SHARP THRESHOLD FOR THE BALLISTICITY OF THE RANDOM WALK ON THE EXCLUSION PROCESS

We present our main results with minimal formalism in the model case where the environment is the (simple) symmetric exclusion process, abbreviated (S)SEP in the sequel, and refer to Sections 2 and 6 for full definitions. We will in fact prove more general versions of these results, see Theorem 3.1, which hold under rather broad assumptions on the environment, satisfied for instance by the SEP. Another environment of interest that fulfils these conditions is discussed in Appendix B, and is built using a Poisson system of particles performing independent simple random walks (PCRW, short for Poisson Cloud of Random Walks).

The SEP is the continuous-time Markov process $\eta=(\eta_{t})_{t\geq 0}$ taking values in $\{0,1\}^{\mathbb{Z}}$ and describing the simultaneous evolution of continuous-time simple random walks subject to the exclusion rule. That is, for a given realization of $\eta_{0}$, one puts a particle at those sites $x$ such that $\eta_{0}(x)=1$. Each particle then attempts to move at a given rate $\nu>0$, independently of its previous moves and of the other particles, to one of its two neighbours chosen with equal probability. The move happens if and only if the target site is currently unoccupied; see Section 6.1 for precise definitions. For the purposes of this article, one could simply set $\nu=1$. We have kept the dependence on $\nu$ explicit in anticipation of future applications, for which one may wish to slow down/speed up the environment over time.

On top of the SEP $\eta$, a random walk $X=(X_{n})_{n\geq 0}$ starting from $X_{0}=0$ moves randomly as follows. Fixing two parameters $p_{\bullet},p_{\circ}\in(0,1)$ such that $p_{\bullet}>p_{\circ}$, the process $X$ moves at integer times to a neighbouring site, the right site being chosen with probability $p_{\bullet}$, resp. $p_{\circ}$, depending on whether $X$ is currently located on an occupied or empty site. Formally, given a realization $\eta$ of the SEP, and for all $n\geq 1$,

(notice that the transition probabilities depend on $\eta$ since $p_{\circ}\neq p_{\bullet}$ by assumption). For $\rho\in(0,1)$ we denote by $\mathbb{P}^{\rho}$ the annealed (i.e. averaged over all sources of randomness; see Section 2.3 for details) law of $(\eta,X)$ whereby $\eta_{0}$ is sampled according to a product Bernoulli measure $((1-\rho)\delta_{0}+\rho\delta_{1})^{\otimes\mathbb{Z}}$, with $\delta_{x}$ a Dirac measure at $x$, which is an invariant measure for the SEP, see Lemma 6.1. Incidentally, one can check that except for the trivial cases $\rho\in\{0,1\}$, none of the invariant measures for the SEP (cf. [51, Chap. III, Cor. 1.11]) is invariant for the environment viewed from the walker (to see this, one can for instance consider the probability that the origin and one of its neighbours are both occupied). In writing $\mathbb{P}^{\rho}$, we leave the dependence on $\nu,p_{\bullet}$ and $p_{\circ}$ implicit, which are regarded as fixed, and we will focus on the dependence of quantities on $\rho$, the main parameter of interest. For a mental picture, the reader is invited to think of $X$ as evolving in a half-plane, with space being horizontal and time running upwards. All figures below will follow this convention.

We now describe our main results, using a language that will make analogies to critical phenomena apparent. Our first main result is simplest to formulate in terms of the fast-tracking probability, defined for $n\geq 0$ and $\rho\in(0,1)$ as

where $H_{k}=\inf\{n\geq 0:X_{n}=k\}$. In words $\theta_{n}(\rho)$ is the probability that $X$ visits $n$ before $-1$. Analogous results can be proved for a corresponding back-tracking probability, involving the event $\{H_{-n}<H_{1}\}$ instead; see the end of Section 1.1 for more on this. The events in (1.2) being decreasing in $n$, the following limit

is well-defined and constitutes an order parameter (in the parlance of statistical physics) for the model. Indeed, it is not difficult to see from (1.1) and monotonicity properties of the environment (see (P.3) in Section 2.2 and Lemma 2.2), that the functions $\theta_{n}(\cdot)$, $n\geq 0$, and hence $\theta(\cdot)$ are non-decreasing, i.e. that $\theta(\rho)\leq\theta(\rho^{\prime})$ whenever $\rho\leq\rho^{\prime}$. One thus naturally associates to the function $\theta(\cdot)$ a corresponding critical threshold

