Non-trivial linear bounds for a random walk driven by a simple symmetric exclusion process

Non-trivial linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalization methods of Kesten and Sidoravicius.

1 Introduction, results and motivation 1

.1 Setup
In this note, we discuss linear scaling properties of a random walk in a dynamic random environment (RWDRE), where the role of the random environment is taken by a one-dimensional simple symmetric exclusion process (SSEP). The latter is the càdlàg Markov process ξ = (ξ t ) t≥0 with state space E := {0, 1} Z whose infinitesimal generator L acts on bounded local functions f in the following manner: where η ∈ {0, 1} Z and η x,y is defined by For a detailed description, we refer the reader to Liggett [11], Chapter VIII. We say that the site x is occupied by a particle at time t if ξ t (x) = 1 and is vacant (alternatively, occupied by a hole) if ξ t (0) = 0.
For a fixed realization of ξ, the random walk in dynamic random environment W = (W t ) t≥0 is the time-inhomogeneous Markov process that starts at 0 and, given that W t = x, jumps to where α i , β i ∈ (0, ∞), i ∈ {0, 1}. We will assume that α 0 + β 0 = α 1 + β 1 =: γ (1. 4) and v 1 > v 0 with v 0 := α 0 − β 0 and v 1 := α 1 − β 1 , (1. 5) i.e., the total jump rate is constant and equal to γ, and the local drift is larger on particles than on holes. The latter is made w.l.o.g., since the SSEP is invariant under reflection through the origin. We will denote by P η the joint law of W and ξ when ξ 0 = η. We will draw ξ 0 from a Bernoulli product measure ν ρ with ρ ∈ (0, 1); these are known to be the only non-trivial extremal invariant measures for the SSEP. While many results for RWDRE have been obtained in the past few years for random environments exhibiting uniform and fast enough mixing (see e.g. Avena [1], and dos Santos [7]), very little is known when the random environment mixes in a non-uniform way, as happens in the SSEP. For example, there are still no general laws of large numbers available for such cases. In particular, for the model described here, the law of large numbers has only been proven under the restriction that v 1 > v 0 > 1 (see Avena, dos Santos and Völlering [2]). Another recent result is the paper by den Hollander, Kesten and Sidoravicius [5], where an approximate law of large numbers is proven when the random environment is a high-density Poisson field of independent random walks.

Main result
It is easy to see, with a coupling argument, that W lies between two homogeneous random walks with drifts v 0 and v 1 . In particular, any subsequential limit of t −1 W t as t → ∞ lies in the interval [v 0 , v 1 ]. But would it be possible, even along a subsequence, for W to travel at one of the extremal speeds? For the case of the SSEP, the following theorem answers this question in the negative.
While this result is "intuitively obvious", it does not seem a trivial fact to prove. For dynamic random environments consisting of single-site spin-flips with bounded flip rates, there is a simple proof strategy since particles and holes can be found locally "around" the random walk. For the supercritical contact process, the proof by den Hollander and dos Santos [6] that W cannot travel with speed v 0 is already non-trivial and relies on model-specific features. The proof of Theorem 1.1 given here is based on the multiscale analysis scheme put forth by Kesten in Sidoravicius [8], and seems exceedingly heavy for such a simple fact. It has however the advantage of being easier to generalize; while several technical facts are verified here specifically for the SSEP, the overall proof strategy should work in much greater generality. For example, the analogous result for the supercritical contact process can be reobtained with this approach.

