Law of large numbers for a class of random walks in dynamic random environments

In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.


Introduction and main results
In Section 1 we give a brief introduction to the subject, we define the random walk in dynamic random environment, introduce a space-time mixing property for the random environment called cone-mixing, and state our law of large numbers for the random walk subject to cone-mixing. In Section 2 we give the proof of the law of large numbers with the help of a space-time regenerationtime argument. In Section 3 we assume a stronger space-time mixing property, namely, exponential mixing, and derive a series expansion for the global speed of the random walk in powers of the size of the local drifts. This series expansion converges for small enough local drifts and its first term allows us to determine the sign of the global speed. (The perturbation argument underlying the series expansion provides an alternative proof of the law of large numbers.) In Appendix A we give examples of random environments that are cone-mixing. In Appendix B we compute the first three terms in the expansion for an independent spin-flip dynamics.

Background and motivation
In the past forty years, models of Random Walk in Random Environment (RWRE) have been intensively studied by the physics and the mathematics community, giving rise to an important and still lively research area that is part of the field of disordered systems. RWRE on Z d is a Random Walk (RW) evolving according to a random transition kernel, i.e., its transition probabilities depend on a random field or a random process ξ on Z d called Random Environment (RE). The RE can be either static or dynamic. We refer to static RE if ξ is chosen at random at time zero and is kept fixed throughout the time evolution of the RW, while we refer to dynamic RE when ξ changes in time according to some stochastic dynamics. For static RE, in one dimension the picture is fairly well understood: recurrence criteria, laws of large numbers, invariance principles and refined large deviation estimates have been obtained in the literature. In higher dimensions many powerful results have been obtained as well, while many questions still remain open. For a review on these results and related questions we refer the reader to [21,24,25]. In dynamic RE the state of the art is rather modest even in one dimension, in particular when the RE has dependencies in space and time.
RW in dynamic RE in dimension d can be viewed as RW in static RE in dimension d + 1, by considering time as an additional dimension. Consequently, we may expect to be able to adapt tools developed for the static case to deal with the dynamic case as well. Indeed, the proof of our Law of Large Numbers (LLN) in Theorem 1.2 below uses the regeneration technique developed by Comets and Zeitouni for the static case [8] and adapts it to the dynamic case. A number of technicalities become simpler, due to the directedness of time, while a number of other technicalities become harder, due to the lack of ellipticity in the time direction.
For an extended list of references in classes (1) and (2), we refer the reader to [2]. Class (3) is clearly the most challenging.
In this paper we focus on models in which the RE are constituted by Interacting Particle Systems (IPS's). Indeed, IPS's constitute a well-established research area (see e.g. [15]), and are natural (and physically interesting) examples of RE belonging to class (3). Moreover, results and techniques from IPS's theory can be used in the present context as in our Theorem 1.3 (see also [2,9]).
Most known results for dynamic RWRE (like LLN's, annealed and quenched invariance principles, decay of correlations) have been derived under suitable extra assumptions. Typically, it is assumed either that the random environment has a strong space-time mixing property and/or that the transition probabilities of the random walk are close to constant, i.e., are small perturbation of a homogeneous RW. Our LLN in Theorem 1.2 is a successful attempt to move away from these restrictions. Cone-mixing is one of the weakest mixing conditions under which we may expect to be able to derive a LLN via regeneration times: no rate of mixing is imposed. Still, it is not optimal because it is a uniform mixing condition (see (1.11)). For instance, the exclusion process, due to the conservation of particles, is not cone-mixing.
Our expansion of the global speed in Theorem 1.3 below, which concerns a perturbation of a homogeneous RW falls into class (3), but, unlike what was done in previous works, it offers an explicit control on the coefficients and on the domain of convergence of the expansion.

