ξ ) ≥ s ¢ . Further, for any integer n ∈ t γ ( ρ − δ , ρ + δ ) we have that b P µ , o ð°°°° n X i = ˜ O i − t ρ u °°°°° ≥ t ǫ ! ≤ b P µ , o ð°°°° n X i = ˜ O i − nu °°°°° ≥ t ( ǫ − k u k δ ) ! . (3.25)Observe that, by taking δ > ǫ −k u k δ >
0. In this case, since( W t ) has finite exponential moments, (3.25) is exponentially small (in n , hence in t ) by CramérsTheorem applied to ( ˜ O i ). Applying this estimate to (3.24), taking δ small, yields that for some C , c > b P µ , o ð°°°° ρ ( N t , ξ ) X i = ˜ O i − t ρ u °°°°° ≥ t ǫ ! ≤ C e − c t . (3.26)Noting that the same argument can be used to yield, for some C , c > b P µ , o ð°°°° N t − ρ ( N t , ξ ) X j = ˜ V j − t (1 − ρ ) u °°°°° ≥ t ǫ ! ≤ C e − c t , (3.27)completes the proof of the theorem. A càdlàg version of the contact process can be constructed from a graphical representation in thefollowing standard way. For this, let H : = ( H ( x )) x ∈ Z d and I : = ( I ( x , e )) x , e ∈ Z d : k e k = be two indepen-dent collections of i.i.d Poisson processes with rate 1 and λ , respectively. On Z d × [0, ∞ ), draw theevents of H ( x ) as crosses over x and the events of I ( x , e ) as arrows from x to x + e .For x , y ∈ Z d and 0 ≤ s ≤ t , we say that ( x , s ) and ( y , t ) are connected, written ( x , s ) ↔ ( y , t ),if and only if there exists a directed path in Z d × [0, ∞ ) starting at ( x , s ), ending at ( y , t ) and goingeither forward in time without hitting crosses or “sideways” following arrows in the prescribeddirection. For A ⊂ Z d and s ∈ [0, ∞ ), define the set at time t > s connected to ( A , s ) in the graphicalrepresentation by C t ( A , s ) : = n y ∈ Z d : there exist x ∈ A such that ( x , s ) ↔ ( y , t ) o . (4.1)16hen A = { x } for some x ∈ Z d , we write C t ( x , s ) for simplicity. Note that this construction allows usto couple copies of the contact processes starting from different configurations. For each A ⊂ Z d denote by ( ξ At ) t ≥ the process with initial configuration ξ A ( x ) = A and, for all y ∈ Z d and t > ξ At ( y ) = C t ( A ,0) ( y ). (4.2)Let ( ξ A , λ t ) t ≥ denote the contact process with starting configuration A ⊂ Z d and infection parame-ter λ > Lemma 4.1 (Monotonicity property) . The contact process ( ξ A , λ t ) t ≥ has an attractive graphical rep-resentation coupling, which is stochastically monotone in A and in λ . We next recall the self-duality property which is used in the proof of Theorem 1.4a) and b). Forthis, define the backwards process ( ˆ ξ A , ts ) ≤ s ≤ t given A ⊂ Z d and t > ξ A , ts ( x ) = ( y ∈ A such that y ∈ C t ( x , t − s );0 otherwise. (4.3)Then, the distribution of ( ˆ ξ A , ts ( x )) s ≥ is the same as that of the contact process with the same ini-tial configuration. Moreover, the backwards process and the contact process satisfy the dualityequation. Namely, ξ At ∩ B
6= ; if and only if A ∩ ˆ ξ B , tt
6= ; , for any A , B ⊂ Z d . (4.4) Theorem 1.4a) for the contact process started from ¯1 is an immediate consequence of Theorem 1.1and Lemma 4.1. We next show how to extend this to all measures stochastically dominating ¯ ν λ .The proof goes by showing that (1.16) in Theorem 1.2 holds. Actually, our proof of (1.16) is moregeneral and applies to the contact process started from any measure stochastically dominating anon-trivial Bernoulli-product measure. For this, we first state and prove two lemmas. Lemma 4.2.