The transition from the subcritical ($\rho<\rho_{c}$) to the supercritical ($\rho>\rho_{c}$) regime corresponds to the onset of a phase where the walk has a positive chance to escape to the right. In fact it will do so at linear speed, as will follow from our second main result. To begin with, in view of (1.4) one naturally aims to quantify the behavior of $\theta_{n}(\rho)$ for large $n$. Our first theorem exhibits a sharp transition for its decay.

The proof of Theorem 1.1 appears at the end of Section 3.2. Since $\theta=\inf_{n}\theta_{n}$, item (ii) is an immediate consequence of (1.4). The crux of Theorem 1.1 is therefore to exhibit the rapid decay of item (i) in the full subcritical regime, and not just perturbatively in $\rho\ll 1$, which is the status quo, cf. Section 1.2. In the context of critical phenomena, this is the famed question of (subcritical) sharpness, see for instance [53, 1, 36, 76] for sample results of this kind in the context of Ising and Bernoulli percolation models. Plausibly, the true order of decay in item (i) is in fact exponential in $n$ (for instance, a more intricate renormalisation following [40] should already provide a stretched-exponential bound in $n$). We will not delve further into this question in the present article.

Our approach to proving Theorem 1.1 is loosely inspired by the recent sharpness results [31] and [33, 34, 32] concerning percolation of the Gaussian free field and the vacant set of random interlacements, respectively, which both exhibit long-range dependence (somewhat akin to the SEP). Our model is nonetheless very different, and our proof strategy, outlined below in Section 1.3, vastly differs from these works. In particular, it does not rely on differential formulas (in $\rho$), nor does it involve the OSSS inequality or sharp threshold techniques, which have all proved useful in the context of percolation, see e.g. [35, 38, 17, 23]. One similarity with [33], and to some extent also with [76] (in the present context though, stochastic domination results may well be too much to ask for), is our extensive use of couplings. Developing these couplings represents one of the most challenging technical aspects of our work; we return to this in Section 1.3. It is also interesting to note that, as with statistical physics models, the regime $\rho\approx\rho_{c}$ near and at the critical density encompasses a host of very natural (and mostly open) questions that seem difficult to answer and point towards interesting phenomena; see Section 1.2 and Corollary 1.3 for more on this.

Our second main result concerns the asymptotic speed of the random walk, and addresses an open problem of [40] which inspired our work. In [40], the authors prove the existence of a deterministic non-decreasing function $v:(0,1)\to\mathbb{R}$ such that, for all $\rho\in(0,1)\setminus\{\rho_{-},\rho_{+}\}$,

where

leaving open whether $\rho_{+}>\rho_{-}$ or not, and with it the possible existence of an extended (critical) zero-speed regime. Our second main result yields the strict monotonicity of $v(\cdot)$ and provides the answer to this question. Recall that our results hold for any choice of the parameters $0<p_{\circ}<p_{\bullet}<1$ and $\nu>0$.

The proof of Theorem 1.2 is given in Section 3.2 below. It will follow from a more general result, Theorem 3.1, which applies to a class of environments $\eta$ satisfying certain natural conditions. These will be shown to hold when $\eta$ is the SEP.

Let us now briefly relate Theorems 1.1 and 1.2. Loosely speaking, Theorem 1.1 indicates that $v(\rho)>0$ whenever $\rho>\rho_{c}$, whence $\rho_{+}\leq\rho_{c}$ in view of (1.6), which is morally half of Theorem 1.2. One can in fact derive a result akin to Theorem 1.1, but concerning the order parameter $\widehat{\theta}(\rho)=\lim_{n}\mathbb{P}^{\rho}(H_{-n}<H_{1})$ associated to the back-tracking probability instead. Defining $\widehat{\rho}_{c}$ in the same way as (1.4) but with $\widehat{\theta}$ in place of $\theta$, our results imply that $\widehat{\rho}_{c}=\rho_{c}$ and that the transition for the back-tracking probability has similar sharpness features as in Theorem 1.1, but in opposite directions – the sub-critical regime is now for $\rho>\widehat{\rho}_{c}(=\rho_{c})$. Intuitively, this corresponds to the other inequality $\rho_{-}\geq\rho_{c}$, which together with $\rho_{+}\leq\rho_{c}$, yields (1.7).

