Phase transition for a non-attractive infection process in heterogeneous environment
Marinus Gottschau, Markus Heydenreich, Kilian Matzke, Cristina Toninelli
PPHASE TRANSITION FOR A NON-ATTRACTIVE INFECTION PROCESSIN HETEROGENEOUS ENVIRONMENT
MARINUS GOTTSCHAU, MARKUS HEYDENREICH, KILIAN MATZKE, AND CRISTINA TONINELLI
Abstract.
We consider a non-attractive three state contact process on Z and prove that thereexists a regime of survival as well as a regime of extinction. In more detail, the process can beregarded as an infection process in a dynamic environment, where non-infected sites are eitherhealthy or passive. Infected sites can recover only if they have a healthy site nearby, whereasnon-infected sites may become infected only if there is no healthy and at least one infectedsite nearby. The transition probabilities are governed by a global parameter q : for large q , theinfection dies out, and for small enough q , we observe its survival. The result is obtained by acoupling to a discrete time Markov chain, using its drift properties in the respective regimes. Introduction
History.
The classical contact process, as introduced by Harris in 1974 [10], has been acentral topic of research in interacting particle systems. It is formally defined as { , } Z d -valuedspin system, where 1’s flip to 0’s at rate 1, and flips from 0 to 1 occur at rate λ times the numberof neighbors in state 1, where λ > infected (i.e., in state 1) or healthy (i.e., in state 0).Many fundamental questions have been settled for this model, the results are summarized inthe monographs by Liggett and Durrett in [7, 12, 13].Among the most important results are the existence of a phase transition for survival of asingle infected particle, the complete convergence theorem, and extinction of the critical contactprocess. Much more refined results have appeared in recent years. In view of these successes, itmay seem surprising that results are considerably sparse as soon as multitype contact processesare considered. Results have only be achieved in very specific situations, examples are thearticles by Cox and Schinazi [4], Durrett and Neuhauser [5], Durrett and Swindle [6], Konno etal. [11], Neuhauser [14], and Remenik [15] for various models.Our focus here is on the contact process with three types, and this carries already severecomplications. A fair number of models considered in the literature stems from a biologicalcontext (either evolvement of biological species or vegetation models); typical questions thathave been considered are coexistence versus extinction and phase transitions. Examples are thework of Broman [3] and Remenik [15].There are two features that are shared by all of these models: they are monotonic and theyare (self-)dual (we refer to [13] for a definition of these terms). These two properties are crucialingredients in the analysis; if they fail, then most of the known tools fail. This might beillustrated by looking at Model A in [1], which is a certain 3-type contact process. Even thoughthere are positive rates for transitions between the various states of this model and apparentmonotonicity, the lack of any usable duality relation prevented all efforts in proving convergenceto equilibrium for that model.For the model considered in the present paper, it appears that there is no duality relationthat we can exploit and monotonicity is restricted to a very particular situation only. Yet we areable to prove the occurrence of a phase transition by means of coupling to certain discrete-time Date : October 16, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Contact process, phase transition, survival versus extinction.C.Toninelli acknowledges the support by the ERC Starting Grant 680275 MALIG and ANR-15-CE40-0020-03. a r X i v : . [ m a t h . P R ] J un arkov chains and analyzing drift properties of these chains. We believe that the techniquepresented here is useful in greater generality. A motivation for studying this process stemsfrom the connection with the out of equilibrium dynamics of kinetically constrained models, aswe will explain in detail in Section 1.3. We believe that the proof techniques apply in similarsituations.1.2. The model.