Model
Let is an interacting particle system taking values in Ω, i.e., a Feller process on Ω, with ξ t (x) = 0 meaning that site x is vacant at time t and ξ t (x) = 1 that it is occupied. The paths of ξ take values in D Ω [0, ∞). The law of ξ starting from ξ 0 = η is denoted by P η . The law of ξ when ξ 0 is drawn from µ ∈ (Ω) is denoted by P µ , and is given by Through the sequel we will assume that P µ is stationary and ergodic under space-time shifts.
Conditional on ξ, let be the random walk with local transition rates Thus, on occupied sites the random walk has a local drift to the right while on vacant sites it has a local drift to the left, of the same size. Note that the sum of the jump rates equals α + β and is thus independent of ξ. Let P ξ 0 denote the law of X starting from X 0 = 0 conditional on ξ, which is the quenched law of X . The annealed law of X is (1.9)

Cone-mixing and law of large numbers
In what follows we will need a mixing property for the law P µ of ξ. Let (·, ·) and · denote the inner product, respectively, the Euclidean norm on R 2 . Put = (0, 1). For θ ∈ (0, 1 2 π) and t ≥ 0, let be the cone whose tip is at t = (0, t) and whose wedge opens up in the direction with an angle θ on either side (see Figure 1). Note that if θ = 1 2 π (θ = 1 4 π), then the cone is the half-plane (quarter-plane) above t . (1.12) In Appendix A we give examples of interacting particle systems that are cone-mixing.
We are now ready to formulate our law of large numbers (LLN).
The proof of Theorem 1.2 is given in Section 2, and is based on a regeneration-time argument originally developed by Comets and Zeitouni [8] for static random environments (based on earlier work by Sznitman and Zerner [22]).

Global speed for small local drifts
For small α − β, X is a perturbation of simple random walk. In that case it is possible to derive an expansion of v in powers of α − β, provided P µ satisfies an exponential space-time mixing property referred to as M < ε (Liggett [15], Section I.3). Under this mixing property, µ is even uniquely ergodic.
Suppose that ξ has shift-invariant local transition rates i.e., c(A, η) is the rate in the configuration η to change the states at the sites in A, and c( where η x is the configuration obtained from x by changing the state at site x. The interpretation of (1.15) is that M is a measure for the maximal dependence of the transition rates on the states of single sites, while ε is a measure for the minimal rate at which the states of single sites change. See Liggett [15], Section I. 4, for examples. where c 1 = 2ρ − 1 and c n ∈ R, n ∈ N\{1}, are given by a recursive formula (see Section 3.3).
The proof of Theorem 1.3 is given in Section 3, and is based on an analysis of the semigroup associated with the environment process, i.e., the environment as seen relative to the random walk. The generator of this process turns out to be a sum of a large part and a small part, which allows for a perturbation argument. In Appendix A we show that M < ε implies cone-mixing for spin-flip systems, i.e., systems for which c(A, η) = 0 when |A| ≥ 2.
It follows from Theorem 1.3 that for α − β small enough the global speed v changes sign at ρ = 1 2 : We will see in Section 3.3 that c 2 = 0 when µ is a reversible equilibrium, in which case the error In Appendix B we consider an independent spin-flip dynamics such that 0 changes to 1 at rate γ and 1 changes to 0 at rate δ, where 0 < γ, δ < ∞. By reversibility, c 2 = 0. We show that c 3 → 0 as γ + δ → ∞ for fixed ρ = 1 2 and fixed α + β. (1.19) If ρ = 1 2 , then the dynamics is invariant under swapping the states 0 and 1, so that v = 0. If ρ > 1 2 , then v > 0 for α − β > 0 small enough, but v is smaller in the random environment than in the average environment, for which v = (2ρ − 1)(α − β) ("slow-down phenomenon"). In the limit γ + δ → ∞ the walk sees the average environment.

Extensions
Both Theorem 1.2 and 1.3 are easily extended to higher dimensions (with the obvious generalization of cone-mixing), and to random walks whose step rates are local functions of the environment, i.e., in (1.7) replace ξ t (x) by R(τ x ξ t ), with τ x the shift over x and R any cylinder function on Ω. It is even possible to allow for steps with a finite range. All that is needed is that the total jump rate is independent of the random environment. The reader is invited to take a look at the proofs in Sections 2 and 3 to see why. In particular, in the context of Theorem 1.
Moreover, as shown in [1], the LLN in Theorem 1.2 can be extended to an annealed Central Limit Theorem (CLT) under a stronger mixing assumption on the environment (see Chapter 3 therein). In the perturbative regime of Theorem 1.3, a CLT follows easily without further assumptions, by a martingale approximation argument (see [1], Section 3.3).

