Random walk on random walks: higher dimensions

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].


Introduction
Random walks on random environments are models for the movement of a tracer particle in a disordered medium, and have been the subject of intense research for over 40 years. The seminal works [22,33,34], concerning one-dimensional random walk in static random environment (i.e., constant in time), established a rich spectrum of asymptotic behaviours that can be very different from that of usual random walks. In higher dimensions, important questions remain open despite much investigation. For excellent expositions on this topic, see [35,37]. The dynamic version of the model, i.e., when the random environment is allowed to evolve in time, has been also studied for over three decades (see e.g. [14,27]). However, models with both space and time correlations have been only considered relatively recently. For an overview, we refer to the PhD theses [1,31]. We will abbreviate "RWRE" for random walk in static random environment, and "RWDRE" for random walk in dynamic random environment.
Asymptotic results for RWDRE under general conditions were derived e.g. in [5,6,12,15,19,29,30], often requiring uniform mixing conditions on the random environment (implying e.g. that the conditional distribution of the environment at the origin given the initial state uniformly approaches a fixed law for large times). This uniformity can be relaxed in particular examples, e.g. [10,18,28] (supercritical contact process), or under additional assumptions, e.g. [2,3] (spectral gap, weakly non-invariant) and [11] (attractivity). But arguably, some of the most challenging random environments are given by conservative particle systems, due to their poor mixing properties. Such cases have been considered in [4,7,8,21,32] (simple symmetric exclusion), and in [16,17] (independent random walks). Each of these works imposes additional conditions and explores very specific properties of the environment in question. In particular, the works [16,17,21] introduce perturbative approaches, where parameters of the system are driven to a limiting value where the behaviour is known.
In the present paper, we consider as in [16] dynamic random environments given by systems of independent simple symmetric random walks. As mentioned above, asymptotic results for this model are challenging since the random environment is conservative and has slow and non-uniform mixing. We extend the results of [16] to higher dimensions and more general transition kernels. Additional difficulties arise in this setting due to the loss of monotonicity properties present in the one-dimensional, nearest-neighbour case. Our main results are a strong law of large numbers, a functional central limit theorem and large deviation bounds for the position of the random walker under the annealed law in a high density regime. As an additional application of our methods, we re-obtain a (slightly improved) ballisticity condition for (the discrete-time version of) the infection-spread model considered in [23]. Some tools developed in the present paper will be also used in the accompanying article [13].

Definition of the model and main results
Denote by N = {1, 2, . . .} the set of positive integers and let Z + := {0} ∪ N. Fix d ∈ N and let N = (N (x, t)) x∈Z d ,t∈Z+ be a random process with each N (x, t) taking values in Z + , which we call the random environment. Let α : Z + × Z d → [0, 1] satisfy x∈Z d α(k, x) = 1 for every k ∈ Z + . (1.1) We are interested in the case where N is given by the occupation numbers of a system of simple symmetric random walks in equilibrium. More precisely, fix ρ ∈ (0, ∞) and let (N (x, 0)) x∈Z d be an i.i.d. collection of Poisson(ρ) random variables. From each site x ∈ Z d , start N (x, 0) independent simple symmetric random walks (which can be lazy or not). The value of N (x, t), t > 0 is then defined as the number of random walks present at x at time t. The process N (·, t) is a Markov chain in equilibrium on the state-space (Z + ) Z d . As already mentioned, N has relatively poor mixing properties; for example, it can be shown that Cov(N (0, t), N (0, 0)) decays as t −d/2 when t → ∞.
Let | · | denote the 1 -norm on Z d . We will make the following assumptions on α: Assumption (S): The set of possible steps is finite. We set R := max x∈S |x|, which we call the range of the random walk.