Our state space is Ω = { , , } Z , equipped with the product topology (whichmakes Ω compact). Further, q ∈ [0 ,
1] is a parameter and ( η t ) t ≥ is a Markov process on Ω. Wesay that at time t , site x is healthy if η t ( x ) = 0 , passive if η t ( x ) = 1 and infected if η t ( x ) = 2 . Informally, we can describe the process as follows. Each site x independently waits an expo-nential time with intensity 1 and then updates its state according to the following rules: • If at least one neighboring site is healthy, then x becomes healthy with probability q and passive w.p. 1 − q . • If at least one neighbor is infected and none is healthy, then a previously healthy x becomes infected w.p. 1 − q , a previously passive x becomes infected w.p. q and remainsin its state otherwise.For a more formal description, the process can be characterized by its probability generator,which is the closure of the operator L f ( η ) = (cid:88) x ∈ Z (cid:104) c x ( η ) q ( f ( η x, ) − f ( η )) + c x ( η )(1 − q )( f ( η x, ) − f ( η ))+ ¯ c x ( η ) (cid:2) q { η ( x )=1 } + (1 − q ) { η ( x )=0 } (cid:3) ( f ( η x, ) − f ( η )) (cid:105) ,f ∈ (cid:8) f : Ω → R cont. : lim x →∞ sup {| f ( η ) − f ( η (cid:48) ) | : η, η (cid:48) ∈ Ω , η ( y ) = η (cid:48) ( y ) for all y (cid:54) = x } = 0 (cid:9) . Here, c x ( η ) = { η ( x − · η ( x +1)=0 } and ¯ c x ( η ) = { η ( x − · η ( x +1) ≥ } . Furthermore, η x,i is the config-uration where η x,i ( y ) = η ( y ) for all y (cid:54) = x and η x,i ( x ) = i , x, y ∈ Z , i ∈ { , , } .For an initial configuration η ∈ Ω, we denote by P η the corresponding probability measure.This superscript will be dropped for the sake of convenience if context permits.As we wrote earlier, monotonicity is an important tool in the analysis of such processes. Onemonotonicity property the ( η t ) process exhibits is the following. Claim 1.
For arbitrary η ∈ Ω and x ∈ Z , we have that P η (cid:48) (cid:104) η t / ∈ { , } Z for all t ≥ (cid:105) ≥ P η (cid:48)(cid:48) (cid:104) η t / ∈ { , } Z for all t ≥ (cid:105) , where η (cid:48) = η x, and η (cid:48)(cid:48) ∈ { η x, , η x, } . In words, additional infected sites cannot decrease the chance of the infection’s survival.However, the same is not necessarily true anymore for η (cid:48) = η x, and η (cid:48)(cid:48) = η x, . Proof.
If we couple the two processes with respective initial measures, we claim that, almostsurely, η (cid:48) t ( x ) ∈ { η (cid:48)(cid:48) t ( x ) , } for all t ≥ x ∈ Z . This is a consequence of the definition of thedynamics and corresponding transition rates. (cid:3) Results and discussion.
Our main result is a phase transition for ( η t ) in the parameter q : if q is very close to 0, then any number of initially infected sites survives with positiveprobability, whereas if q is close to 1, then the infection dies out with probability 1. Theorem 2.
There exist values < q < q < such that(i) for any initial configuration η / ∈ { , } Z , we have P η (cid:104) η t / ∈ { , } Z for all t ≥ (cid:105) > for all q ≤ q , ii) and for any initial configuration η with sup x ∈ Z inf y ∈ Z {| x − y | : η ( y ) = 0 } < ∞ , we have P η (cid:104) η t ∈ { , } Z (cid:105) t →∞ −−−→ a.s. for all q ≥ q . We thus prove the existence of different regimes without relying on duality properties. Sincethere is no monotonicity that can be exploited here, we can not rule out that there are morethan one transitions between the regimes “the infection dies out” and “the infection survives”.However, we conjecture the following statement to be true.
Conjecture.
The function q (cid:55)→ P η (cid:2) η t / ∈ { , } Z for all t ≥ (cid:3) is decreasing in q ∈ (0 , . This would imply a critical value q c such that if q < q c the infection survives with positiveprobability, while if q > q c the infection dies out with probability 1.Note that the case q = 0 is degenerate and of little interest, as it admits traps: If there is asite x ∈ Z and a time t ≥ η t ( x ) , η t ( x + 1) , η t ( x + 2)) = (1 , , t (cid:48) ≥ t .A very related process to the one just introduced is the simpler version for which, informally,the second condition is altered to: “If at least one neighbor of x is infected and none is healthy,then x becomes infected.” It is clear that the set of infected sites in this version dominatesour process. However, the same proof techniques used below yield similar results to Theorem 2(namely, also a phase transition). Connections to kinetically constrained models.