For λ > λ c ( d ) there exist constants a , C , c > such that for all t > , b P ( | C t ( o , 0) | ≤ at | C t ( o , 0)
6= ; ) ≤ C e − ct , (4.5) Proof.
By [17, Theorem I.2.30], for some constants C , c > b P µ sup s ≥ | C s ( o , 0) | < ∞ | C t ( o , 0)
6= ; ¶ ≤ C e − c t . (4.6)Let c > e − c = + λ ), where e − c < | C t ( o , 0) | ≤ at and each of the particlesin C t ( o , 0) dies before producing further offsprings. Since the contact process is Markovian andattractive, b P µ sup s ≥ | C s ( o , 0) | < ∞ | C t ( o , 0)
6= ; ¶ ≥ b P ( | C t ( o , 0) | ≤ at | C t ( o , 0)
6= ; ) e − c at . (4.7)Combining the two bounds (4.6) and (4.7), we obtain the result for c = c − ac , and c > a < c / c . 17sing the duality property together with Lemma 4.2 we obtain the following estimate. Lemma 4.3 (Coupling of the contact process) . Let µ ρ ∈ P ( Ω ) be the Bernoulli product measurewith density ρ > . For the contact process with λ > λ c ( d ) , there exist constants C , c > such that b P ³ ξ ( µ ρ ) t ( o ) ξ ( δ ¯1 ) t ( o ) for some t ∈ [ T , T + ´ ≤ C e − cT . (4.8) Proof.
By attractiveness, ξ ( µ ) T ( o ) ≤ ξ (¯1) T ( o ) for all µ ∈ P ( Ω ). In particular, we have that ξ ( µ ρ ) T ( o ) ξ (¯1) T ( o ) if and only if the connected set of the dual process at time T started at ( o , T ), denoted byˆ C T ( o , T ), is non-empty and ξ ( µ )0 ( x ) = x ∈ ˆ C T ( o , T ). That is, b P ³ ξ ( µ ρ ) T ( o ) ξ ( δ ¯1 ) T ( o ) ´ = b P ³© ˆ C T ( o , T )
6= ; ª ∩ n ξ ( µ ρ )0 ( x ) = ∀ x ∈ ˆ C T ( o , T ) o´ . (4.9)By Lemma 4.2 and self-duality of the contact process, we can estimate the size of ˆ C T ( o , T ). Inparticular, for certain constants C , c , C , c > b P ³ ξ ( µ ρ )0 ( x ) = ∀ x ∈ ˆ C T ( o , T ) | ˆ C T ( o , T )
6= ; ´ ≤ b P ¡ | ˆ C T ( o , T ) ∩ B ( o , r T ) | ≤ aT | ˆ C T ( o , T )
6= ; ¢ + b P ³ ξ ( µ ρ )0 ( x ) = ∀ x ∈ ˆ C T ( o , T ) | | ˆ C T ( o , T ) ∩ B ( o , r T ) | ≥ aT ´ ≤ C e − c T + C e − c T .Thus, the l.h.s. of (4.9) decays exponentially (in T ). To conclude (4.8), by the graphical represen-tation, it is sufficient to control the times at which there is an arrow events I ( e , o ) from e ∈ Z d with k e k =
1. Note that the number of such events is Poisson distributed with parameter 2 d λ . Inparticular, b P ³ ξ ( µ ρ ) t ( o ) ξ ( δ ¯1 ) t ( o ) for some t ∈ [ T , T + ´ ≤ (1 + d λ ) ¡ C e − c T + C e − c T ¢ .The proof of Theorem 1.4a) follows as a consequence of Lemma 4.3 and Theorem 1.2. Proof of Theorem 1.4a).