Assumption (R):
There exists x • ∈ S satisfying x • · e 1 > 0 and lim inf k→∞ α(k, x • ) > 0. (1.4) Assumption (D) means that, for sufficiently high particle density, the random walker has a local drift in direction e 1 . Assumptions (S) and (R) are technical; (S) simplifies the execution of many technical steps while (R) ensures some regularity for α(k, ·) over large enough k ∈ N. Note that (R) follows from (D) if either α(k, ·) is constant for sufficiently large k, or the random walker moves by nearest-neighbour steps, i.e., S ⊂ {x ∈ Z d : |x| ≤ 1}. Denote by P ρ the joint law of N and X and by E ρ the corresponding expectation. We can now state the main result of the present paper.
(iii) (Large deviation bounds) For every ε > 0, there exists c > 0 such that P ρ X t t − v > ε ≤ c −1 exp{−c(log t) 3/2 } for all t ∈ N. (1.7) EJP 24 (2019), paper 80. Theorem 1.1 may be interpreted as follows: Assumption (D) ensures that the random walker has a positive local drift in direction e 1 inside densely occupied regions of Z d . Theorem 1.1 shows that, when the density ρ is large enough, this behaviour "takes over", i.e., the random walker exhibits a macroscopic drift in direction e 1 , which introduces enough mixing for a law of large numbers and a central limit theorem to hold.
Note that the matrix Σ in item (ii) above might be zero; indeed, our assumptions on α do not exclude the case that X is deterministic. However, Σ will be non-zero as soon as X is non-trivial, and it will be non-singular under mild ellipticity assumptions such as e.g. sup k∈Z+ α(k, ±e i ) > 0 for all 1 ≤ i ≤ d; see (5.17). The speed of the decay in (1.7) is not optimal, and only reflects the limitations of our methods.
As previously mentioned, one of the biggest obstacles to obtain Theorem 1.1 are the poor space-time mixing properties of the random environment. A method to overcome this difficulty in ballistic situations was developed in [16] for the high density regime in one dimension, see also [21] for a similar approach when the random environment is given by a one-dimensional simple symmetric exclusion process. However, these results rely on monotonicity properties of the random walker that are in general not valid in higher-dimensional and/or non-nearest neighbour settings. A coupling method (cf. [20], [11]) can sometimes be used to deal with this problem, but is limited to cases where α belongs to a set of at most two transition kernels. Here we follow a different approach, exploiting properties of the random environment through more general renormalization and renewal schemes that also bypass the requirement of uniform ellipticity.
As another application of our methods, we provide a short proof of ballisticity for the one-dimensional discrete-time version of the model for the spread of an infection studied in [23]. In this model, particles can be of two types: healthy or infected. Fix ρ ∈ (0, ∞). At time zero, we place on each site of Z an independent number of particles, each distributed as a Poisson(ρ) random variable. Given the assignment of particles to sites, we declare all particles to the right of the origin to be healthy and all particles to its left, including those on the origin, to be infected. Then the system evolves as follows: each particle, regardless of its state, moves independently as a discrete-time simple symmetric random walk (with a fixed random walk transition kernel), and any healthy particle sharing a site with an infected particle becomes immediately infected.
We are interested in the positionX t of the rightmost infected particle at time t ∈ Z + . Still denoting by P ρ the underlying probability measure, we obtain: Proposition 1.2. For any ρ > 0, there exist v > 0 and c > 0 such that (1.8) The above proposition offers a slight improvement to the deviation bound given in [23], which is an important ingredient in establishing finer results about the infection front. For example, a similar statement was used in [24] to prove a law of large numbers, and in [9] to establish a central limit theorem for (the continuous-time version of)X t .
The rest of the paper is organized as follows. Section 1.2 below contains a short heuristic description of ideas used in our proofs. In Section 2, we give a particular construction of our model with convenient properties. In Section 3, we develop a renormalization procedure for general classes of observables, relying on a key decoupling result for the environment (Theorem 3.4 below) whose proof is given in Appendix A. Applications of the renormalization scheme to show ballisticity of the random walker and of the infection front, including the proof of Proposition 1.2, are discussed in Section 4. Finally, in Section 5 we define and control a regeneration structure for the random walker path and finish the proof of Theorem 1.1.
Throughout the text, we denote by c a generic positive constant whose value may change at each appearance. These constants may depend on all model parameters EJP 24 (2019), paper 80. discussed above but, in Section 3 and in Appendix A, they will not be allowed to depend on ρ, as we recall in the beginning of these sections.

Proof ideas
The proof of Theorem 1.1 is split into two main steps that can be informally described as: a ballisticity condition and a renewal decomposition. They are performed respectively in Sections 4 and 5. Let us now describe them in more detail.
Our first result for the random walker described above is reminiscent of the (T )condition of Sznitman (see [35]): in Theorem 4.1, we show that, for ρ large enough, X t · e 1 diverges to infinity in a strong sense, i.e., the random walker is ballistic. This is done via a renormalization argument. Once ballisticity has been established, our intuition tells us that, as time passes, the random walker will see "fresh environments" since the particles of the random environment have no drift; this informal description is made precise by defining a regeneration structure for the path of the random walker. Here this step must be performed differently from [16] because of the higher dimensions and non-nearest neighbour transition kernels. Moreover, because of the lack of monotonicity, the tail of the regeneration time must also be controlled differently. Proposition 1.2 is proved using a similar argument as for Theorem 4.1. First, the problem is reduced to showing that, with large probability, we can frequently find particles nearX t , cf. Lemma 4.4. Indeed, this implies the existence of a density of times whereX t behaves as a random walk with a drift, which is enough because it always dominates a simple symmetric random walk. The reduced problem can then be tackled using the renormalization procedure, driving not ρ to infinity but the size of the window aroundX t where we look for particles. See Section 4.2.