This model has an indirect connectionwith Frederickson-Andersen 1 spin facilitated model (FA1f) [2, 8, 9]. In this case, the configura-tion space is { , } Z and the dynamics are defined as follows: a site x with occupation variable0 flips to 1 at rate 1 − q iff at least one among its nearest neighbors is in state zero; a site x with occupation variable 1 flips to 0 at rate q iff at least one among its nearest neighbors is instate zero. Note that the constraint for the 0 → → µ with µ ( η ( x ) = 0) = q . Notealso that the dynamics of our contact process coincide with the FA1f dynamics if we start froma configuration which does not contain infected sites.A non trivial problem for FA1f dynamics is to determine convergence to the equilibriummeasure µ for some reasonable initial measure, e.g. an initial product measure with density ofhealthy sites different from q [2]. We will now explain how our results provide an alternativeapproach to prove convergence to equilibrium in a restricted density regime. A possible strategyto prove convergence to equilibrium for FA1f dynamics started from an initial configuration η is to couple it with some ˜ η distributed according to µ . This gives rise to a process with 4states { , , ↓ , ↑ } . Here, 0 represent sites where both configurations are 0; 1 sites where bothconfigurations are 1; 2 ↓ sites where η is 0 and ˜ η is 1; and 2 ↑ sites where η is 1 and ˜ η is 0. If wenow denote the union of sites in state 2 ↓ and 2 ↑ as ”infected sites”, then if infection dies out,the original process started in η is distributed with the equilibrium measure (since there are nomore discrepancies with the process evolved from ˜ η which is at equilibrium at any time). It isnot difficult to verify that the dynamics of the 4 state contact process induced by the standardcoupling among two configurations evolving with FA1f dynamics are such that the union ofsites in state 2 ↓ and 2 ↑ is dominated by the infected sites of our 3-state contact process. Thuswhen infection dies out for our process it also dies out for the 4-state contact process and fromour Theorem 2 (ii) we get convergence to equilibrium for q ≥ q for the FA1f dynamics. Thisresult was already proven by a completely different technique in Blondel et al. [2] for parameter q > /
2. Notice that convergence to equilibrium is expected to hold for FA1f dynamics at all q > η satisfying the hypothesis of our Theorem 2 (ii), namely infection shouldalways disappear in the 4 state contact process. This is certainly not the case for our 3 statecontact process which has a survival extinction transition, as proved by Theorem 2 (i). . The small q regime In this section, we prove assertion (i) of Theorem 2. First, We defineΩ ∗ = { η ∈ Ω : ∃ a ≤ b ∈ Z : { x : η ( x ) = 2 } = [ a, b ] ∩ Z } , the set of configurations where infected sites form a finite, nonempty interval. Proposition 3.
Consider some η ∈ Ω ∗ . Then there exists < q such that P η (cid:104) η t / ∈ { , } Z for all t ≥ (cid:105) > for all q ≤ q , For the proof, we observe first that the set of sites in state 2, which we call the infectedcluster , is always connected. We would like to focus on the behavior of the infected boundarysites and so, due to symmetry, on I ( t ) := sup { x ∈ Z : η t ( x ) = 2 } , the position of the rightmost infected site. If there is only one infected site (thus, leftmostand rightmost infected site coincide), both with positive probability the next change in numberof infected sites might result in zero (extinction of the infection) or two infected sites. If thenumber of infected sites is at least two, only the status on the sites to the right of the rightmostinfected site have direct influence on the ‘movement’ of I ( t ).In (an informal) summary, if the infection shrinks to size one, it recovers with positive proba-bility to size at least two. If we show that from there, infection spreads with positive probability,we obtain our result. Therefore, we focus on this latter regime in the following.With this in mind, we now introduce a Markov chain, which can be interpreted as a simplifiedmodel of the rightmost infected site and its local right neighborhood, and prove a drift propertyfor it. This shall turn out to be useful when coupling this auxiliary Markov chain to our originalprocess in section 2.2.2.1. An auxiliary Markov chain.