Let µ ρ ∈ P ( Ω ) be a non-trivial Bernoulli product measure, let m , T ∈ (0, ∞ )and consider V m ( T ) : = V m ∩ ¡ Z d × [ T , T + ¢ with V m as defined in (1.12). Since the contact processis attractive and translation invariant, b P ³ ∃ ( x , t ) ∈ V m ( T ) : ξ ( µ ) t ( x ) ξ ( δ ¯1 ) t ( x ) ´ ≤ ( m ( T + d b P ³ ξ ( µ ) t ( o ) ξ ( δ ¯1 ) t ( o ) for some t ∈ [ T , T + ´ .Hence, by Lemma 4.3, for some constant C > b P ³ ∃ ( x , t ) ∈ V m ∩ ³ Z d × [ T , ∞ ) ´ : ξ ( µ ) t ( x ) ξ ( δ ¯1 ) t ( x ) ´ ≤ C ∞ X k = ( T + k ) d e − c ( T + k ) ,and this vanishes as T → ∞ . Hence, since m was arbitrary chosen, the contact process startedfrom µ ρ satisfies equation (1.16) for any m ∈ [0, ∞ ). Evoking Theorem 1.2 this completes the proofof Theorem 1.4a), noting that ¯ ν λ stochastically dominates a non-trivial Bernoulli product measure,as shown in [18, Corollary 4.1], and by using Lemma 4.1.18 .3 Proof of Theorem 1.4b) In the remaining part of this article we present the proof of Theorem 1.4b). We start by showingthat ρ ( · ) is non-decreasing and right-continuous. Proof of monotonicity and right continuity of λ ρ ( λ ) . Monotonicity of λ ρ ( λ ) follows directlyby the coupling construction in Section 2.1 and the graphical representation of the contact process,Lemma 2.1 and Lemma 4.1.For right-continuity, let λ ∈ (0, ∞ ) and denote by ξ ( λ ) the corresponding contact process. For T >
0, let f ( T , λ ) : = T b E δ ¯1 , o £ ρ T ( ξ ( λ )) ¤ . (4.10)By Theorem 3.2, it follows that f ( T , λ ) ↓ ρ ( λ ) as T → ∞ . Moreover, by Lemma 2.1 and Lemma 4.1, f ( T , λ ) is also non-decreasing in λ . Hence, λ ρ ( λ ) is right-continuous as the decreasing limit ofnon-decreasing continuous functions provided that λ f ( T , λ ) is continuous for any fixed T > f ( T , λ ) is continuous in λ , note first that in order to determine the behaviour of W t for t ∈ [0, T ] we only need to consider the contact process in a finite space-time box. This follows bylarge deviation estimates on N T and our restriction on the transition rates, i.e. k u k , k u k < ∞ . Bythe weak law of large numbers, this suffices to conclude that the probability of the walker escapinga box of size L within time T converges to 0 as L → ∞ . Continuity of f ( T , λ ) now follows by usingthe graphical representation of the contact process and standard arguments for functions on thecontact process in a finite space-time region (see e.g. the discussion on page 40 in [17]).We continue with the proof of (1.20). It is not difficult to see that ρ ( λ ) < λ ∈ (0, ∞ ),since vacant sites appear independently. This was already observed in den Hollander and dosSantos [12] for d =
1, and their argument transfers directly to higher dimensions.However, in order to show that ρ ( λ ) >
0, the arguments in [12] do not carry over. In brief,they essentially use that there is a positive density of “waves” of particles moving from the right tothe left in space-time. Using the ordering of Z and monotonicity in the displacement of a near-est neighbour random walk, the random walk cannot escape less than a positive proportion ofthe waves. Due to the one-dimensional nature of this argument it does not carry over to generaldimensions nor does it extend beyond nearest neighbour jumps.Our proof is based on monotonicity of ρ t ( ξ ). We propose here a simple strategy which also gen-eralises to many other monotone dynamics. In particular, by applying the theory in [5], our argu-ment for d ≥ d = Proof of (1.20) for d = . Let λ > λ c (1) and consider the contact process ( ξ t ) on Z with infectionparameter λ and initial configuration drawn according to µ ∈ P ( Ω ), where µ is the distribution of η = η ′ · ( −∞ ,0) , where η ′ ∼ ¯ ν λ .Further, for 0 ≤ s ≤ t and z ∈ Z , denote by r s , t ( z ) : = sup{ y ∈ Z : ξ s ( x ) = x ≤ z and ( x , s ) ↔ ( y , t )} (4.11)the rightmost site that is occupied at time t by a particle and connected to a site to the left of z attime s . It is well known that there exists α > λ ) such thatlim t →∞ t r t ( o ) = α P µ − a . s . (4.12)19ee [16, Theorem VI.2.19] for a proof of (4.12) with respect to P δ ¯1 . The extension to P µ follows fromthis statement and standard coupling arguments (e.g. [16, Theorem VI.2.2]).