Construction
In this section, we introduce a construction of the environment of simple random walks in terms of a Poisson point process of trajectories as in [16]. This construction provides a convenient way to explore certain independence properties of the environment. We also provide a construction for the random walker and discuss positive correlations of certain monotone observables of the environment (cf. Proposition 2.2 below).
Define the set of trajectories Note that the trajectories in W are allowed to jump in any canonical direction, as well as to stay put. We endow the set W with the σ-algebra W generated by the canonical coordinates w → w(i), i ∈ Z.
Let (S z,i ) z∈Z d ,i∈N be a collection of independent random elements of W , with each S z,i = (S z,i ) ∈Z distributed as a double-sided simple symmetric random walk on Z d started at z, i.e., the past (S z,i − ) ≥0 and future (S z,i ) ≥0 are independent and distributed as a simple symmetric random walk on Z d started at z (lazy or not).
For a subset K ⊂ Z d × Z, denote by W K the set of trajectories in W that intersect K, i.e., W K := {w ∈ W : ∃ i ∈ Z, (w(i), i) ∈ K}. This allows us to define the space of point measures Ω = ω = i δ wi ; w i ∈ W and ω(W {y} ) < ∞ for every y ∈ Z d × Z , (2.2) endowed with the σ-algebra generated by the evaluation maps ω → ω(W K ), K ⊂ Z d × Z. Fix ρ ∈ (0, ∞) and let N (x, 0), x ∈ Z d be i.i.d. Poisson(ρ) random variables. Defining the random element ω ∈ Ω by it is straightforward to check that ω is a Poisson point process on Ω with intensity measure ρµ, where µ = and P z is the law of S z,1 as an element of W . Setting then we may verify that N has the distribution described in Section 1. We enlarge our probability space to support i.i.d. random variables U y , y ∈ Z d × Z sampled independently from ω, where each U y is uniformly distributed in the interval [0, 1]. We then define P ρ to be the joint law of ω and U = (U y ) y∈Z d ×Z . Our configuration space may be thus identified as Ω := Ω × [0, 1] Z d ×Z , equipped with the product σ-algebra.
To define our random walker, recall Assumption (R) and let x , ≥ 0. For y = (x, t) ∈ Z d × Z we write X y for the projection of Y y into Z d , i.e., Y y = Y (x,t) = (X y , + t). When y = 0 we omit it from the notation. One may verify that the random walker X = (X ) ∈Z+ is indeed distributed as described in Section 1.
We discuss next an important property of our random environment: the FKG inequality (cf. e.g. [26] Corollary 2.12 p. 78). It states that monotone functions of ω are positively correlated. This result will be used in the proof of Lemma 5.4, which is an important ingredient to control the tail of the regeneration time constructed in Section 5. We first need the following definition.

Definition 2.3.
A measurable function f : Ω → R is called non-decreasing, nonincreasing or monotone if, for all U ∈ [0, 1] Z d , the function ω → f (ω, U ) satisfies the same property in the sense of Definition 2.1, and analogously for events in σ(ω, U ). We give next a few other useful definitions. For a measurable function g : Ω → E (with E some measurable space), we will abuse notation by writing g to refer also to the random variable g(ω, U ), distributed according to the push-forward of P ρ . Definition 2.5. We say that the function g : For y = (x, n) ∈ Z d × Z and w ∈ W , define the space-time translation θ y w as (2.10) For A ⊂ W , θ y A is defined analogously, i.e., θ y A := w∈A {θ y w}. We may then define space-time translations operating on Ω as follows. For (ω, U ) ∈ Ω, let The translations of a measurable function g : Ω → E are then defined by setting g y = θ y g := g • θ y . (2.12) Note that θ y N (u) = N (y + u), and that the law of (ω, U ) is invariant with respect to the space-time translations, i.e., θ y (ω, U ) is distributed as (ω, U ) under P ρ for any y ∈ Z d × Z. In particular, the law of Y y − y in (2.7) does not depend on y since Y y = y + θ y Y .

Renormalization
In this section, we develop an important tool in the analysis of our model, namely, a multi-scale renormalization scheme. We will keep the setup reasonably general so that it may be used in future applications. An important consequence of the technique developed here is the ballisticity of the random walker (cf. Theorem 4.1), which is an essential ingredient for proving Theorem 1.1. All constants in this section will be independent of ρ, but may depend on other parameters of the model.