We define a Markov chain ( Y i ) i ≥ living in the (countable)state space S = Z × { , } . We denote its first coordinate as the chain’s level or state and thuscan partition S into its n -states S n := { ( ω , ω , ω , ω ) ∈ S : ω = n } for any integer n . The Markov chain is defined by its transition graph shown in Figure 1.The subgraphs induced by S n are isomorphic, and furthermore, two states from S n and S m for | m − n | ≥ S n , with additional states in S n ± along with theirrespective transition probabilities. We denote the probability measure of this Markov chain by P (we trust that this causes no confusion with the measure of the interacting particle system).We define the stopping time τ to be the first time the Markov chain changes its level: τ := min { i ∈ N : ∃ n ∈ Z : Y ∈ S n , Y i ∈ S n ± } . Using Y mi (for 0 ≤ m ≤
3) to access the m th component of the state which Y is in at time i ,say that Y i is a progressive step (progress) if Y i = Y i − + 1 and similarly call Y i a regressivestep (regress) if Y i = Y i − −
1. We say that a natural number i is a step time (step) if Y i iseither a progressive or a regressive step. Lemma 4.
Let n ∈ Z and ε > . Then there exists < q < such that − ε < P [ Y τ ∈ S n +1 | Y = ( n, , , <
47 + ε, − ε < P [ Y τ ∈ S n +1 | Y = ( n, , , for all q < q . − q − q − q − q q − q q − q q − q − q − q − q − q − q − q
12 121 − q − q q − q q − qq q − q − q Y : − q ( n, , , n, , , n, , , n + 1 , , ,
1) ( n, , , n, , ,
0) ( n − , , , n, , , n, , , n, , ,
1) ( n + 1 , , ,
1) ( n − , , , n − , , , n − , , , − q q − q − q − q − q − q Figure 1.
Transition subgraph of Y induced by S n and its neighboring states. Proof.
The proof proceeds by counting paths in the transition graph. We define θ := P [ Y τ ∈ S n +1 | Y = ( n, , , ,θ := P [ Y τ ∈ S n +1 | Y = ( n, , , ,θ := P [ Y τ ∈ S n +1 | Y = ( n, , , . We also set a := − q − q ) to be the weight of the 2-cycle between states ( n, , ,
1) and ( n, , , θ = 12 (1 + θ ) ,θ ≥ − q − q θ + 1 − q − q θ ,θ ≥ (cid:18) θ + 1 − q a (cid:19) (cid:88) k ≥ a k = 11 − a (cid:18) θ + 1 − q a (cid:19) , using the strong Markov property. This leads to the explicit lower bounds θ ≥ − q + 3 q − q − q ,θ ≥ − q + 4 q − q − q ,θ ≥ − q + q q . For small q , all of these values are strictly larger than , except for θ , where we have θ (cid:37) as q →
0. Finally set b := − q − q ) to be the weight of a 2-cycle between states ( n, , ,
1) and( n, , ,
0) and observe that, by counting paths ending in S n − , we have1 − θ ≥ (cid:18) − q ) + 1 − q − q ) (cid:19) (cid:88) n ≥ n (cid:88) k =0 (cid:18) nk (cid:19) a k b n − k = 6 − q + q − q . n the first parenthesis, the first term comes from paths ending in ( n − , , ,
0) and ( n − , , , n − , , ,
1) as well as ( n − , , , q sufficiently small. (cid:3) Lemma 5.
There exists < q < such that for all < q < q , we have E (cid:2) Y τ − Y (cid:3) > . Proof.
We start by defining τ := min { i > τ : ∃ n ∈ Z : Y τ ∈ S n , Y i ∈ S n ± } to be the first level change after τ and actually prove E (cid:2) Y τ − Y (cid:3) >
0. Noting that after twolevel changes, Y will either have increased or decreased by 2 or not changed at all, the lemmafollows from proving P [ Y τ ∈ S n +2 | Y ∈ S n ] > P [ Y τ ∈ S n − | Y ∈ S n ] . (1)for any integer n . To this end, we make the following observation, which is an immediateconsequence of the definition of the Markov chain dynamics. Observation 1.