Proof of Theorem 1.2
In this section we prove Theorem 1.2 by adapting the proof of the LLN for random walks in static random environments developed by Comets and Zeitouni [8]. The proof proceeds in seven steps. In Section 2.1 we look at a discrete-time random walk X on Z in a dynamic random environment and show that it is equivalent to a discrete-time random walk Y on the half-plane in a static random environment that is directed in the vertical direction. In Section 2.2 we show that Y in turn is equivalent to a discrete-time random walk Z on H that suffers time lapses, i.e., random times intervals during which it does not observe the random environment and does not move in the horizontal direction. Because of the cone-mixing property of the random environment, these time lapses have the effect of wiping out the memory. In Section 2.3 we introduce regeneration times at which, roughly speaking, the future of Z becomes independent of its past. Because Z is directed, these regeneration times are stopping times. In Section 2.4 we derive a bound on the moments of the gaps between the regeneration times. In Section 2.5 we recall a basic coupling property for sequences of random variables that are weakly dependent. In Section 2.6, we collect the various ingredients and prove the LLN for Z, which will immediately imply the LLN for X . In Section 2.7, finally, we show how the LLN for X can be extended from discrete time to continuous time.
The main ideas in the proof all come from [8]. In fact, by exploiting the directedness we are able to simplify the argument in [8] considerably.

Space-time embedding
Conditional on ξ, we define a discrete-time random walk on Z with transition probabilities and P ξ 0 denotes the law of X starting from X 0 = 0 conditional on ξ. This is the discrete-time version of the random walk defined in (1.6-1.7), with p and q taking over the role of α/(α + β) and β/(α + β). Note that the walk observes the environment at the moment when it jumps. As in Section 1.2, we write P ξ 0 to denote the quenched law of X and P µ,0 to denote the annealed law of X . Our interacting particle system ξ is assumed to start from an equilibrium measure µ such that the path measure P µ is stationary and ergodic under space-time shifts and is cone-mixing. Given a realization of ξ, we observe the values of ξ at integer times n ∈ Z, and introduce a random walk on By construction, Y is the random walk on H that moves inside the cone with tip at (0, 0) and angle 1 4 π, and jumps in the directions either l + or l − , such that We refer to P ξ (0,0) as the quenched law of Y and to as the annealed law of Y . If we manage to prove that there exists a u = (u 1 , then, by (2.6), u 2 = 1, and the LLN for the discrete-time process Y holds with v = u 1 . In Section 2.7 we show how to pass in continuous time to obtain Theorem 1.2.

The quenched and annealed laws of Z defined bȳ
In words, Z becomes Y when the average over ε is taken. The importance of (2.13) is two-fold. First, to prove the LLN for Y in (2.8) it suffices to prove the LLN for Z. Second, Z suffers time lapses during which its transitions are dictated by ε rather than ξ. By the cone-mixing property of ξ, these time lapses will allow ξ to steadily lose memory, which will be a crucial element in the proof of the LLN for Z.