Finally, Theorem 1.2 implies that $v(\rho)\neq 0$ whenever $\rho\neq\rho_{c}$. Determining whether a law of large number holds at $\rho=\rho_{c}$ holds or not, let alone whether the limiting speed $v(\rho_{c})$ vanishes when $0<\rho_{c}<1$, is in general a difficult question, to which we hope to return elsewhere; we discuss this and related matters in more detail below in Section 1.2.

One noticeable exception occurs in the presence of additional symmetry, as we now explain. In [40] the authors could prove that, at the ‘self-dual’ point $p_{\bullet}=1-p_{\circ}$ (for any given value of $p_{\bullet}\in(1/2,1)$), one has $v(1/2)=0$ and the law of large numbers holds with limit speed $0$, but they could not prove that $v$ was non-zero for $\rho\neq 1/2$, or equivalently that $\rho_{+}=\rho_{-}=1/2$. Notice that the value $p_{\bullet}=1-p_{\circ}$ is special, since in this case (cf. (1.1)) $X$ has the same law under $\mathbb{P}^{\rho}$ as $-X$ under $\mathbb{P}^{1-\rho}$, and in particular $X\stackrel{{\scriptstyle\text{law}}}{{=}}-X$ when $\rho=\frac{1}{2}$. In the result below, which is an easy consequence of Theorem 1.2 and the results of [40], we summarize the situation in the symmetric case. We include the (short) proof here. The following statement is of course reminiscent of a celebrated result of Kesten concerning percolation on the square lattice [45]; see also [23, 17, 58] in related contexts.

We now place the above results in broader context and contrast our findings with existing results. We then discuss a few open questions in relation with Theorems 1.1 and 1.2.

We give here a relatively thorough overview of the proof, intended to help the reader navigate the upcoming sections. For concreteness, we focus on Theorem 1.2 (see below its statement as to how Theorem 1.1 relates to it), and specifically on the equalities (1.7), in the case of SEP (although our results are more general; see Section 3). The assertion (1.7) is somewhat reminiscent of the chain of equalities $\bar{u}=u_{*}=u_{**}$ associated to the phase transition of random interlacements that was recently proved in [34, 32, 33], and draws loose inspiration from the interpolation technique of [2], see also [30]. Similarly to [32] in the context of interlacements, and as is seemingly often the case in situations lacking key structural features (in the present case, a notion of reversibility or more generally self-adjointness, cf. [24, 75]), our methods rely extensively on the use of couplings.

The proof essentially consists of two parts, which we detail individually below. In the first part, we compare our model with a finite-range version, in which we fully re-sample the environment at times multiple of some large integer parameter $L$ (see e.g. [32] for a similar truncation to tame the long-range dependence). One obtains easily that for any $L$ and any density $\rho\in(0,1)$, the random walk in this environment satisfies a strong Law of Large Numbers with some speed $v_{L}(\rho)$ (Lemma 3.2), and we show that in fact,

for some $\delta_{L},\varepsilon_{L}$ that are quantitative and satisfy $\delta_{L},\varepsilon_{L}=o_{L}(1)$ as $L\to\infty$ (see Proposition 3.3).

In the second part, we show that for any fixed $\rho,\rho+\varepsilon\in(0,1)$, we have

for $L$ large enough (Proposition 3.4). Together, (1.10) and (1.11) readily imply that $v(\rho+\varepsilon)>v(\rho)$, and (1.7) follows. Let us now give more details.