Let Y ∈ S n . Then from Y τ ∈ S n +1 , it follows that Y τ = ( n + 1 , , , . On theother hand, if Y τ ∈ S n − , then Y τ , with probability q , is one of the two states { ( n − , , , , ( n − , , , } and, with probability − q , is one of the two states { ( n − , , , , ( n − , , , } . We can thus restrict ourselves to proving (1) for Y being one of the two ‘ good ’ n -states G n := { ( n, , , , ( n, , , } , as we are allowed to choose q sufficiently small. CombiningObservation 1 with Lemma 4, we have˜ α := P [ Y τ ∈ S n +2 | Y ∈ G n ] ≥ min ω ∈G n ( P [ Y τ ∈ G n +1 | Y = ω ]) > (cid:18) − ε (cid:19) for some ε > q appropriately small. Recalling that a := − q − q ) was the weight of a2-cycle between states ( n, , ,
1) and ( n, , ,
1) and b := − q − q ) the value of a 2-cycle betweenstates ( n, , ,
1) and ( n, , , ω = ( n, , , ω (cid:48) = ( n, , ,
1) as well as ω (cid:48)(cid:48) =( n − , , , κ := P (cid:2) Y τ ∈ G n − , Y τ − = ω (cid:48) | Y = ω (cid:3) ≥ − q − q ) (cid:88) m ≥ m (cid:88) k =0 (cid:18) mk (cid:19) a k b m − k = 1 − q − q ) · − a − b = 2 − q + q − q , with the bound obtained simply by counting paths from ω to ω (cid:48)(cid:48) which pass through ω (cid:48) intheir second to last step. With ε small enough ( ε < /
100 say), we are now able to bound α := P [ Y τ ∈ S n − | Y = ω ], the probability of double regress from ω , as follows: α = P [ { Y τ ∈ S n − } ∩ { Y τ ∈ S n − } | Y = ω ] ≤ q + (1 − q ) · P (cid:2) { Y τ ∈ S n − } ∩ { Y τ ∈ G n − } ∩ { Y τ − = ω (cid:48) } | Y = ω (cid:3) + q + (1 − q ) · P (cid:2) { Y τ ∈ S n − } ∩ { Y τ ∈ G n − } ∩ { Y τ − (cid:54) = ω (cid:48) } | Y = ω (cid:3) . earranging by defining B = { Y τ ∈ G n − } ∩ { Y τ − = ω (cid:48) } and B (cid:48) = { Y τ ∈ G n − } ∩ { Y τ − (cid:54) = ω (cid:48) } and observing that the event B implies that Y τ = ω (cid:48)(cid:48) , we continue to find that α ≤ q + (1 − q ) (cid:88) A ∈{ B,B (cid:48) } P [ Y τ ∈ S n − | A ] · P [ A | Y = ω ] ≤ q + (1 − q ) · P (cid:2) Y τ ∈ S n − | Y τ = ω (cid:48)(cid:48) (cid:3) · κ + (1 − q ) · P [ Y τ ∈ S n − | Y τ = ω ] · ( P [ Y τ ∈ G n − | Y = ω ] − κ ) ≤ q + (1 − q ) ( κ (1 − θ ) + (1 − q )(1 − θ )(1 − θ − κ )) < q + (1 − θ ) + κ ( θ − θ ) < q + (cid:18)
12 + ε (cid:19) + (cid:18) − ε (cid:19) (cid:18) − (cid:19) < q + ˜ α − (cid:18) − ε (cid:19) < ˜ α for our chosen ε and q sufficiently small, where θ i have been defined in the proof of Lemma 4.Note that again we make heavy use of the strong Markov property as well as the bounds fromLemma 4. (cid:3) The coupling.