General procedure
To describe the renormalization procedure, we introduce the sequence of scales L 0 = 10 50 and The choice of constants 10 50 and 1/2 appearing above is not crucial; many other choices would have been equally good for our purposes. Note that Fix R ∈ N. In the relevant applications, R will be taken as in Assumption (S). Given a scale k, we will consider translations of the space-time boxes     Having this in place, we introduce the following definition. We aim to bound the probability of certain events A m inductively in k. For this, we will need another definition, concerning the occurrence of A m in consecutive scales.
where m 1 ↔ m m 2 stands for pairs of indices m 1 , m 2 ∈ M k such that B m1 , B m2 ⊆ B m and such that the vertical distance between the boxes B m1 , B m2 is at least L k .
In the definition above, if m 1 = (k, y 1 ) and m 2 = (k, y 2 ) with y 1 = (x 1 , t), y 2 = (x 2 , s) ∈ Z d × Z we say that the vertical distance between the boxes B m1 and B m2 is equal to Intuitively speaking, the above definition says that the occurrence of A m implies that two similar events happened in well-separated boxes of the smaller scale. The imposition that the boxes indexed by m 1 and m 2 in (3.8) are vertically separated will be useful to decouple the events A m1 and A m2 via Theorem 3.4. Examples of cascading events will be given in Section 3.2.
Let us also denote ιk := exp 2 The next theorem is the main result that we will use in order to bound p k (ρ).
∈ Z + such that the following holds. Fixk ≥ k o and a collection (A m ) m∈M ≥k that is adapted and cascading. Assume that the A m 's are either all non-increasing or all non-decreasing and that, for somê Then, writing ρ * := ιkρ and ρ * * := ι −1 kρ , for all k ≥k we have in the non-increasing case, ∀ ρ ≤ ρ * * in the non-decreasing case. (3.12) The upper bound 3/2 appearing in Theorem 3.3 is not sharp; any number β satisfying The statement of the previous theorem has two different cases, depending on whether the events A m are non-increasing or non-decreasing. All applications considered in this paper concern non-increasing events, but we choose to keep the exposition general in order to be able to use our results in the accompanying paper [13].
One of the main ingredients for the proof of Theorem 3.3 is a recursion inequality for p k , cf. Lemma 3.5 below. As the cascading property suggests, the key to obtain such a recursion is to decouple pairs of events A m1 and A m2 supported in boxes that are well-separated in time. Recall however that the environment of simple random walks, being conservative, presents poor mixing properties, which makes decoupling hard. We overcome this difficulty using a "sprinkling technique", which consists in performing a change in the density of particles in the environment in order to blur the dependency between such events. Thus, up to an error term, we bound Pρ(A m1 ∩ A m2 ) by the product P ρ (A m1 )P ρ (A m2 ), where ρ is slightly different fromρ. This is the content of Theorem 3.4 below, which has different statements for the cases where the events A m are non-increasing or non-decreasing.    The proof of Theorem 3.4 is given in the Appendix A, and is very similar to the proof of Theorem C.1 in [16].
We may now identify the constant k o appearing in Theorem 3.3. Fix γ ∈ (1, 3/2] and let n o , C o , c o as given by Theorem 3.4.
As anticipated, Theorem 3.4 leads to the following recursion inequality for p k .
Proof. We start with the case when the A m 's are all non-increasing. Using that the A m 's are adapted and cascading and that, by (3.15), L k ≥ n o , we apply Theorem 3.4 to the indicator functions of A m1 , A m2 to obtain This finishes the proof of (3.18) in the first case. Now assume that the events A m are all non-decreasing. As before, we can estimate Since, by the definition of L 0 ,ρ ≥ ρ/2, (3.18) follows.
Now that we know how large the sprinkling should be as we move from scale k + 1 to k in order to obtain a good recursive inequality for the p k 's, we will introduce a sequence of densities ρ k .
Given ρk > 0, define ρ k for k ≥k recursively by setting This shows that the sequence of densities ρ k is not asymptotically trivial.
We are now in position to prove Theorem 3.3.
Proof of Theorem 3.3. Given γ ∈ (1, 3/2], take k o as in (3.15)-(3.16). The first step in the proof is to show how one can use (3.18) to transport the bound p k (ρ k ) ≤ exp{−(log L k ) γ } to the scale k + 1. This can be summarized by saying that: To see why this is true, let us first use (3.18) in order to estimate which is smaller than 1 by (3.16), proving (3.24).
To this end, we first claim that, for all k ≥k, and our definition of ρk while, if the A m 's are all non-decreasing, then (3.28) follows by induction using (3.21), (3.1) and the assumption thatρ ≥ L Let us now prove (3.27) by induction on k. The case k =k holds by hypothesis. Assume now that (3.27) holds for some k ≥k. Noting that ρ k+1 ≥ L −1/16 k+1 by (3.28) and that (3.18) holds for ρ k and ρ k+1 replacing ρ andρ respectively (because the relation between ρ k+1 and ρ k is exactly as forρ and ρ in Lemma 3.5) we conclude by (3.24), that (3.27) also holds with k + 1 replacing k. This concludes the induction step and the proof of the theorem.