(Recall the general construction of ( W t ) in Section 2.2). We next specify the family of Booleanfunctions ( f k ) k ≥ , which is defined by an iterative procedure involving a second process of ele-ments in Z denoted by ( R k ). Assume w.l.o.g. that u ≤
0. Let R = r J ( o ) be the position of therightmost particle of the contact process at the first jump time of the random walk. Recall that S = o , and define iteratively for k ≥ f k : = ( ξ J k ( S k ) if S k ≤ R k ;0 otherwise. (4.13) R k + : = ( r J k , J k + ( S k ) if S k ≤ R k ; r J k , J k + ( R k ) otherwise. (4.14)Hence, if r J ( o ) ≥
0, then R is assigned the position of the rightmost particle which at time J is connected to an occupied site to the left of o at time J , the first jump time. Further, in thiscase, f is assigned the value of the contact process at the location of the random walk at the 1’stjump time. Thus, the random walk “observes” the environment and chooses its transition kernelaccordingly. Otherwise, if r J ( o ) < R is assigned the position of the rightmost particle which attime J is connected to an occupied site to the left of r J ( o ) at time J and f is assigned the value0. Consequently, for this latter case, the random walk jumps as if it had observed a vacant site.For arbitrary k ≥
0, if S k ≤ R k , then R k + is assigned the location of the rightmost site at the( k + S k at the ( k + f k is assigned the value ξ J k ( S k ). On the other hand, if S k − > R k − , R k + is a prolongation of R k and we set f k =
0. By construction, and using that Z is ordered and thecontact process has nearest neighbour interactions, the following holds: at times k for which S k ≤ R k there is a connected path ( ω t ) ≤ t ≤ J k such that ξ ( ω ) = ω l ≤ S l for all l ≤ k − ω k ≥ S k . Furthermore, since f k is the product of an indicator function and ξ J k ( S k ), we have ρ (1) ( k , ξ ) ≤ ρ ( k , ξ ), where ρ (1) ( k , ξ ) = P k − i = f i and ρ ( k , ξ ) is as in Section 2.1. By the last observation, in order toprove the first part of (1.20), it is sufficient to show that there exists ρ (1) >
0, depending on λ , suchthat lim inf k →∞ k ρ (1) ( k , ξ ) ≥ ρ (1) , b P µ , o − a . s . (4.15)For this, let T = k ≥
1, let T k : = inf{ n > T k − : S n − ≤ R n − } (4.16)denote the k ’th time that the random walk observes the environment and set τ k : = T k − T k − , k ≥
1. (4.17)As noted above, at times T k , there is a connected path ( ω t ) ≤ t ≤ J k such that ξ ( ω ) = ω l ≤ S l forall l ≤ k − ω k ≥ S k . At such times, the law of ( ξ J k − ( x )) x ≤ z stochastically dominates ¯ ν λ , asshown in [12, Lemma 4.1]. Consequently, since τ k is decreasing with respect to the configurationof the contact process on sites strictly to the left of S T k − , the times ( τ k ) k ≥ are dominated by ani.i.d. sequence of τ distributed random variables. Furthermore, for the same reason, at times T k ,the random walk has a probability of at least ¯ ν λ ( η ( o ) = > b E ν λ , o ( τ ) < ∞ . (4.18)20or this, note that, since { τ ≥ n } = { S k − ≥ R k − for all k ≤ n }, b P ( τ ≥ n ) ≤ b P ¡ S n − ≥ β ( n − τ ≥ n ¢ + b P ¡ R n − ≤ β ( n − ¢ , (4.19)for any β >
0. For any β < α (with α as in (4.12)), the rightmost term in (4.19) decays exponentially(in n ) due to large deviation estimates for r t ( o ); see [16, Corollary 3.22]. Moreover, for β >