First part: from infinite to finite range. For a density $\rho\in(0,1)$ the environment $\eta$ of the range-$L$ model is the SEP starting from $\eta_{0}\sim\text{Ber}(\rho)^{\otimes\mathbb{Z}}$ during the time interval $[0,L)$. Then for every integer $k\geq 1$, at time $kL$ we sample $\eta_{kL}$ again as $\text{Ber}(\rho)^{\otimes\mathbb{Z}}$, independently from the past, and let it evolve as the SEP during the interval $[kL,(k+1)L)$. The random walk $X$ on this environment is still defined formally as in (1.1), and we denote $\mathbb{P}^{\rho,L}$ the associated annealed probability measure. Clearly, the increments $(X_{kL}-X_{(k-1)L})_{k\geq 1}$ are i.i.d. (and bounded), so that by the strong LLN, $X_{n}/n\rightarrow v_{L}(\rho):=\mathbb{E}^{\rho}[X_{L}/L]$, a.s. as $n\rightarrow\infty$.

Now, to relate this to our original ’infinite-range’ ($L=\infty$) model, the main idea is to compare the range-$L$ with the range-$2L$, and more generally the range-$2^{k}L$ with the range-$2^{k+1}L$ model for all $k\geq 0$ via successive couplings. The key is to prove a chain of inequalities of the kind

where the sequences $(\varepsilon_{k,L})_{k\geq 1}$ of and $(\delta_{k,L})_{k\geq 1}$ are increasing, with $\varepsilon_{L}:=\lim_{k\rightarrow\infty}\varepsilon_{k,L}=o_{L}(1)$ and $\delta_{L}:=\lim_{k\rightarrow\infty}\delta_{k,L}=o_{L}(1)$. In other words, we manage to pass from a scale $2^{k}L$ to a larger scale $2^{k+1}L$, at the expense of losing a bit of speed $(\delta_{k+1,L}-\delta_{k,L})$, and using a little sprinkling in the density $(\varepsilon_{k,L}-\varepsilon_{k-1,L})$, with $\delta_{0,L}=\varepsilon_{0,L}=0$. From (1.12) (and a converse inequality proved in a similar way), standard arguments allow us to deduce (1.10). Let us mention that such a renormalization scheme, which trades scaling against sprinkling in one or several parameters, is by now a standard tool in percolation theory, see for instance [73, 67, 30].

We now describe how we couple the range-$L$ and the range-$2L$ models in order to obtain the first inequality, the other couplings being identical up to a scaling factor. We do this in detail in Lemma 4.1, where for technical purposes, we need in fact more refined estimates than only the first moment, but they follow from this same coupling. The coupling works roughly as follows (cf. Fig. 1). We consider two random walks $X^{(1)}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathbb{P}^{\rho,L}$ and $X^{(2)}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathbb{P}^{\rho+\varepsilon_{L},2L}$ coupled via their respective environments $\eta^{(1)}$ and $\eta^{(2)}$ such that we retain good control (in a sense explained below) on the relative positions of their respective particles during the time interval $[0,2L]$, except during a short time interval after time $L$ after renewing the particles of $\eta^{(1)}$. In some sense, we want to dominate $\eta^{(1)}$ by $\eta^{(2)}$ ‘as much as possible’, and use it in combination with the following monotonicity property of the walk: since $p_{\bullet}>p_{\circ}$, $X^{(2)}$ cannot be overtaken by $X^{(1)}$ if $\eta^{(2)}$ covers $\eta^{(1)}$ (cf. (1.1); see also Lemma 2.2 for a precise formulation). Roughly speaking, we will use this strategy from time $0$ to $L$, then lose control for short time before recovering a domination and using the same argument but with a lateral shift (in space), as illustrated in Figure 1. The key for recovering a domination in as little time as possible is a property of the following flavour.

Roughly, for $t$ poly-logarithmic in $L$ the right-hand side of (1.13) will be as close to 1 as we need. We do not give precise requirements on the scales $\ell,t,L$ (nor a definition of empirical density), except that they must satisfy some minimal power ratio due to the diffusivity of the particles. We will formalize this statement in Section 3.1, see in particular condition (C.2.2), with exponents and constants that are likely not optimal but sufficient for our purposes. Even if the environment were to mix in super-quadratic (but still polynomial) time, our methods would still apply, up to changing the exponents in our conditions (C.1)-(C.3).

The only statistic we control is the empirical density, i.e. the number of particles over intervals of a given length. The relevant control is formalised in Condition (C.1). In particular, our environment couplings have to hold with a quenched initial data, and previous annealed couplings in the literature (see e.g. [15]) are not sufficient for this purpose (even after applying the usual annealed-to-quenched tricks), essentially due to the difficulty of controlling the environment around the walker in the second part of the proof, as we explain in Remark 5.5; cf. also [64, 4, 65] for related issues in other contexts.