Constructing cascading events
We provide in this section a systematic way to construct certain collections of cascading events based on averages of functions of the random environment along Lipschitz paths. Our ultimate goal is to obtain Corollary 3.11 below, which provides in this context a short-cut to ballisticity-type results with minimal reference to the bulkier technical setup of the previous section.
Let us first describe the type of paths that we will consider. We say that a function We will further restrict the class of paths using a function H : Given such a function H and a path σ : The interpretation is that h σ (t) = 1 if and only if the jump σ(t + 1) − σ(t) is allowed by the random environment according to the rule H. The formal definition is as follows.
Definition 3.6. Given a box index m ∈ M k , we say that an R-Lipschitz function σ :  .7)). In addition, if for H : Remark 3.7. In the remainder of this paper, we will only be interested in applications where H ≡ 1, which implies that every R-Lipschitz function starting in I m is an (m, H)crossing. The more general set-up to be used in [13] will allow us to consider only paths σ that coincide with the trajectory performed by the random walker. The following definition plays a central role in our construction.
and an integer k ≥k, define the events Note that the events defined by (3.34) are not necessarily adapted or monotone. However, as already anticipated, we have the following. Proof. Fix k ≥k and recall that we have assumed vk ≥ L −1/16 k . The first thing we note is that this inequality holds for all k ≥k. This indeed follows by induction using the definition of v k exactly as for (3.28).
Next we claim that: if A m occurs for some m ∈ M k+1 , then there exist at least three elements satisfying s i = s j when i = j and such that A mi occurs for i = 1, 2, 3. Let us denote by σ j , j ∈ {1, . . . , J}, the restriction of σ to [n + (j − 1)L k , n + jL k ) which is again an (m j , H)-crossing for an appropriate index m j in M k , with B mj ⊂ B m (see Figure 2). We now estimate where, in the first inequality, we used the fact that, if A mi does not occur for some i ∈ {1, . . . , J}, where for the second inequality we use L k ≥ 10 50 (cf. (3.1)). Substituting this into (3.38), we get χ g σ ≥ v k+1 so that A m cannot occur. This proves the claim (3.35). Thus, on the event A m , we may assume that there exist m 1 = (k, y 1 ), m 3 = (k, y 3 ) in M k where y 1 = (x 1 , s 1 ) and y 3 = (x 3 , s 3 ) with s 3 ≥ s 1 + 2 (meaning that the vertical distance between B m3 and B m1 is at least L k ) and such that both A m1 and A m3 occur.
This finishes the proof of the lemma.
The events defined by (3.34) may be analysed with the help of Theorem 3.3 whenever they are adapted and monotone. We next give a complementary result stating that, whenever the conclusion of Theorem 3.3 holds for (A m ) m∈M ≥k , it can be extended by interpolation to boxes of length L ∈ N (not necessarily of the form L k ). We first need to extend the above definitions.  Given a function H : Ω × Z d → {0, 1} and y = (x, n), we define a (y, L, H)-crossing to be an R-Lipschitz function σ : [n, ∞) ∩ Z → Z d such that (σ(n), n) ∈ I y,L and h σ (j) = 1 for every j ∈ [n, n + L) ∩ Z. If the function H is identically equal to one we simply say that σ is a (y, L)-crossing.
Finally, given σ a (y, L, H)-crossing, we let Our interpolation result reads as follows.  Then, for every ε > 0, there exists c > 0 such that where v ∞ is given by (3.33).
Proof. We follow the proof of Lemma 3.5 in [16]. We may assume L to be so large that, definingǩ by 2Lǩ +2 ≤ L < 2Lǩ +3 , thenǩ ≥k and L k+1 L k > 1 + 2/ε for all k ≥ǩ. We first consider multiples of L k , k ≥ǩ. Define  Let us now see that Bǩ has high probability. Indeed, where for the last inequality we used that Lǩ ≥ L . We now claim that: on Bǩ, for any L ∈ Jǩ and any (0, L , h)-crossing σ, we have χ g σ ≥ v ∞ . (3.49) Let us prove this for fixed k ≥ǩ by writing L = lL k and fixing a (0, lL k , h)-crossing σ.
Before we proceed to the proof, a few words about Corollary 3.11. Assumption (3.53) can be interpreted as a triggering condition, i.e., an a-priori estimate that must be provided in order to start the renormalisation procedure. The measurability and monotonicity assumptions must be checked in each case. Note that measurability follows whenever g and H(·, x) (for all x ∈ Z d ) are local (i.e., supported in a finite set in the sense of Definition 2.9) and instantaneous, where we say that a function f : Ω → R is instantaneous if f (ω, U ) ∈ σ(N (z, 0), U (z,0) : z ∈ Z d ), i.e., f depends only on one time slice of the random environment.
Proof of Corollary 3.11. Let k o ∈ N be as in the statement of Theorem 3.3, and fixk ≥ k o satisfying Lk ≥ L * and (3.53). Setting vk := v(Lk), define v k , k >k as in (3.32) and v ∞ as in (3.33). For k ≥k, let A m be defined as in (3.34) and p k (ρ) as in (3.9). Note that the events A m , for m ∈ M ≥k as above are cascading, adapted and non-decreasing (resp. non-increasing) according to Lemma 3.9 and our assumptions.
The conclusion then follows from Proposition 3.10.

Applications
This section is dedicated to applying the renormalization setup developed in Section 3 to show ballistic behavior of two processes. Namely, for a random walker in the environment of simple random walks and for the front of an infection process.
Proof. Take k o as in the statement of Theorem 3.3 for γ = 3/2, and letk ≥ k o be large enough such that k≥k (1 − L −1/16 k ) ≥ 1 − ε/2. Choose vk := 1, g := 1 {N (0,0)≥K} , H ≡ 1 (thus, we will say only m-crossing instead of (m, H)-crossing). Define the family (A m ) m∈M ≥k as in (3.34) and note that it is adapted and that each A m is non-increasing. For a fixedρ > 0, consider the crude bound pk(ρ) ≤ Pρ ∃(y, n) ∈ B Lk such that N (y, n) < K For fixed K,k, we can chooseρ ≥ L −1/16 k such that the right-hand side of (4.4) is less than exp(−(log Lk) 3/2 ). Therefore, by Theorem 3.3 and Proposition 3.10, there exists c > 0 such that, for all ρ ≥ ρ(K, ε) := ιkρ (with ιk as in (3.10)) and all ≥ 1, The proposition follows by noticing that the first steps of any σ ∈ S form a (0, )crossing, and then applying a union bound.
Our second proposition is a quenched deviation estimate for the position of the random walk. Intuitively speaking, it says that if all paths spend a large proportion of their time in sites with many particles, then the random walker itself has to move ballistically.
For technical reasons we first have to restrict our attention to the collection of paths that behave well in a certain sense. For L ∈ N and v ∈ (0, R], let S v,L be those paths of S that never touch H v,L . More precisely S v,L := {σ ∈ S : (σ(i), i) / ∈ H v,L ∀ i ∈ Z + } .
Note that, under P ρ (· |ω), the process is a zero-mean martingale with respect to the filtration σ(X y 0 , . . . , X y ), and has increments bounded by 2R. Therefore, by Azuma's inequality and a union bound, there exists a c > 0 such that  Indeed, let 0 ∈ N be the smallest time satisfying Y y 0 − y ∈ H v,L . Then 0 ≥ L/(R + v) ≥ L/(2R). Setting σ = X y − x up to time 0 − 1 and equal to an arbitrary R-Lipschitz path that does not touch H v,L for times greater than 0 , then σ ∈ S v,L and we obtain by (4.11) that, on (A L,v,K,ε y ) c , D ω 0 ≥ (v + δ ) 0 . If additionally |M 0 | < δ 0 would hold, then we would have a contradiction since This shows (4.14), and the conclusion follows by (4.13). 1 {N (y+(σ(k),k))≥K} < (1 − ε) (4.16) and that the probability of the right-hand side of (4.16) does not depend on y.