0, theleftmost term of (4.19) decays like n − as n → ∞ . This follows by applying Chebyshev’s inequality,using that ( W t ) has finite second moments, and since, under τ > n , the random walk ( S k ) ≤ k ≤ n behaves as a simple random walk in a 0-homogeneous environment. Consequently, by setting0 < β < α , (4.18) holds and this concludes the first part of (1.20).To conclude the second part of (1.20), we note that α = α ( λ ) in (4.12) diverges to ∞ as λ → ∞ .In particular, by reasoning as above and choosing β = α /2, we have that lim λ →∞ b P ( τ ≥ = λ →∞ ¯ ν λ ( η ( o ) = =
1, it follows that ρ (1) in (4.15) approaches 1 as λ → ∞ fromwhich, by the remark above (4.15), we conclude the proof. We proceed with the proof of (1.20) for the case d ≥ . For this, we make use of the fact that thesupercritical contact process survives in certain space-time slabs, as first shown in Bezuidenhoutand Grimmett [6]. For the cases when u o their result suffices. However, in order to also treatthe special case when u = o , we use an extension of their theorem to certain tilted slabs.For K ∈ N , L ∈ R and A ⊂ Z d , denote by ¡ LK ξ At ¢ t the truncated contact process defined via thegraphical representation by LK ξ At ( x ) = ( y , 0) : y ∈ A } is connected to ( x , t ) within S K , L ;0 otherwise. (4.20)Here, S K , L : = n ( x , t ) ∈ Z d × [0, ∞ ) : x ∈ [ − K , K ] × Z d − + [ Lt , 0, . . . , 0] o , (4.21)and ( A , 0) is connected to ( x , t ) within S K , L if ( A , 0) is connected to ( x , t ) in the graphical represen-tation without using arrows outside S K , L . We say that S K , L is a tilted slab if L
0, and that it is nottilted if L = Proposition 4.4 (Survival in tilted slabs) . Let d ≥ . i) For λ > λ c ( d ) , there exist K ( λ ) ∈ N , L ( λ ) > such that for all K > K ( λ ) and L ∈ ( − L ( λ ), L ( λ )) , b P ³ LK ξ { o } t
6= ; ∀ t ≥ ´ >
0. (4.22) ii)
There exists K ∈ N such that for all L > , lim λ →∞ b P ³ LK ξ { o } t
6= ; ∀ t ≥ ´ =
1. (4.23)The proof of survival in non-tilted slabs proceeds via a block argument and comparison with acertain (dependent) oriented percolation model. As pointed out by Bezuidenhout and Grimmett[6], there is a certain freedom in the spatial location of these blocks. The proof of Proposition 4.4is achieved by adapting the proof of [6] in a way where the blocks are organised in a tilted way. Asketch of the proof of Proposition 4.4 is given at the end of this section.Using that infections can spread fast in a small time interval, we note that also the followingcorollary of Proposition 4.4 holds. 21 orollary 4.5.
Let λ > λ c ( d ) and consider the same parameters as in Proposition 4.4. Then, for any δ > small enough there is an ǫ > , depending on all parameters, such that for all ( x , t ) ∈ S K , L with ( x , t − δ ) ∈ S K , L , we have that b P ³ LK ξ [ − K , K ] × Z d − t ( x ) = ´ > ǫ , and ǫ = ǫ ( λ ) approaches as λ → ∞ . IfL = , then the claim also holds for δ = .Proof. First note that, since the process is started from all sites in [ − K , K ] × Z d − occupied and thespace-time slab S K , L is translation invariant in all coordinate directions besides the first one, wemay assume w.l.o.g. that x = ( x , 0, . . . , 0). Further, by Proposition 4.4i), we have that b P ³ LK ξ [ − K , K ] × Z d − t − δ ( y ) = ´ > y ∈ [ − K + L ( t − δ ), K + L ( t − δ )] × {0, . . . , 0}.The claim thus follows by the Markov property and since an infection at ( y , t − δ ) can spread to each( x , t ) with x ∈ [ − K + Lt , K + Lt ] × {0, . . . , 0} with positive probability. In particular, since x is such thatalso ( x , t − δ ) ∈ S K , L and the set [ − K + Lt , K + Lt ] × {0, . . ., 0} is finite, this probability is uniformlybounded away from 0.We next present the proof of (1.20) in Theorem 1.4 for the case d ≥
2, assuming Proposition 4.4to be true.