Essential enhancements
Our question can also be formulated in terms of essential enhancements, in analogy with percolation theory (see e.g. Grimmett [4], Chapter 3). From this point of view, W is seen as a perturbation of a homogeneous random walk with drift v 0 , and ρ is the intensity of the perturbation. The question then becomes: is this perturbation, for any ρ > 0, an "essential enhancement" in the sense that it changes the linear scaling of W ?
Let us look at what can happen for random walks in static one-dimensional random environments. For these models, there are criteria for recurrence/transience as well as laws of large numbers proven under very general assumptions (see e.g. Zeitouni [12]).
If v 0 = 0 < v 1 , then the random walk is always transient to the right in any ergodic random environment with a positive density of particles. But random walks in static random environments can exhibit slow-down phenomena; for example, there are regimes where the random walk can be transient to the right with zero speed. In the case of i.i.d. static random environments, the latter can only happen when v 0 < 0 < v 1 . Therefore, if v 0 = 0 < v 1 , then the perturbation given by a static i.i.d. random environment is always an essential enhancement, as long as the density of 1's is positive.
Consider, however, the following example of a stationary and ergodic static random environment with positive particle density that does not result in an essential enhancement. Let L be an N-valued random variable with finite first moment but infinite second moment. Partition Z into intervals in a translation-invariant way such that the length of each interval is independent and distributed as L. Let η be obtained by coloring each interval with 1's or 0's according to independent fair coin tosses. On top of this static random environment, put a random walk with β 0 = α 0 = 1/2, β 1 = 0 and α 1 = 1. As discussed above, this random walk is transient to the right; therefore, it eventually reaches a point where there is an interval full of 1's to its left (into which it cannot backtrack) and an interval to its right whose law is still independent of the past. In other words, the times when W crosses the boundary between an interval full of 1's and the next interval are regeneration times. This observation allows us to estimate the speed of W by a constant times the ratio between the expectation of L and the expected time required by W to cross one interval, given that the interval to the left is occupied. The latter turns out to be infinite, so that W has speed 0 = v 0 . Therefore, in this example the random environment is not an essential enhancement, despite having particle density equal to 1/2.

Outline
The rest of the paper is organized as follows. In Section 2, we construct particular versions of the SSEP and of the random walk. In Section 3, we give the proof of Theorem 1.1 with the help of a proposition (Proposition 3.1 below) concerning rarefied and turbulent regions in the SSEP. In Section 4, we lay out the basic tools that will be used to prove Proposition 3.1 in Section 5, where all constructions and estimates specific to the SSEP are carried out.

Construction of the model
In Section 2.1 we construct the SSEP and, in Section 2.2, the random walk on top of the SSEP.

Graphical construction of the SSEP
It will be convenient to have a graphical construction of the SSEP including negative times. Let E be the set of edges of Z, i.e., all unordered pairs of neighbouring sites, and let A = (A e ) e∈E be a collection of independent Poisson point processes on R with intensity 1. Draw each event of A e in space-time as an arrow between the two sites connected by e. This gives rise to a system of random paths in Z × R as follows. For each (x, t) ∈ Z × R, there exists a.s. a unique doubly infinite right-continuous path that goes either vertically in time or (forcibly) across arrows of A. For s ∈ R, let ζ t s (x) denote the position of this path at time s. Given η ∈ {0, 1} Z , we will define the SSEP ξ = (ξ t ) t∈R by i.e., a space-time point (x, t) is occupied if and only if the path going through it hits an occupied site at time 0. If we take η to be distributed as ν ρ , ρ ∈ (0, 1), then we may check that this construction indeed results in a stationary process with the correct distribution. To verify this, we only need to note that ξ t (x) = ξ s (ζ t s (x)) for any s, t ∈ R and that, by the product structure and exchangeability of ν ρ , ξ s is independent of A.