Let us now track the coupling a bit more precisely to witness the quantitative speed loss $\delta_{L}$ incurred by (1.13). We start by sampling $\eta^{(1)}_{0}$ and $\eta^{(2)}_{0}$ such that a.s., $\eta^{(1)}_{0}(x)\leq\eta^{(2)}_{0}(x)$ for all $x\in\mathbb{Z}$, which is possible by stochastic domination. In plain terms, wherever there is a particle of $\eta^{(1)}$, there is one of $\eta^{(2)}$. Then, we can let $\eta^{(1)}$ and $\eta^{(2)}$ evolve during $[0,L]$ such that this domination is deterministically preserved over time (this feature is common to numerous particle systems, including both SSEP and the PCRW). As mentioned above, conditionally on such a realization of the environments, one can then couple $X^{(1)}$ and $X^{(2)}$ such that $X^{(2)}_{s}\geq X^{(1)}_{s}$ for all $s\in[0,L]$. The intuition for this is that ’at worst’, $X^{(1)}$ and $X^{(2)}$ sit on the same spot, where $X^{(2)}$ might see a particle and $X^{(1)}$ an empty site (hence giving a chance for $X^{(2)}$ to jump to the right, and for $X^{(1)}$ to the left), but not the other way around (recall to this effect that $p_{\bullet}>p_{\circ}$).

At time $L$, we have to renew the particles of $\eta^{(1)}$, hence losing track of the domination of $\eta^{(1)}$ by $\eta^{(2)}$. We intend to recover this domination within a time $t:=(\log L)^{100}$ by coupling the particles of $\eta^{(1)}_{L}$ with those of $\eta^{(2)}_{L}$ using (1.13) (at least over a space interval $[-3L,3L]$, that neither $X^{(1)}$ nor $X^{(2)}$ can leave during $[0,2L]$). Since during that time, $X^{(1)}$ could at worst drift of $t$ steps to the right (and $X^{(2)}$ make $t$ steps to the left), we couple de facto the shifted configurations $\eta^{(1)}(\cdot)$ with $\eta^{(2)}(\cdot-2t)$. Hence, we obtain that $\eta^{(1)}_{L+t}(x)\leq\eta^{(2)}_{L+t}(x-2t)$ for all $x\in[-3L+ct,3L+ct]$ with high probability as given by (1.13). As a result, all the particles of $\eta^{(1)}_{L+t}$ are covered by particles of $\eta^{(2)}_{L+t}$, when observed from the worst-case positions of $X^{(1)}_{L+t}$ and $X^{(2)}_{L+t},$ respectively.

Finally, during $[L+t,2L]$ we proceed similarly as we did during $[0,L]$, coupling $\eta^{(1)}$ with $\eta^{(2)}(\cdot-2t)$ so as to preserve the domination, and two random walks starting respectively from the rightmost possible position for $X^{(1)}_{L+t}$, and the leftmost possible one for $X^{(2)}_{L+t}$. The gap between $X^{(1)}$ and $X^{(2)}$ thus cannot increase, and we get that $X^{(1)}_{2L}\leq X^{(2)}_{2L}+2t$, except on a set of very small probability where (1.13) fails. Dividing by $2L$ and taking expectations, we get

for suitable $\varepsilon>0$. The exponent $100$ is somewhat arbitrary, but importantly, owing to the coupling time of the two processes (and the diffusivity of the SEP particles), this method could not work with a value of $\delta$ smaller than $(\log^{2}L)/L$. In particular, the exponent of the logarithm must be larger than one, and this prevents potentially simpler solutions for the second part.

Second part: quantitative speed increase at finite range. We now sketch the proof of (1.11), which is a quantitative estimate on the monotonicity of $v_{L}$, equivalently stating that for $\rho\in(0,1)$ and $\varepsilon\in(0,1-\rho)$,

with $\delta_{L}$ explicit and supplied by the first part. This is the most difficult part of the proof, as we have to show that a denser environment actually yields a positive gain for the displacement of the random walk, and not just to limit the loss as in the first part. The main issue when adding an extra density $\varepsilon$ of particles is that whenever $X$ is on top of one such particle, and supposedly makes a step to the right instead of one to the left, it could soon after be on top of an empty site that would drive it to the left, and cancel its previous gain.