We are now ready to return to our process. Recall that Y should bethought of as a model of the right neighborhood of the rightmost infected site in the originalprocess. Intuitively speaking, we want to find a coupling such that Y ≤ I ( t ) at any giventime—this, however, is ill-defined. To make it more precise, let us first formally build towardsthe discrete version of the segment of the process that is of interest (i.e., the right neighborhoodof the rightmost infected site). For ( η t ) t ≥ a realization of the process in Ω ∗ , we define the mapΦ : Ω ∗ → Z × { , } as Φ( η t ) = (cid:16) I ( t ) , ( η t ( I ( t ) + i )) i =1 (cid:17) . Hence, (Φ t ) t ≥ = (Φ( η t )) t ≥ is the segment of the process we are interested in. Let S ( x ) =( s i ( x )) i ∈ N be the sequence of clock rings for site x . That is, s ∼ Exp(1) and ( s i +1 ( x ) − s i ( x )) ∼ Exp(1) for all i ∈ N . This allows us to define ( R i ) i ∈ N , the sequence of times of clock rings ofthe process restricted to ( η t ( I ( t ) + i )) i =0 , as R = 0 and R i +1 = inf { s j ( x ) : s j ( x ) > R i , x ∈ {I ( R i ) + l : 0 ≤ l ≤ } , j ∈ N } for all i ≥
0. We are interested in the process ( X i ) i ∈ N , a subset of ( X ) i ∈ N , where X = X =Φ( η R ) and X i = Φ( η R i ) ,X i = Φ( η R l ) , l = inf { k ≥ i : Φ( η R k ) (cid:54) = Φ( η R i − ) } for all i ≥
1. In words, ( X ) i is the embedded discrete time chain of (Φ) t , and X is the chainobtained from X by removing all of the self-loops. Process X is the one which, in certain timewindows, behaves very much like Y . To make this precise, we define ( R (cid:48) i ) i ∈ N with R (cid:48) = 0 and R (cid:48) i +1 = inf (cid:8)(cid:8) t > R (cid:48) i : I ( R (cid:48) i ) (cid:54) = I ( t ) (cid:9) ∪ { s j ( I ( R (cid:48) i ) + 4) : s j ( I ( R (cid:48) i ) + 4) > R (cid:48) i , j ∈ N } (cid:9) to be the times when either the position of the rightmost infected particle changes or the clockat the site determining the boundary condition, I ( · ) + 4, rings. We call W i := [ R (cid:48) i , R (cid:48) i +1 ) the stable windows for all i ≥
0. A stable window closes whenever the boundary site rings or theinfected site moves. We can now proceed to describe the behavior of X in a stable window. Aswe did for Y , we can partition the state space of X into its levels and, as no confusion arisesthis way, call them S n as well. The dynamics within W i depend only on Φ( η R (cid:48) i ), namely theinitial state also encoding the boundary conditions, and are therefore Markovian. Given thisinitial state for W , we can depict the transition graph in a very similar way as the one for Y ,as the subgraphs induced by the levels are again isomorphic, the states in neighboring levels − q − q − q − q q − q q − q q − q − q − q − q − q − q − q
12 121 − q − q
12 14 − q − q q − q − q ( X ) = 1 − q ( n, , , n, , , n, , , S n +1 ( n, , , n, , ,
0) ( n, , , n, , , n, , , S n +1 S n − S n − Figure 2.
Transition subgraph of X in window W induced by S n for passiveinitial boundary conditions. Note that entering S n ± closes W . − q − q − q − q q − q q − q q q − q − q − q − q q q − q − q − q q − q − q − q q
12 12 q q q q − q − q
12 14 − q − q − q qq − q − q − q ( X ) = 0 − q − q ( n, , , n, , , n, , , S n +1 ( n, , , n, , ,
0) ( n, , , n, , , n, , , S n +1 S n − S n − Figure 3.
Transition subgraph of X in window W induced by S n for healthyinitial boundary conditions.are terminal as they ‘close’ the window W . Conditional on the boundary conditions, the twotransition graphs are shown in Figures 2 and 3. Lemma 6.
We have that for any i ≥ P (cid:104) X R (cid:48) ∈ S n +1 | X R (cid:48) ∈ S n , X R (cid:48) / ∈ S n (cid:105) ≥ P [ Y τ ∈ S n +1 | Y ∈ S n ] for any integer n . In words, conditioned on W closing due to a level change, the probability ofprogress in X is bounded from below by the probability of progress in Y . roof. Knowing that X does not change its fifth coordinate (its boundary conditions as a clockring would close the window) within W allows us to couple the first four coordinates of X with Y while considering both cases of X ’s boundary conditions and aim for the desired domination.It is not hard to see that X dominates Y under passive boundary conditions: Note that thetransition probabilities as well as the states one ends up in after regress are the same except forstate ( n, , , Y .Let us justify why the same holds for healthy boundary conditions by showing that for anypath leading to progress in Y , we can find a union of heavier paths in X (with those unions beingdisjoint). It is clear that for all q < / Y last visiting ( n, , , X using the edges between states ( n, , ,
0) and ( n, , , X with healthy boundary conditions, we have additional edges betweenstates ( n, , ,
1) and ( n, , , Y leading to progress and last visiting either oneof these two states may make use of these edges and then progress in X . Defining c := q (1 − q )(2+ q )(3 − q ) to be the weight of a 2-cycle between these states we have that, starting from state ( n, , , (cid:18) − q q + c (cid:19) (cid:88) k ≥ c k = 1 − q , so these extra paths add up precisely to the weight of the edges from ( n, , ,
1) to S n +1 . Ananalogous computation gives the same result when starting from state ( n, , , (cid:3) Proof of Proposition 3.