The random walk on top of the SSEP
We next give a particular construction of the random walk model described in the introduction. Take a Poisson process N = (N t ) t≥0 with rate γ and two sequences +1}-valued random variables taking the value +1 with probability α 1 /γ and α 0 /γ , respectively. These random variables are taken such that ξ, N, J 1 , J 0 are jointly independent.
The random walk W is a functional of (ξ, N, J 1 , J 0 ) obtained as follows. We set W 0 := 0. At a time t > 0, W jumps if and only if N jumps, and the increment is is the state of the exclusion process at the position of W just before the jump. Setting then N 0 t + N 1 t = N t and we see that W has the following representation: , are discrete-time simple random walks that jump to the right with probability α i /γ. From this we immediately get 3 Proof of Theorem 1.1 Since the holes of a SSEP under P νρ have the same distribution as the particles of a SSEP under P ν 1−ρ , we may w.l.o.g. restrict ourselves to proving the statement for the lim inf in (1.6).
The main idea in the proof of Theorem 1.1 is that, because the jump rates are positive and bounded, the random walk can spend time on top of particles whenever it is in a region of the environment that is not too rough, namely, neither too rarefied nor too turbulent. A rarefied region is one where the density of the environment is atypically low. A turbulent region is one where the environment is moving atypically fast. It is of course not possible to control such deviations of the environment in all space and time simultaneously, but, as we will see in Proposition 3.1 below, it is possible to show that, in most of the regions accessible to the random walk, the environment cannot deviate too much from its typical behaviour.
In Section 3.1 we state Proposition 3.1. In Section 3.2, we use this proposition to prove Theorem 1.1. The proof of Proposition 3.1 is given in Section 5.
The key ingredient in the proof of Theorem 1.1 is the following proposition.
Proposition 3.1. For any ρ ∈ (0, 1), there exist (∆ r , ω r , ρ r , r ) r∈N as above such that, P νρ -a.s., Part (a) will be proved using a multiscale renormalization scheme developed by Kesten and Sidoravicius (see [8]; we also borrow some ideas from [9]). The adaptation is straightforward, and some simplifications are possible in our setting. Nevertheless, for completeness, we include all the details. The main new ingredient is a comparison between the SSEP and a system of independent random walkers, which is due to Liggett. The proof of part (b) uses a similar strategy, but is much simpler.
To simplify the exposition, we present the proof in dimension one only. There are no technical issues to extend it to higher dimensions. Small complications arise in the proof of Lemma 5.2 below, but they can be dealt with straightforwardly.

Proof of Theorem 1.1
Proof. Fix ρ ∈ (0, 1) and recall the definition of N 1 in (2.2). By (2.4), it is enough to prove the existence of a δ 0 > 0 such that Regard (W s ) s∈[0,t] as a path in R 2 and denote its length by t = t + N t . Recall that N is a Poisson process with rate γ > 0, independent of ξ. Using Proposition 3.1, fix * ∈ (1, ∞) and r * ∈ N such that Let B * t (W ) be the unique r * -block containing the spacetime point (W t , t). We call B * t (W ) rough if it is either r * -rarified or turbulent, and call it smooth otherwise. For t ≥ 0, let denote the total number of integer times between 0 and t at which W is inside a rough block. Since W can spend at most ∆ r * time units in each rough block, if t ≥ * , then by (3.7) we have Note that N 1 s+1 > N 1 s if and only if W jumps at least once from a particle in the time interval (s, s + 1]. Since W has uniformly positive jump rates, for any s ≥ 0, r ∈ N, > 0 and j ∈ [W s − r, W s + r], P νρ W jumps once from j in the time interval (s, s + ) | (W u ) u∈[0,s] , ξ ≥ δ (3.11) for some δ = δ(r, ) > 0. Therefore, if B * s (W ) is smooth, then since there is at time s at least one particle in [W s −r * , W s +r * ] that does not move before time s + r * . Therefore we can couple Y with an i.i.d. sequence (Ỹ s ) s∈N of Bernoulli(δ * ) random variables such that Y s+1 ≥Ỹ s+1 if B * s (W ) is smooth. Using these observations, we can write, for t ≥ * , By (3.9), the lim inf as t → ∞ of the term in parentheses in the r.h.s. of (3.13) is at least 1/2. The remaining term converges to δ * , since the number of integer times s in [1, t] for which B * s−1 (W ) is smooth is unbounded. Thus (3.6) holds with δ 0 = δ * /2.

Block percolation and partitioned systems
In this section we present a percolation result, Proposition 4.3 below, which will play an important role in the proof of Proposition 3.1 in Section 5.