Infection
In this subsection, we prove Proposition 1.2 regarding the front of the infection process described in the introduction. We start with a precise construction of the model. Fix ρ > 0, d = 1 and let N (z, 0) and S z,i be as in Section 2, i.e., (N (z, 0)) z∈Z are i.i.d. Poisson(ρ) random variables and (S z,i − z) z∈Z,i∈N are i.i.d., each distributed as a double-sided simple symmetric random walk on Z started at 0.
We also introduce random variables η(z, i, n) ∈ {0, 1} to indicate whether the particle corresponding to S z,i is healthy (η(z, i, n) = 0) or infected (η(z, i, n) = 1) at time n. We will define them recursively as follows. Set the initial configuration to be η(z, i, 0) = 1 if z ≤ 0 and i ≤ N (z, 0), (4.17) η(z, i, 0) = 0 otherwise. (4.18) Supposing that, for some n ≥ 0, η(z, i, n) is defined for all z ∈ Z, i ∈ N, we set This definition means that, whenever a collection of particles share the same site at time n, if one of them is infected then they will all become infected at time n + 1.
We are interested in the processX = (X n ) n∈Z+ defined bȳ X n = max{S z,i n : η(z, i, n) = 1}, (4.20) i.e.,X n is the rightmost infected particle at time n. We callX the front of the infection. Note that the process η differs slightly from that described in the introduction, where particles sharing a site with an infected one were required to become immediately infected. However, it is easy to check that the processX is not affected by this difference, and we choose to work with η for simplicity.
Our first result towards Proposition 1.2 is a reduction step, stating that it suffices to find, with high probability, enough times n when the frontX n of the infection process is close to another particle. For this we fix r ≥ 0 and define g r by that is, g r is the indicator function of the event that, at time zero, there are at least two particles at even sites within distance r from the origin. Our lemma reads as follows.
Proof. One can check from the definition of the rightmost infected particle that the incrementX n+1 −X n always dominates that of a symmetric random walk on Z. At some steps, however, this increment has a drift to the right, namely when there is more than one particle atX n . The idea of the proof will be to bound the number of times at which such positive drift is observed. We first note that the front starts close to the origin. Indeed, Now, at every time n at which there is another particle at distance at most r from the frontX n at a site with the same parity asX n , we can use the Markov property to see that, with uniformly positive probability, this additional particle will reach the front within the next r steps. This means that, if n is such a time, the incrementX n +r+1 −X n stochastically dominates (under the conditional law given (N (·, )) ≤n ) a random variable ζ with positive expectation satisfying |ζ| ≤ r + 1. We will show that v = hE[ζ]/(3(r + 1)) fulfills (1.8).
Consider the 1-Lipschitz path given by the front (X ) L =0 . Denote by D the intersection of the event appearing in (4.22) L}. On D, we see that, for at least hL steps between times zero and L, the frontX is r-close to another particle. Therefore, the same happens for at least k L := hL /(r + 1) steps that are at least r + 1 time units apart from each other, and we can estimate using the Markov property , where the ζ i 's are i.i.d. and distributed as ζ. Applying standard large deviation estimates to the sum of the ζ i 's and to S 0,1 , we see that (4.24) finishing the proof of the lemma.
We next present the proof of Proposition 1.2. In light of Lemma 4.4, all we need to prove is (4.22), and for this we will use the renormalization procedure developed in Section 3. One might try to obtain (4.22) by direct application of Theorem 3.3, defining the events A m in a natural way and then taking r large enough. There is however a serious problem with this approach: for large values of r, the family A m will no longer be adapted in the sense of Definition 3.1. To circumvent this issue, we define intermediate classes of events that will certainly be adapted, although not necessarily cascading. The details are carried out next.  In the definition of A m we have used the local function g L k , which means that we are looking for particles on even sites at distance at most L k from the origin. Intuitively speaking, this task will become easier and easier to accomplish as k grows. This is made precise in the following claim: there exists a c > 0 such that  As mentioned above, the family A m may not be cascading, however it is clearly adapted. We now define another collection that will indeed be cascading. Let  which satisfies that A m ⊂ A m . Note that the local function g Lk is fixed, i.e. it does not depend on the scale k associated with m. This allows us to employ Lemma 3.9 and conclude that the family (A m ) m∈M ≥k is cascading. (4.29) Moreover, this collection is adapted and composed of non-increasing events.