Lemma 6 is valid regardless of the boundary condition, so we can gluetogether stable windows until the event X R (cid:48) / ∈ S n is satisfied—that is, until a window endswith a level change. In this case, there are two possibilities, namely X R (cid:48) ∈ S n +1 (progress) or X R (cid:48) ∈ S n − (regress).In the canonical coupling of the first four coordinates of X and Y within W , we obtainthat after regress, X and Y end up in the same state (w.r.t. to the first four coordinates of X ),whereas after progress, X is in one of the states ( n +1 , , , , z ) , ( n +1 , , , , z ) , ( n +1 , , , , z )or ( n + 1 , , , , z ) (with z ∈ { , } determining the new boundary condition), while Y will finditself in ( n + 1 , , , Y is in a state from which progress in less likely.In summary, Lemma 5 shows that Y dominates a random walk on Z with positive drift andso Y has a positive drift. Due to the coupling obtained from Lemma 6, this drift carries over to X . Hence, the law or large number for a random walk with drift yields the claimed statement.Finally, Theorem 2 (i) follows from Proposition 3 via Claim 1. (cid:3) Extinction for large q We import some notation from Section 2. Namely, let ( X i ) i ∈ N ⊂ Z × { , } be the discretetime process describing the rightmost infected site and its neighbors and let S n denote all n -levels of its state space. Similar to how τ and τ were defined for the Markov chain Y in thatsection, we define τ i for i ≥ τ i +1 = inf { j ≥ τ i : ∃ n ∈ Z : X τ i ∈ S n , X j ∈ S n ± } , where we set τ = 0, to be the sequence of level changes of X . We abbreviate τ = τ when it isconvenient. We next define two stopping times describing the length of consecutive progressiveand regressive steps, respectively. That is, we set (cid:42) τ := sup { i ∈ N : ∃ n ∈ Z : X ∈ S n , X τ i ∈ S n + i } , (cid:40) τ := sup { i ∈ N : ∃ n ∈ Z : X ∈ S n , X τ i ∈ S n − i } and call (cid:42) τ a progressive and (cid:40) τ a regressive interval, respectively. It is clear that we can partition X into alternating progressive and regressive intervals. Our aim is to prove that the length ofa progressive interval is, in expectation, less than the length of a regressive one. Note that if τ s a regressive step, then X τ ∈ G n for some integer n , where G n = { ( n, z , , z , z ) : z i ∈ { , } for i = 2 , , } . Similarly, if τ is a progressive step, then X τ ∈ B n for some n , with B n = { ( n, , z , z , z ) : z i ∈ { , } for i = 3 , , } . The following lemma is the main step in the proof of Theorem 2 (ii).
Lemma 7.
In the above notation, we have that E [ (cid:42) τ | X ∈ G n ] < E [ (cid:40) τ | X ∈ B n ] for n ∈ Z und q sufficiently large.Proof of Theorem 2 (ii). As observed above, starting at τ , any progressive interval must startfrom a G state, whereas any regressive interval must start from a B state. Hence, the conditioningin Lemma 7 is not a restriction and the rightmost infected site is dominated by a Z -valuedrandom walk with negative drift, which yields the claimed result. (cid:3) Turning towards the proof of Lemma 7, a key observation is the fact that, when q is sufficientlylarge, healthy sites drift towards each other. More precisely, given a connected set of passivesites with healthy boundary conditions, we expect the size of this set to decrease with time.With this in mind we define ξ x ( η ) = inf {| y − x | : y ∈ Z , η ( y ) = 0 } , for some η ∈ Ω and x ∈ Z , i.e. the distance of x to the next healthy site in η . Lemma 8.
Let q > / . Given any initial distribution ν taking values in { , } Z . Assume that κ := E [ ξ x ( ν )] < ∞ for some site x ∈ Z . Then for the process ( η t ) with η ∼ ν , we have E ν [ ξ x ( η t )] ≤ max { , κ + t (1 − q ) } ∀ t ≥ . Proof.