Percolative systems
be the block in R d of side length ∆ with lower-left corner at k. A collection of random variables  In order to control Ψ( ), we need to restrict the class of allowed percolative systems. We will call a PS Υ homogeneous with parameter p ∈ (0, 1) if Υ(k) has distribution Bernoulli(p) for each k ∈ ∆Z d . We call it (finitely) partitioned if there exists a finite partition P of ∆Z d such that, for each I ∈ P, (Υ(k)) k∈I are jointly independent. (4.6) In other words, Υ is partitioned if its dependence graph has a finite chromatic number.
In that case, we let |P| := #P. In the following, we use the abbreviation p-PPS for "homogeneous partitioned percolative system with parameter p".

Key lemma
The following lemma is the key to the proof of Proposition 4.3 below.
Lemma 4.1. There exist constants c 1 , c 2 ∈ (0, ∞) depending on d only such that, for any percolative system Υ with scale ∆ that is stochastically dominated by a p-PPS with partition P, Our proof of Lemma 4.1 is an adaptation of the proof of Lemma 8 in [8]. It is based on geometric constraints of R d and an application of Bernstein's inequality, which we recall for the case of i.i.d. bounded random variables.

Lemma 4.2. (Bernstein's inequality)
Suppose that (X i ) i∈N is an i.i.d. sequence of a.s. bounded random variables with joint law P . Then For a proof of Lemma 4.2, see e.g. Chow and Teicher [3], Exercise 4.3.14.
Proof of Lemma 4.1. There exist K 1 , K 2 ∈ N, depending on d only, with the following properties. For any and ∆, the total number of ∆-blocks intersecting any path in W is at most K 1 /∆ and, for any n and ∆, the number of connected subsets of R d that are unions of exactly n ∆-blocks and contain the origin is at most e K 2 n . We will show that (4.7) holds with c 2 := 2 d K 1 K 2 and c 1 := 16c 2 . Since Ψ( ) does not decrease if additional ∆-blocks are opened, we may suppose that Υ is a p-PPS with partition P. Let L := θ −1 , N := K 1 /(L∆) . (4.9) As discussed in the first paragraph, N is an upper bound for the number of L∆-blocks intersected by any path in W . If /(L∆) < 1 2 , then θ /∆ < 1 and (4.7) holds trivially. Therefore, we may assume that /(L∆) ≥ 1 2 , in which case N ≤ 3K 1 /(L∆). Letting C N L := { connected subsets of R d containing the origin that are the union of N distinct L∆-blocks}, (4.10) we can estimate, for x > 0, To estimate for a fixed C ∈ C N L the corresponding term in (4.11), we use the partition.
where Bin(N L d , p) is a Binomial random variable and (4.12) is justified by (4.6) and the fact that each C ∈ C N L is the union of exactly N L d ∆-blocks. By the definition of L and our choice of c 1 , we can check that pN L d < 1 2 c 1 θ /∆. Therefore, substituting x in (4.12) by |P|c 1 θ /∆ and applying Bernstein's inequality (4.8), we obtain Since N ≤ 3K 1 /(L∆) ≤ 3K 1 θ/∆, we have K 2 N < c 2 θ /∆. Hence, combining (4.11) and (4.13), we get (4.14)

Sequences of percolative systems
The following proposition concerns sequences of percolative systems, and will be used in Section 5 in the proof of Proposition 3.1.
Proposition 4.3. Let (Υ r ) r∈N be a sequence of percolative systems in R d with with scales ∆ r , defined jointly in the same probability space through an arbitrary coupling. Suppose that, for each r ∈ N, Υ r is stochastically dominated by a p r -PPS with partition P r such that the following hold: where Ψ r ( ) is defined for Υ r as in (4.4)-(4.5).
Proof. Let 0 < ε < 1 2 (1/κ − md) and put θ r := d √ p r ∨ e −br with b = m(d − 1) + ε. By (i), there exists K 1 ∈ (0, ∞) such that sup r |P r | ≤ K 1 and, by (ii), there exist K 2 ∈ (0, ∞) and r 0 ∈ N such that ∆ r ≤ K 2 e (m+ε)r whenever r ≥ r 0 . Hence, by Lemma 4.1, where a := 1 − κ(m + ε + b) > 0 by our choice of ε and b. Thus, (4.17) is summable in . By the Borel-Cantelli lemma, a.s. for n ≥ r 0 and large enough we may estimate Most of the work is concentrated in Section 5.1, where the renormalization scheme for rarefied blocks is defined and analyzed using the results from Section 4. Central to this work are estimates for systems of independent simple random walks, stated in Lemma 5.3 below, which are used for comparison with the system of holes of the SSEP via a result due to Liggett. These estimates are used to control a recursive formula that, roughly speaking, transfers properties from larger to smaller scales, allowing us to deduce microscopic properties from mesoscopic and macroscopic properties.
In Section 5.2, a similar approach is used to analyze turbulent blocks from the point of view of Section 4. There the construction and estimates are much simpler.