Regeneration times
Fix L ∈ 2N and define the L-vector which means that the stretch of walk Z n , . . . , Z n+L travels in the vertical direction irrespective of ξ.
Define regeneration times Note that these are stopping times w.r.t. the filtration = ( n ) n∈N given by Also note that, by the product structure of W = w ⊗N defined in (2.9), we have τ Recall Definition 1.1 and put Cone-mixing is the property that lim t→∞ Φ(t) = 0 (for all cone angles θ ∈ (0, 1 2 π), in particular, for θ = 1 4 π needed here). Let This sequence of sigma-fields allows us to keep track of the walk, the time lapses and the environment up to each regeneration time. Our main result in the section is the following.
Proof. We give the proof for k = 1. Let A ∈ σ(H N 0 ) be arbitrary, and abbreviate 1 A = 1 {Z∈A} . Let h be any 1 -measurable non-negative random variable. Then, for all x ∈ H and n ∈ N, there exists a random variable h x,n , measurable w.r.t. the sigma-field Let E P µ ⊗W and Cov P µ ⊗W denote expectation and covariance w.r.t. P µ ⊗ W , and write θ n to denote the shift of time over n. Then and (2.26) Combining (2.23) and (2.26), we get (2.28) Therefore, since (2.28) is uniform in B, (2.28) holdsP µ,(0,0) -a.s. when B is replaced by 1 . Moreover, (2.28) holdsP µ,(0,0) -a.s. (with 1 in place of B), simultaneously for all measurable cylinder sets A. Since the total variation norm is defined over cylinder sets, we can take the supremum over A to get the claim for k = 1.
The extension to k ∈ N is straightforward.

Gaps between regeneration times
Recall (2.16) and that r = 1 2 q. Define In this section we prove two lemmas that control the moments of these increments.

Lemma 2.2. For every
Thus, to get the claim it suffices to show that, for a small enough, By (2.9), I is geometrically distributed with parameter r L . Moreover, τ (L) with the sum convergent for 0 < a < (1/r L L) log[1/(1−r L )] and tending to zero as L → ∞ (because r < 1). Hence we can choose a small enough so that (2.34) holds. This Markov chain moves up one unit with probability r, drops to 0 with probability p + r when it is even, and drops to 0 or 1 with probability p, respectively, r when it is odd. Since τ 1 is bounded from below by a sum of independent random variables, each bounded from below by 1, whose number is geometrically distributed with parameter r L−1 . HenceP This proves the claim.

A coupling property for random sequences
In this section we recall a technical lemma that will be needed in Section 2.6. The proof of this lemma is a standard coupling argument (see e.g. Berbee [4], Lemma 2.1).

Lemma 2.4.
Let (U i ) i∈N be a sequence of random variables whose joint probability law P is such that, for some marginal probability law µ, Then there exists a sequence of random variables

LLN for Y
Similarly as in (2.29), define In this section we prove the LLN for these increments and this will imply the LLN in (2.8).
Proof. By Lemma 2.1, we have is Bernoulli distributed with parameter Φ(L), and also a sequence of random variables is a mean-zero martingale w.r.t. the filtration n . By the Burkholder-Davis-Gundy inequality (Williams [23], (14.18)), it follows that, for β = α ∧ 2,  for some constant C 2 < ∞ and L large enough. This implies the existence of the limit lim L→∞ z L /t L , as well as the fact that lim n→∞ Z n /n = uP µ,(0,0) -a.s., which in view of (2.13) is equivalent to the statement in (2.8) with u = (v, 1). Let χ = (χ n ) n∈N 0 denote the jump times of the continuous-time random walk X = (X t ) t≥0 (with χ 0 = 0). Let Q denote the law of χ. The increments of χ are i.i.d. random variables, independent of ξ, whose distribution is exponential with mean 1/(α + β). Define
Proof. Fix θ > arctan(α + β), and put c = c(θ ) = cot θ < 1/(α + β). Recall from (1.10) that C θ t is the cone with angle θ whose tip is at (0, t). For M ∈ N, let C θ t,M be the cone obtained from C θ t by extending the tip to a rectangle with base M , i.e., Because ξ is cone-mixing with angle θ , and

66)
ξ is cone-mixing with angle θ and base M , i.e., (1.11) holds with C θ t replaced by C θ t,M . This is true for every M ∈ N.
Define, for t ≥ 0 and M ∈ N,

(2.73)
Since D n is independent of A, B and P(D n ) ≥ 1 − 2δ, it follows that I ≤ 2δ and I I I ≤ 2δ uniformly in A and B. To bound I I, we use (2.70) to estimate But the r.h.s. is bounded from above by because, for every B ∈ θ cn,M and C ∈ n , where we use that C is independent of A, B .