Let η = η be state of the process at time 0. Due to translation invariance and symmetry,we shall consider site x = 0 and assume the closest healthy site is located at ξ t = ξ ( η t ) (cid:29) t . Since we are only interested in an upper bound, we always assume that η ( ξ t +1) = 0.In doing this, we obtain a process whose ξ t -value dominates the original one. We thus end upwith the following simplification: • If site ξ t updates, then with probability 1 − q , it becomes passive and ξ t + = ξ t + 1, • if site ξ t − q , it becomes healthy and ξ t + = ξ t − ξ t . Hence, the expected change of η t after an update is 1 − q <
0. The number of updates in [0 , t ] of these two sites is 2 Poisson( t )-distributed, and with probability 1 /
2, an update yields a change of position, so N t , the numberof position changes in [0 , t ], is Poisson( t )-distributed. Hence, as all of this remains true for ξ t ≥
1, the statement follows by Wald’s lemma. (cid:3)
Note that Lemma 8 is very much in the spirit of Proposition 4.1 in [2], even though we needa much weaker statement to prove Lemma 7, namely that E ν [ ξ x ( η t )] is not increasing. Proof of Lemma 7.
We begin by considering (cid:40) τ and noting that, no matter the boundary con-ditions, P [ X τ ∈ S n − | X τ ∈ S n , X ∈ S n +1 ] = P [ X τ ∈ S n − | X ∈ G n ] ≥ α ( q )= min (cid:26) q − q · − q , − q (1 + q − q ) (cid:27) q → −−−→ . In words, following a regressive step, we witness another regressive step with probability at least α →
1. That is because from G n , X ends up in another regressive step within three steps orless, regardless of a change of boundary conditions during that time. As a direct consequence, [ (cid:40) τ | X ∈ G n ] ≥ (1 − α ( q )) − gets arbitrarily large for q →
1. On the other hand, we havethat there exists β < P [ X τ ∈ S n +1 | X ∈ B n \{ ( n, , , , } ] ≤ β for all q not too small ( q > / E (cid:104) (cid:42) τ | X ∈ B n \{ ( n, , , , } (cid:105) ≤ β (cid:16) E (cid:104) (cid:42) τ | X ∈ B n (cid:105)(cid:17) ≤ β (cid:16) E (cid:104) (cid:42) τ | X = ( n, , , , (cid:105)(cid:17) . So if we can bound the last quantity by some constant, we are done. This is where Lemma 8comes in. We bound this expectation by “jumping” to the closest healthy site, infecting allpassive sites on the way. More precisely, we progress the infection by force until reaching astate in G n . E (cid:104) (cid:42) τ | X = ( n, , , , (cid:105) ≤ ∞ (cid:88) i =0 (cid:16) E (cid:104) (cid:42) τ | X ∈ G n , ξ I (0) ( η ) = i + 2 (cid:105) + i (cid:17) P (cid:104) ξ I (0) ( η ) = i + 2 (cid:105) ≤ ∞ (cid:88) i =0 i P (cid:104) ξ I (0) ( η ) = i (cid:105) + E (cid:104) (cid:42) τ | X ∈ G n (cid:105) ∞ (cid:88) i =0 P (cid:104) ξ I (0) ( η ) = i + 2 (cid:105) ≤ E (cid:104) ξ I (0) ( η ) (cid:105) + E (cid:104) (cid:42) τ | X ∈ G n (cid:105) , which is bounded by a constant, combining Lemma 8 with the fact that the second term goesto 0 as q → (cid:3) References [1] J. van den Berg, J.E. Bj¨ornberg, and M. Heydenreich,
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TUM School of Management and Department of Mathematics, TechnischeUniversit¨at M¨unchen, Arcisstraße 21, 80333 M¨unchen, Germany
E-mail address : [email protected] (Kilian Matzke, Markus Heydenreich) Mathematisches Institut, Ludwig-Maximilians-Universit¨at M¨unchen,Theresienstraße 39, 80333 M¨unchen, Germany
E-mail address : [email protected], [email protected] (Cristina Toninelli) CNRS, Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Univ. Paris VI etVII, Bˆatiment Sophie Germain, Case courrier 7012, 75205 Paris Cedex 13, France
E-mail address : [email protected]@upmc.fr