Bad blocks
Fix ρ − ∈ (0, ρ), let N 0 ∈ N be large enough such that The parameters r will be defined in Section 5.2. Set alsoρ := 1−ρ and, for η ∈ {0, 1} Z , defineη byη (x) := 1 − η(x). (5.4) In the following, we will also need r-superblocks, defined as We call the r-block B r (k, s) bad if B r (k, s) is r-rarefied. Thus, any r-rarefied r-block is bad. We call r-dense any set in R 2 that is not r-rarefied. Proof. Since the product Bernoulli measure ν ρ is a translation-invariant equilibrium, for any r ∈ N, x ∈ Z and t ∈ R, we have where Bin(ω r , ρ) is a Binomial random variable and ε > 0. The last step can be justified e.g. by using Bernstein's inequality (4.2). Therefore, for any (k, s) ∈ ∆ r Z 2 , for some C ∈ (0, ∞). Since at most (2 + 1) 2 r-blocks intersect [− , ] 2 , we can estimate P νρ ∃ r > κ log( ) and a bad r-block intersecting [− , ] 2 which is summable in , and so the claim follows by the Borel-Cantelli lemma.
r (x, t) < ρ r ω r . Being locally spoiled means that, in the scale ∆ r+1 , the (r + 1)-block "has good conditions", meaning that the base of its (r + 1)-superblock is (r + 1)-dense, but nonetheless there are not enough particles transfered locally (i.e., inside B r+1 (k, s)) to ensure that in the finer scale ∆ r the neighbourhood B r+1 (k, s) is r-dense (which would in turn guarantee that B r+1 (k, s) contains no bad r-blocks). We will see below that, with our choice of parameters, being locally spoiled is an extremely unlikely event. Define a percolative system Υ r with scale ∆ r by Υ r (k, s) := 1 {Br(k,s) is locally spoiled} , and, for each a = (a 1 , a 2 ) ∈ B r (0, 2∆ r ) ∩ ∆ r Z 2 , let I a := {(z 1 , z 2 ) ∈ ∆Z 2 : z 1 ≡ a 1 (mod 11) and z 2 ≡ a 2 (mod 3)}. (5.14) Then is a partition of ∆ r Z 2 with |P r | = 33.
Lemma 5.2. For all large enough r ∈ N, Υ r is stochastically dominated by a p r -PPS with partition P r , where p r tends to 0 super-exponentially fast as r → ∞.
The proof of Lemma 5.2 requires quite a bit of work, including estimates for systems of simple random walks for comparison with the SSEP. Therefore, we postpone it to Section 5.1.5, and show first how it is used to prove Proposition 3.1(a).
We claim that, for all r ∈ N, Indeed, if an r-block is bad, then the unique (r + 1)-block containing it is either bad or locally spoiled, and the number of r-blocks inside any given (r + 1)-block is equal to N 12 0 . By induction we get, for R ≥ r + 1, For κ ∈ (0, (12 log(N 0 )) −1 ), take 0 as in Lemma 5.1 and R = κ log( ) . Then, for ≥ 0 , we may estimate The rest of this section is dedicated to the proof of Lemma 5.2. In Section 5.1.4 we derive some estimates for systems of independent simple random walks. These are used in Lemma 5.4 below for comparison with the system of holes of the SSEP. The latter lemma is used in Section 5.1.5 to prove Lemma 5.2.