Remark on the cone-mixing assumption
We could have tried to follow a shorter approach to deriving the strong LLN in Theorem 1.2, avoiding the technicalities of Sections 2.5 and 2.6. Indeed, with the help of the cone-mixing assumption and the auxiliary random process Z introduced in Section 2.2, it seems possible to deduce that the environment process, i.e., the environment as seen relative to the random walk (see Definition 3.1), admits a mixing equilibrium measure µ e . Consequently, a weak law of large numbers, L 2 -convergence, as well as almost-sure convergence with respect to µ e can be inferred. If we could subsequently show that the equilibrium measure µ is absolutely continuous with respect to µ e , then Theorem 1.2 would follow. A similar approach has been successfully used in several papers for static and dynamic environments under somewhat stronger assumptions than cone-mixing (see e.g. [18,10]). In the present generality it is not trivial to show the absolutely continuity of µ e with respect to µ.

Series expansion for M < ε
Throughout this section we assume that the dynamic random environment ξ falls in the regime for which M < ε (recall (1.15)). In Section 3.1 we define the environment process, i.e., the environment as seen relative to the position of the random walk. In Section 3.2 we prove that this environment process has a unique ergodic equilibrium µ e , and we derive a series expansion for µ e in powers of α − β that converges when α − β < 1 2 (ε − M ). In Section 3.3 we use the latter to derive a series expansion for the global speed v of the random walk.

Definition of the environment process
Let X = (X t ) t≥0 be the random walk defined in (1.6-1.7). For x ∈ Z, let τ x denote the shift of space over x.
Definition 3.1. The environment process is the Markov process ζ = (ζ t ) t≥0 with state space Ω given by Equivalently, if ξ has generator L IPS , then ζ has generator L given by where f is an arbitrary cylinder function on Ω and Let S = (S(t)) t≥0 be the semigroup associated with the generator L. Suppose that we manage to prove that ζ is ergodic, i.e., there exists a unique probability measure µ e on Ω such that, for any cylinder function f on Ω, lim where 〈·〉 µ e denotes expectation w.r.t. µ e . Then, picking f = φ 0 with φ 0 (η) = η(0), η ∈ Ω, we have for some ρ ∈ [0, 1], which represents the limiting probability that X is on an occupied site given ).
Next, let N + t and N − t be the number of shifts to the right, respectively, left up to time t in the environment process.
are martingales with stationary and ergodic increments, we have with M t = M + t − M − t a martingle with stationary and ergodic bounded increments. It follows from (3.6-3.7) that lim In Section 3.2 we prove the existence of µ e , and show that it can be expanded in powers of α − β when α − β < 1 2 (ε − M ). In particular, it follows from this expansion (see e.g. (3.40)) that µ e is absolutely continuous with respect to µ. In Section 3.3 we use this expansion to obtain an expansion of ρ.

Unique ergodic equilibrium measure for the environment process
In Section 3.2.1 we prove four lemmas controlling the evolution of ζ. In Section 3.2.2 we use these lemmas to show that ζ has a unique ergodic equilibrium measure µ e that can be expanded in powers of α − β, provided α − β < 1 2 (ε − M ). We need some notation. Let · ∞ be the sup-norm on C(Ω). Let · be the triple norm on Ω defined as follows. For x ∈ Z and a cylinder function f on Ω, let be the maximum variation of f at x, where η x is the configuration obtained from η by flipping the state at site x, and put It is easy to check that, for arbitrary cylinder functions f and g on Ω, where (

3.13)
Then L 0 is the generator of a Markov process that still has µ as an equilibrium, and that satisfies and where S 0 = (S 0 (t)) t≥0 is the semigroup associated with the generator L 0 , c = ε − M , and C < ∞ is a positive constant.
Proof. Note that L SRW and L IPS commute. Therefore, for an arbitrary cylinder function f on Ω, we have where the first inequality uses that e t L SRW is a contraction semigroup, and the second inequality follows from the fact that ξ falls in the regime M < ε (see Liggett [15], Theorem I.3.9). The inequality in (3.15) follows by a similar argument. Indeed, where the last inequality again uses that ξ falls in the regime M < ε (see Liggett [15], Theorem I.4.1). The fact that µ is an equilibrium measure is trivial, since L SRW only acts on η by shifting it.
Note that L SRW is the generator of simple random walk on Z jumping at rate α + β. We view L 0 as the generator of an unperturbed Markov process and L * as a perturbation of L 0 . The following lemma gives us control of the latter.