Proof of (1.20) for d ≥ . We consider first the case where u o , for which we do not need thenotion of tilted slabs in order to prove the l.h.s. of (1.20). Moreover, in this case, by translationinvariance of the contact process, we assume w.l.o.g. that u · e <
0. Let λ > λ c ( d ), and let K ∈ N besuch that Proposition 4.4i) is satisfied with L =
0. Partition Z d × [0, ∞ ) into slabs Π i = K i + S K ,0 , i ∈ Z , and consider ( ζ ( i ) t ) i ∈ Z consisting of independent copies of the process ( K ξ [ − K , K ] × Z d − t ). Further,denote by ( ζ t ) the process on Ω where, for ( x , t ) ∈ Π i , we set ζ t ( x ) = ζ ( i ) t ( θ K i · e x ).We next specify the family of Boolean functions ( f k ) k ≥ . Let R =
1, recall that S = o , and defineiteratively for k ≥ f k : = ( ζ J k ( S k ) if ( S k , J k ) ∈ Π i for some i < R k ;0 otherwise. (4.24) R k + : = ( i if ( S k , J k ) ∈ Π i for some i < R k ; R k otherwise. (4.25)That is, f = ζ J ( o ) and R =
0. Further, for arbitrary k , R k records the label of the leftmost slab therandom walk has “observed” at jump times J , . . . , J k . If, at a jump time, the random walk finds itselfinside a slab which is at the left of all the slabs it previously has observed, then, by the definition of f k , the random walk “observes” the environment. Otherwise, f k =
0, and the random walk acts asif it had seen a vacant site. In particular, we have that ρ ( d ) ( k , ζ ) ≤ ρ ( k , ζ ), where ρ ( d ) ( k , ζ ) = P k − i = f i and ρ ( k , ζ ) is as in Section 2.1. As in the d = ζ t ) ≤ ( ξ t ), it is sufficient to show thatthere exists ρ ( d ) > k →∞ k ρ ( d ) ( k , ζ ) ≥ ρ ( d ) b P δ ¯1 , o − a . s . (4.26)For this, let T = k ≥
1, let T k : = inf{ n > T k − : R n < R n − } (4.27)22enote the k ’th jump time at which the random walk is in a slab to the left of the origin which itpreviously has not observed, and let τ k : = T k − T k − , k ≥
1, (4.28)be the number of jumps it takes before the random walk observes a new slab. By Corollary 4.5(which holds with δ = L = τ k ), the random walk has a positive probabilityto observe an occupied site, uniform in k . We conclude (4.26), since the times τ k have finite mean,uniformly in k . Indeed, the latter follows since u · e < W t ) has finite second moments.We continue with the case u = o , for which we need to make certain modifications to theapproach above. Choose L > K ∈ N such that Proposition 4.4i) holds and partition Z d × [0, ∞ ) into tilted slabs ˜ Π i = K i + S K , L . For i ∈ Z , denote by ( ˜ ζ ( i ) t ) i ∈ Z independent copies of theprocess ( LK ξ [ − K , K ] × Z d − t ). Further, denote by ( ˜ ζ t ) the process on Ω where, for ( x , t ) ∈ ˜ Π i , we set ˜ ζ t ( x ) = ˜ ζ ( i ) t ( θ K i · e x ). Next, let ˜ R = S = o . Fix δ > k ≥ f k : = ( ˜ ζ J k ( S k ), if { S k } × [ J k − δ , J k ) ∈ ˜ Π i for some i < ˜ R k ;0, otherwise. (4.29)˜ R k + : = ( i if ( S k , J k ) ∈ ˜ Π i for some i < ˜ R k ; R k otherwise. (4.30)Next, define the variables ˜ T k and ˜ τ k similar to the non-tilted case, by replacing R by ˜ R and T by˜ T . Note that, by our choice of δ >
0, with strictly positive probability (uniformly in k ), it holds that˜ ζ J τ k ( S k ) =
1. It hence suffices to show that the times τ k have finite mean, which implies that (4.26)holds and thus the l.h.s. of (1.20). To see that this indeed is the case, first note that, since the jumptimes ( N t ) are continuous, each time the random walk enters a new slabs there is the possibilitythat it satisfies { S k } × [ J k − δ , J k ) ∈ Π i . Lastly, the number of new slabs that ( W t ) observes is a positivefraction of its jumping times. This follows similarly as for the case that u o , by using that now u · e = L >
0. In particular, by again using Chebyshev’s inequality we obtain sufficientestimates on the time it takes until a new slab is observed. By this we conclude that the times τ k have finite mean, and thus the l.h.s. of (1.20).Note that the approach with tilted slabs also applies in the case when u · e < K and L ). In order to conclude ρ ( λ ) → λ → ∞ for both cases, we argue based on this approach.First note that, by Proposition 4.4ii), we may take L large (keeping K fixed) by choosing λ largeenough. Consequently, we may chose L large and δ > τ k ) have meanbounded from above by 1 + c , uniformly in k and for any fixed c >
0. Subsequently, for each such K , L and δ , the probability ˜ ζ J τ k ( S k ) = k , by againtuning λ large. This yields the proof of (1.20) for d ≥ Sketch of the proof of Proposition 4.4.