Estimates for systems of independent random walks
It will be useful to compare the systemξ of the holes of the exclusion process with a system of independent simple random walks, which we define next.
Let (S z ) z∈Z be a collection of independent simple random walks on Z, with S z 0 = z for each z ∈ Z. For η ∈ {0, 1} Z , define the process ξ The first term in the r.h.s. of (5.27) can be estimated by for r large enough, where we use (5.25) and the fact that t ≥ ∆ r+1 . Decompose A x t into disjoint intervals I 1 , . . . , I n with length exactly ω r+1 , and let z i ∈ I i be the maximizer of z → P (S z t ∈ [x, x + ω r )) in I i . Then the second term in (5.27) is at most The last double sum in the r.h.s. of (5.29) is bounded by The first term in (5.30) can be estimated by and, via (5.26), the second term in (5.30) by for large enough r, where for the second inequality we use that ∆ r+1 ≤ t ≤ 3∆ r+1 . Now (5.23) follows by combining (5.27)-(5.32) sinceρ r+1 ≤ 1.

Proof of Lemma 5.2
In this section we give the proof of Lemma 5.2. The first step is to compareξ with a system of independent simple random walks and use the estimates of Lemma 5.3 to show that, if Λ r+1 (k, s) is (r + 1)-dense, then it is extremely unlikely for B r+1 (k, s) to be r-rarefied. This will also imply that the probability to have a locally spoiled B r+1 (k, s) is extremely low, since particles in the SSEP, with large probability, do not travel very large distances in a short time. This is the content of Lemma 5.4 below. We will need the following σ-algebras: Proof. By translation invariance and the Markov property, it is enough to prove (5.34) for (k, s) = (0, 2∆ r+1 ) and under P η for an arbitrary η ∈ {0, 1} Z , under the assumption that Λ r+1 (0, 2∆ r+1 ) is (r + 1)-dense. We will first do this for Σ x r (ξ t ). We claim that, for any η ∈ {0, 1} Z and λ > 0, where ξ • is a system of independent simple random walks as in Lemma 5.3. This can be justified using a result due to Liggett [11], Chapter VIII, Proposition 1.7, by noting that, for any n ∈ N, the function (y 1 , . . . , y n ) → exp λ n i=1 1 [x,x+ωr) (y i ) is symmetric and positive definite. Liggett's result only applies to initial configurations with finitely many particles, but, since Σ x r (ξ t ) is monotone in η, (5.35) follows by the monotone convergence theorem.
To that effect, let (y − t ) t≥0 denote the path starting at time 0 from the point y − 0 := −5∆ r+1 that goes upwards in time and (forcibly) jumps across any arrows of the graphical representation to the right. Likewise, let (y + t ) t≥0 denote the path that starts at y + 0 := 6∆ r+1 − 1 and follows the arrows of the graphical representation to the left. We see that, on the event A := {y − , y + do not hit B r+1 }, no particles can move from outside B r+1 into B r+1 . In particular, Σ r (x, t) = Σ x r (ξ t ) on A if [x, x + ω r ) × {t} ⊂ B r+1 . On the other hand, y − t − y − 0 and y + 0 − y + t are both distributed as a rate 1 Poisson process, and are independent of ξ 0 ; therefore, because of the shape chosen for B r+1 , we have P η (A) ≥ 1 − Ce −ε∆ r+1 (5.39) for some C, ε ∈ (0, ∞), which completes the proof.

Proof of Proposition 3.1(b)
In this section, we use the same proof strategy as in Section 5.1, but the arguments will be technically much simpler.
Set r := e −∆r . We call a point (x, t) ∈ Z × R r-stuck if both Poissonian clocks in the graphical representation that lie to the right and to the left of x fail to ring between times t and t + r . A subset of Z × R is called r-stuck if all its points with integer coordinates are r-stuck. Note that r-turbulent blocks are not r-stuck.
Lemma 5.5. Υ ns r is stochastically dominated by ap r -PPS with partition P ns r , wherep r decays super-exponentially fast in r.