Lemma 3.3.
For any cylinder function f on Ω, and Proof. To prove (3.18), estimate (3.20) To prove (3.19), recall (3.13) and estimate where the second inequality uses (3.11) and the third inequality follows from the fact that f ∞ ≤ f for any f such that 〈 f 〉 µ = 0.
We are now ready to expand the semigroup S of ζ. Henceforth abbreviate Lemma 3.4. Let S 0 = (S 0 (t)) t≥0 be the semigroup associated with the generator L 0 defined in (3.13).
Then, for any t ≥ 0 and any cylinder function f on Ω, Moreover, for all n ∈ N, and
Proof. Since L = L 0 + L * , Dyson's formula gives which, in terms of semigroups, reads The expansion in (3.23-3.24) follows from (3.28) by induction on n.
We next prove (3.26) by induction on n. For n = 1 the claim is immediate. Indeed, by Lemma 3.2 we have the exponential bound Suppose that the statement in (3.26) is true up to n. Then where the third inequality uses (3.19), and the fourth inequality relies on the induction hypothesis.
Using (3.26), we can now prove (3.25). Estimate where the first inequality uses that S 0 (t) is a contraction semigroup, while the second and fourth inequality rely on (3.18) and (3.26).
We next show that the functions in (3.23) are uniformly close to their average value.
Proof. Note that h n (t, f ) = g n (t, f ) for t ≥ 0 and n ∈ N, and estimate where the first inequality uses (3.15), while the second and third inequality rely on (3.19) and (3.26).

Expansion of the equilibrium measure of the environment process
We are finally ready to state the main result of this section.
Theorem 3.6. For α − β < 1 2 c, the environment process ζ has a unique invariant measure µ e . In particular, for any cylinder function f on Ω, (3.35) Proof. By Lemma 3.5, we have (3.36) Since α − β < 1 2 c, we see that the r.h.s. of (3.36) tends to zero as t → ∞. Consequently, the l.h.s. tends to zero uniformly in η, and this is sufficient to conclude that the set of equilibrium measures of the environment process is a singleton, i.e., = {µ e }. Indeed, suppose that there are two equilibrium measures ν, ν ∈ . Then Since the l.h.s. of (3.37) does not depend on t, and the r.h.s. tends to zero as t → ∞, we have ν = ν = µ e . Next, µ e is the unique ergodic measure, meaning that the environment process converges to µ e as t → ∞ no matter what its starting distribution is. Indeed, for any µ , and therefore where the last equality is justified by the bound in (3.25) in combination with the dominated convergence theorem.
We close this section by giving a more transparent description of µ e , more suitable for explicit computation.
where the second equality uses that µ is invariant w.r.t. S 0 , while the fifth equality uses the linearity and continuity of L * in combination with the bound in (3.25).
For general n, the argument runs as follows. First write Iteration shows that the latter expression is equal to (3.46)

Expansion of the global speed
As we argued in (3.8), the global speed of X is given by with ρ = 〈φ 0 〉 µ e . By using Theorem 3.7, we can now expand ρ.
First, if 〈φ 0 〉 µ = ρ is the particle density, then where Ψ n is constructed recursively via (3.41) with f = φ 0 . We have where d n = d n (α + β; P µ ), and the factor (α − β) n−1 comes from the fact that the operator L * is applied n − 1 times to compute Ψ n , as is seen from (3.41). Recall that, in (3.13), L SRW carries the prefactor α + β, while L * carries the prefactor α − β. Combining For n = 2, 3 we have It is possible to compute c 2 and c 3 for appropriate choices of ξ.
If the law of ξ is invariant under reflection w.r.t. the origin, then ξ has the same distribution as ξ defined by ξ (x) = ξ(−x), x ∈ Z. In that case c 2 = 0, and . For examples of interacting particle systems with M < ε, see Liggett [15], Section I.4. Some of these examples have the reflection symmetry property.
An alternative formula for c 2 is (recall (3.13)) is the space-time correlation function of the interacting particle system (with generator L IPS ), and E SRW,i is the expectation over simple random walk Y = (Y t ) t≥0 jumping at rate α+β (with generator L SRW ) starting from i. If µ is a reversible equilibrium, then (recall (1.3)) implying that c 2 = 0.
In Appendix B we compute c 3 for the independent spin-flip dynamics, for which c 2 = 0.