For the proof of Proposition 4.4 we adapt the proof of [17,Theorem 1.2.30]. The idea is to proceed in the same manner, however, instead of comparing thesurvival of the process with the ordinary oriented percolation structure, we compare it to a certaintilted oriented percolation model. 23o make this more precise, consider the following sub-graph of Z d × Z . Fix l ∈ N . Next, set V = {(0, 0, . . . , 0)} and, for n ∈ N , iteratively define V n = ½ V n − + {(0, l , 0, . . . 0, l )} if n is odd, V n − + {(1, 0, . . . , 0, 1), ( −
1, 0, . . . , 0, 1} if n is even.Let the directed graph G = ( V , E ) be given by V = S ∞ i = V i , and, for any pair x , y ∈ V , the (di-rected) edge ( x , y ) ∈ E if and only if x ∈ V n − and y ∈ V n for some n ∈ N with y = ½ x + (0, l , 0, . . . , 0, l ) if n is odd, x + ( ±
1, 0, . . . , 0, 1) if n is even.This produces a tilted oriented percolation graph where the tilting in the second coordinate de-pends on the ratio ll + <
1. As for the ordinary oriented percolation model, let each edge be openwith probability p independently, otherwise closed. By [17, Theorem B26], when p is large enough(depending on l ), the origin lies in an infinite connected component of open edges with positiveprobability.The proof of the first part of Proposition 4.4 now follows similar as the proof of Theorem 2.23 in[17], by constructing a coupling between the contact process and the oriented percolation modelon the above defined graph G . Thus, by choosing ǫ > G percolates when edges are open with probability p = − ǫ . By the couplingconstruction in [17], this also holds for the contact process, depending only on the graphical rep-resentation within S K , L , with K = a + a ll + and L = ll + a b . This completes the proof of the firstpart.The second part follows by monotonicity in λ . By a standard coupling procedure we may cou-ple the systems with infection rates λ and 3 λ such that, if [17, Proposition 2.22] is true for λ andconstants a and b , then it also holds for the system with infection rate 3 λ and constants a and b /3.Hence, given the constants a and b for a fixed λ , by letting λ converge towards infinity we may take b as small as we wish. In particular, we may choose L large and still satisfy Equation (4.22). On theother hand, for fixed K , L >
0, the probability in [17, Proposition 2.22] converges to 1 as λ → ∞ , andtherefore (4.23) follows. Acknowledgments
The authors are indebted to Rob van den Berg for kind support during various stages of this project.They are further grateful to Frank den Hollander, Renato dos Santos and Florian Völlering for fruit-ful discussions, and to the referees for valuable comments and suggestions. This work is supportedby the Netherlands Organization for Scientific Research (NWO).
ReferencesReferences [1] L. Avena and P. Thomann. Continuity and anomalous fluctuations in random walks in dy-namic random environments: numerics, phase diagrams and conjectures.
J. Stat. Phys. , 147(6):1041–1067, 2012. 242] L. Avena, F. den Hollander, and F. Redig. Large deviation principle for one-dimensional ran-dom walk in dynamic random environment: attractive spin-flips and simple symmetric ex-clusion.