Regeneration: proof of Theorem 1.1
In this section, we adapt Section 4 of [16] to our setting using Propositions 4.2 and 4.3. Theorem 1.1 will then follow as a consequence of the resulting renewal structure.
Hereafter, we fix v ∈ (0, v • ) and take k ∈ N and ∈ (0, 1) as in Proposition 4.3. We then define ρ := ρ(k , ) as given by Proposition 4.2 and Theorem 4.1. We will also fix ρ ≥ ρ and write P := P ρ from now on. By Assumption (R) and Proposition 4.3, we may assume that  x · e 1 ≥vn, |x| ≤ Rn}, where R is as in Assumption (S). As in [16], we define the sets of trajectories W ∠ y = trajectories in W that intersect ∠(y) but not ∠ (y), W ∠ y = trajectories in W that intersect ∠ (y) but not ∠(y), W y = trajectories in W that intersect both ∠(y) and ∠ (y). (5.5) Note that W ∠ y , W ∠ y and W y are disjoint, and therefore the sigma-algebras are jointly independent under P. Define also the sigma-algebras and set Note that, for two space-time points y, y ∈ Z d × Z, if y ∈ ∠ (y ) then F y ⊂ F y .
In order to define the regeneration time, we first need to introduce certain record times (R k ) k∈N . The definition here will be different from the one in [16]. To this end, set R 0 := 0 and, recursively for k ∈ N 0 , Define now a filtration (F k ) k≥0 by setting, for k ≥ 0, i.e., the sigma-algebra generated by Finally we define the event in which the walker started at y remains inside ∠(y), the probability measure with corresponding expectation operator E ∠ , the regeneration record index (5.14) The following two theorems are the analogous of Theorems 4.1-4.2 of [16] in our setting.
Theorem 5.2. There exists a constant c > 0 such that E e c(log τ ) 3/2 < ∞ (5.15) and the same holds with E ∠ replacing E.
The proof of Theorem 5.1 follows exactly as that of Theorem 4.1 in [16] and thus we omit it here. Theorem 5.2 will be proved in the next section. From them follows the: Proof of Theorem 1.1. Using Theorems 5.1-5.2, one may follow almost word for word the arguments given in Section 4.3 of [16], with the difference of having now random vectors instead of real-valued random variables. In particular, we obtain the formulas for the velocity v and the covariance matrix Σ, from which the comments made after Theorem 1.1 may be deduced. The fact that v · e 1 ≥ v follows from Theorem 4.1.

Control of the regeneration time
In this section, we prove Theorem 5.2 by adapting Section 4.2 of [16] to our setting. The two most important modifications are as follows. First, in order to bypass the requirement of uniform ellipticity, we do not require the random walker to make jumps in a fixed direction independently of the environment but instead only over points containing enough particles. For this, we need to estimate the probability of certain joint occupation events, cf. Lemma 5.3 below. Second, we need a substitute for Lemma 4.5 of [16], which gave a quenched estimate on the backtrack probability of the random walker and was obtained therein using a monotonicity property only available in one dimension. This is the role of Lemma 5.4 below, obtained with the help of Propositions 4.2-4.3.
In our first lemma, we construct a path for the random walk to follow where all the points have a large number of particles. This has a cost that is at most exponential. Lemma 5.3. There exists c 0 ∈ (0, 1) such that, for all L ∈ N, Proof. We proceed by induction in L. Recall the definition of S 0,1 in Section 2 and let (5.20) Using now that, for any i = 0, . . . , L − 1, the sets of trajectories W ix•,i \ (W Lx•,L ∪ W 0 ) and W Lx•,L \ (W (L+1)x•,L+1 ∪ W Lx•,L ) are disjoint, and using also the translation invariance of P, we see that the right-hand side of (5.20) equals ) by the induction hypothesis, concluding the proof.
Our next result is an estimate on the conditional backtrack probability of the random walker, which as already mentioned can be seen as a substitute for Lemma 4.5 of [16].

Recall the definition of
Lemma 5.4. There exists a constant c 1 > 0 such that  Recall (2.6) and the discussion below it. Put L := (1 − v )L and abbreviate A L y := A L , v ,k ,ε y (cf. (4.6)). Note that B L y , C L y are measurable in F y (L) to obtain, P-a.s., a.s. (5.25) by Proposition 4.3 (recall that v ≤ v ). Substituting this back into (5.24) and using that B L y , A L y (L) ∈ σ(ω), C L y ∈ σ(U ), we obtain that (5.24) is a.s. larger than when L is large enough. Reasoning as for equation (4.16) in [16], we see that, P-a.s., Moreover, since B L y ∩ (A L y (L) ) c is non-decreasing (in the sense of Definition 2.1), its conditional probability given G ∠ y ∨ G y only increases if ω(W y ) = 0. Hence, P-a.s.,  We proceed with the adaptation of Section 4.2 of [16]. As in equation (4.21) therein, we define the influence field h(y) := inf l ∈ Z + : ω(W y ∩ W y+(lx•,l) ) = 0 , y ∈ Z d × Z. where p is as in (5.1) and c 0 as in (5.18). Analogously to equations (4.28)-(4.29) in [16], we set, for T > 1, and we define the local influence field at a space-time point y ∈ Z d × Z to be: Note that our definition is slightly different from that of [16]. As in Lemma 4.4 therein, we obtain: Lemma 5.6. For all T > 1 and all y ∈ Z d × Z, (5.36) where c 2 , c 3 are the same constants as in Lemma 5.5.
As in [16], an important definition is that of a good record time (g.r.t.): With the above definitions and results in place, only minor modifications are required to adapt the rest of Section 4.2 of [16] to our setting. For completeness, we provide below all the details.
The following proposition is the main step in the proof of Theorem 5.2.  Proof of (5.43): we may upper-bound (5.48) by for some constantĈ > 0, where for the first inequality we use Lemmas 5.4 and 5.6 (see also the comment after (5.8)), and for the second we use the definition of . Thus, for T large enough, (5.43) is satisfied with c = c 1 /2. Proof of (5.44): Let B L y as in (5.23) and note that B T y , (U y+(lx•,l) ) l∈Z+ and F y are jointly independent. For B ∈ F k , write where the second equality uses the independence between G ∠ y and F y .
Thus, (5.42) is verified. Since {R k is a g.r.t.} ∈ F k+T , we obtain, for T large enough, P (R k is not a g.r.t. for any k ≤ T ) ≤ P R (2k+1)T is not a g.r.t. for any k ≤ T /3T by our choice of and δ.
The proof of Theorem 5.2 can then be finished as in [16].
Proof of Theorem 5.2. Since P ∠ (·) = P(·|A 0 , ω(W 0 ) = 0) and P(A 0 , ω(W 0 ) = 0) > 0, it is enough to prove the statement under P. To that end, let We are trying to prove (5.57) and for this, pick any trajectory w in W y1 and let us show that it is not in our Poisson point process. Observe that w touches ∠(y 1 ). From the fact that we are in E c 1 , w does not touch ∠ (y −1 ). Using (5.37) we also conclude that w does not touch ∠ (y 0 ), otherwise it would have to touch ∠ (y −1 ). But by (5.39), w has to touch ∠ (y 0 ), contradicting the above. In conclusion, for T large enough we have from which (5.15) follows.