A.1 Spin-flip systems in the regime M < ε
Let ξ be a spin-flip system for which M < ε. We recall that in a spin-flip system only one coordinate changes in a single transition. The rate to flip the spin at site x ∈ Z in configuration η ∈ Ω is c(x, η).
As shown in Steif [20] and in Maes and Shlosman [16], two copies ξ, ξ of the spin-flip system starting from configurations η, η can be coupled such that, uniformly in t and η, η , where P η,η is the Vasershtein coupling (or basic coupling), and Γ is the matrix Γ = (γ(u, v)) u,v∈Z with elements Recall (1.15) to see that Γ is a bounded operator on 1 (Z) with norm M (see also Liggett [15], Section I.3). Define Recall Definition 1.1, fix θ ∈ (0, 1 2 π) and put c = c(θ ) = cot θ . For B ∈ θ t , estimate Since this estimate is uniform in B and η, η , it follows that for the cone-mixing property to hold it suffices that Note that if the supremum in (A.3) is attained at the same pair of starting configurations η, η for all t ≥ 0, then (A.5) amounts to the condition that the average coupling time at the origin for this pair is finite.
In view of the remark made at the end of Section 1.1, we note the following. For the Stochastic Ising Model in dimensions d ≥ 2 exponentially fast decay occurs only at high enough temperature (Martinelli [17], Theorem 4.1). The Voter Model in dimensions d ≥ 3 has non-trivial ergodic equilibria, but none of these is cone-mixing. The same is true for the Contact Process in dimensions d ≥ 2.

A.3 Space-time Gibbs measures
We next give an example of a discrete-time dynamic random environment that is cone-mixing but not Markovian. Accordingly, in (1.12) we must replace 0 by −N 0 = {ξ t (x): x ∈ Z, t ∈ (−N 0 )}. Let σ = {σ(x, y): (x, y) ∈ Z 2 } be a two-dimensional Gibbsian random field in the Dobrushin regime (see Georgii [11], Section 8.2). We can define a discrete-time dynamic random environment ξ on Ω by putting The cone-mixing condition for ξ follows from the mixing condition of σ in the Dobrushin regime.
In particular, the decay of the mixing function Φ in (2.18) is like the decay of the Dobrushin matrix, which can be polynomial.

B Independent spin-flips
Let ξ be the Markov process with generator L ISF given by i.e., 0's flip to 1's at rate γ and 1's flip to 0's at rate δ, independently of each other. Such a ξ is an example of a dynamics with M < ε, for which Theorem 3.7 holds. From the expansion of the global speed in (3.50) we see that c 2 = 0, because the dynamics is invariant under reflection in the origin. We explain the main ingredients that are needed to compute c 3 in (1.18).
We therefore see from (3.51) that we must compute expressions of the form where η is a typical configuration of the environment process ζ = (ζ t ) t≥0 = (τ X t ξ t ) t≥0 (recall with V = γ + δ, and p t (0, y) is the transition kernel of simple random walk on Z jumping at rate 1. Therefore, by (B.5-B.6), we have with G V ( y) = The expression between square brackets can be worked out, because where the latter is derived by using that ∂ ∂ t p U t (0, 0) = 1 2 U p U t (0, 1) + p U t (0, −1) − 2p U t (0, 0) (B.12) and p U t (0, 1) = p U t (0, −1). This leads to (1.18).