Markov Process. Related Fields , 16(1):139–168, 2010.[3] L. Avena, F. den Hollander, and F. Redig. Law of large numbers for a class of random walks indynamic random environments.
Electron. J. Probab. , 16:no. 21, 587–617, 2011.[4] L. Avena, R. dos Santos, and F. Völlering. Transient random walk in symmetric exclusion: limittheorems and an Einstein relation.
ALEA Lat. Am. J. Probab. Math. Stat. , 10(2):693–709, 2013.[5] C. Bezuidenhout and L. Gray. Critical attractive spin systems.
Ann. Probab. , 22(3):1160–1194,1994.[6] C. Bezuidenhout and G. Grimmett. The critical contact process dies out.
Ann. Probab. , 18(4):1462–1482, 1990.[7] A. Drewitz and A. F. Ramírez. Selected topics in random walks in random environment. In
Topics in percolative and disordered systems , pages 23–83. Springer, 2014.[8] R. Durrett. Ten lectures on particle systems. In
Lectures on Probability Theory , pages 97–201,Springer, 1995.[9] G. Grimmett. Large deviations in subadditive processes and first-passage percolation. In
Particle systems, random media and large deviations , pages 175–194, Amer. Math. Soc., 1985.[10] G. Grimmett and H. Kesten. First-passage percolation, network flows and electrical resis-tances.
Z. Wahrsch. Verw. Gebiete , 66(3):335–366, 1984.[11] M. Hilário, F. den Hollander, V. Sidoravicius, R. S. dos Santos, and A. Teixeira. Random walkon random walks.
Electron. J. Probab. , 20:no. 95, 1–35, 2015.[12] F. den Hollander and R. S. dos Santos. Scaling of a random walk on a supercritical contactprocess.
Ann. Inst. H. Poincaré Probab. Statist. , 50(4):1276–1300, 2014.[13] M. Holmes and T. T. Salisbury. Random walks in degenerate random environments.
Canad. J.Math. , 66(5):1050–1077, 2014.[14] F. Huveneers and F. Simenhaus. Random walk driven by simple exclusion process.
Electron. J.Probab. , 20(105): 1–42, 2015.[15] H. Kesten, M. V. Kozlov, and F. Spitzer. A limit law for random walk in a random environment.
Compositio Math. , 30:145–168, 1975.[16] T. M. Liggett.
Interacting particle systems , Springer, 1985.[17] T. M. Liggett.
Stochastic interacting systems: contact, voter and exclusion processes , Springer,1999.[18] T. M. Liggett and J. E. Steif. Stochastic domination: the contact process, Ising models and FKGmeasures.
Ann. Inst. H. Poincaré Probab. Statist. , 42(2):223–243, 2006.[19] T. Mountford and M. E. Vares. Random walks generated by equilibrium contact processes.
Electron. J. Probab. , 20:no. 3, 17, 2015. 2520] Y. Peres, S. Popov, and P. Sousi. On recurrence and transience of self-interacting random walks.
Bulletin of the Brazilian Mathematical Society, New Series , 44(4):841–867, 2013.[21] F. Redig and F. Völlering. Limit theorems for random walks in dynamic random environment.
ArXiv e-prints , 2011. http://arxiv.org/abs/1106.4181 .[22] F. Redig and F. Völlering. Random walks in dynamic random environments: a transferenceprinciple.
Ann. Probab. , 41(5):3157–3180, 2013.[23] R. S. dos Santos. Non-trivial linear bounds for a random walk driven by a simple symmetricexclusion process.
Electron. J. Probab. , 19:no. 49, 1–18, 2014.[24] Ya. G. Sina˘ı. The limit behavior of a one-dimensional random walk in a random environment.
Teor. Veroyatnost. i Primenen. , 27(2):247–258, 1982.[25] F. Solomon. Random walks in a random environment.
Ann. Probability , 3:1–31, 1975.[26] A.-S. Sznitman. Topics in random walks in random environment. In
School and Conferenceon Probability Theory , pages 203–266, Abdus Salam Int. Cent. Theoret. Phys., 2004.[27] O. Zeitouni. Random walks in random environment. In