A Decoupling of space-time boxes
The aim of this section is to prove Theorem 3.4. The proof is very similar to the proof of Theorem C.1 in [16]; only the most important changes are described here. In the following subsections, we will concentrate on intermediate results required for item (b) of Theorem 3.4, i.e., the case where f 1 , f 2 are both non-decreasing. The non-increasing case will be discussed in the proof of Theorem 3.4 itself at the end of this Appendix.
The constants in this section will be all independent of ρ; this is crucial for the perturbative arguments of Section 3.

A.1 Soft local times
We start with a coupling result. For a Polish space Σ and a Radon measure µ on Σ, let m denote the Poisson point process on Σ × R + with intensity measure µ ⊗ dv, where dv is the Lebesgue measure on R + . We write m = i∈N δ zi,vi with (z i , v i ) ∈ Σ × R + .
Fix a sequence of independent Σ-valued random elements Z j , j ∈ N. Assume that the law of Z j is absolutely continuous with respect to µ with density g j . As in Appendix A of [16], we define the soft local times G j : Σ → [0, ∞), j ∈ N by setting ξ 1 = inf t ≥ 0 : tg 1 (z i ) ≥ v i for at least one i ∈ N , G 1 (z) = ξ 1 g 1 (z), . . . ξ k = inf t ≥ 0 : tg j (z i ) + G k−1 (z i ) ≥ v i for at least k indices i ∈ N , G k (z) = ξ 1 g 1 (z) + · · · + ξ k g k (z)  (1). Furthermore, there exists a coupling Q of (Z j ) j∈N and m such that, for any J ∈ N, ρ > 0, for all compact H ⊂ Σ.

A.2 Simple random walks
As in [16], we will need some basic facts about the heat kernel of random walks on Z d . Let p n (x, x ) = P x (S x,1 = x ), x, x ∈ Z d , with P z , S z,i as defined in Section 2. Hereafter we will assume that S 1,0 is lazy; non-lazy S 1,0 are bipartite, and the argument below may adapted as outlined in Remark C.4 of [16]. Lazy S 1,0 are aperiodic in the sense of [25], and thus there exist constants C, c > 0 such that the following hold for all n ∈ N: 2)], while (A.5) follows by an application of e.g. Azuma's inequality.
The above inequalities will be used to prove Lemma A.3 below, regarding the integration of the heat kernel over a sparse cloud of sample points. In order to state it, we need the following definitions. (b) For ρ ∈ (0, ∞), we say that a collection of points (x j ) j∈J ⊂ Z d is ρ-sparse with respect to the L-paving {C i } i∈I when #{j : x j ∈ C i } ≤ ρL d ∀ i ∈ I. In the above definition, by interval, we mean a subset of Z d that is a Cartesian products of intervals of Z.
The next lemma provides an estimate of the sum of the heat kernel over a sparse collection (x j ) j∈J . Lemma A.3. There exists c > 0 such that the following holds. Let {C i } i∈I be an L-paving and (x j ) j∈J be ρ-sparse collection with respect to {C i } i∈I . Then, for all n ≥ L, j∈J p n (0, x j ) ≤ ρ 1 + cL(log n) d √ n .

A.3 Coupling of trajectories
Given a sequence of points (x j ) j∈J in Z d , let (Z j n ) n∈Z+ , j ∈ J, be a sequence of independent simple random walks on Z d starting at x j , and let j∈J P xj denote their joint law. The next lemma, analogous to Lemma B.3 in [16], provides a coupling of (Z j n ) j∈J with a product Poisson measure on Z d . Lemma A.4. There exists a constant c ≥ 1 such that the following holds. Let (x j ) j∈J ⊂ Z d be ρ-sparse with respect to the L-paving {C i } i∈I . Then for any ρ ≥ ρ there exists a coupling Q of ⊗ j∈J P xj and the law of a Poisson point process j ∈J δ Y j on Z d with intensity ρ such that Proof. By Lemma A.1, there exists a coupling Q such that where G J (z) = j∈J ξ j p n (x j , z) with (ξ j ) j i.i.d. Exp (1)  Inserting this estimate into (A.14), we get the claim.

A.4 Proof of Theorem 3.4
We can now finish the: in [16]. The proof of Theorem 3.4(b) is completely analogous, following from Lemma A.4 above as Theorem C.1 in [16] follows from Lemma B.3 therein.