We design a strategy to get around this and preserve gaps, illustrated in Figure 2 (cf. also Figure 5 for the full picture, which is more involved). When coupling two walks $X^{\rho}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathbb{P}^{\rho}$, $X^{\rho+\varepsilon}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathbb{P}^{\rho+\varepsilon}$ with respective environments $\eta^{\rho},\eta^{\rho+\varepsilon}$, our strategy is to spot times when i) $X^{\rho+\varepsilon}$ sees an extra particle (that we call sprinkler) and jumps to the right, while $X^{\rho}$ sees an empty site and jumps to the left, and ii) this gap is extended with $X^{\rho}$ then drifting to the left and $X^{\rho+\varepsilon}$ drifting to the right, for some time $t$ that must necessarily satisfy $t\ll\log L$ – so that this has a chance of happening often during the time interval $[0,L]$.

Once such a gap is created, on a time interval say $[s_{1},s_{2}]$ with $s_{2}-s_{1}=t$, we attempt to recouple the environments $\eta^{\rho},\eta^{\rho+\varepsilon}$ around the respective positions of their walker, so that the local environment seen from $X^{\rho+\varepsilon}$ again dominates the environment seen from $X^{\rho}$, which then allows us to preserve the gap previously created, up to some time $T$ which is a polynomial in $\log L$. As we will explain shortly, this gap arises with not too small a probability, so that repeating the same procedure between times $iT$ and $(i+1)T$ (for $i\leq L/T-1$) will provide us with the discrepancy $3\delta_{L}$ needed.

The re-coupling of the environments mentioned in the previous paragraph must happen in time less than $O(\log L)$ (in fact less than $t$), in order to preserve the gap previously created. This leads to two difficulties:

• •

  in such a short time, we cannot possibly recouple the two environments over a space interval of size comparable to $L$ ($\approx$ the space horizon the walk can explore during $[0,L]$), and

• •

  the coupling will have a probability $\gg L^{-1}$ to fail, hence with high probability, this will actually happen during $[0,L]$! In this case, we lose track of the domination of the environments completely, hence it could even be possible that $X^{\rho}$ overtakes $X^{\rho+\varepsilon}$.

We handle the first of these difficulties with a two-step surgery coupling:

1. 1)

   (Small coupling). We perform a first coupling of $\eta^{\rho}$ and $\eta^{\rho+\varepsilon}$ on an interval of stretched exponential width (think $\exp(t^{1/2})$ for instance) during a time $s_{3}-s_{2}=t/2$ (cf. the orange region in Fig. 2), so as to preserve at least half of the gap created. We call this coupling “small” because $t$ is small (compared to $L$). If that coupling is successful, $X^{\rho+\varepsilon}$ now sees an environment that strictly covers the one seen by $X^{\rho}$, on a spatial interval $I$ of width $\asymp\exp(t^{1/2})\gg t$. Due to the length of $I$, this domination extends in time, on an interval $[s_{3},T]$ of duration say $t^{200}$ (it could even be extended to timescales $\asymp\exp(t^{1/2})$): it could only be broken by SEP particles of $\eta^{\rho}_{s_{3}}$ lying outside of $I$, who travel all the distance to meet the two walkers, but there is a large deviation control on the drift of SEP particles. The space-time zone in which the environment as seen from $X^{\rho+\varepsilon}$ dominates that from $X^{\rho}$ is the green trapezoid depicted in Figure 2.

2. 2)

   (Surgery coupling). When the coupling in 1) is successful, we use the time interval $[s_{3},T]$ to recouple the environments $\eta^{\rho}$ and $\eta^{\rho+\varepsilon}$ on the two outer sides of the trapezoid, on a width of length $10L$ say, so that $\eta^{\rho+\varepsilon}_{T}(\cdot+t/2)$ dominates $\eta^{\rho}_{T}(\cdot-t/2)$ on the entire width, all the while preserving the fact that $\eta^{\rho+\varepsilon}(\cdot+t/2)$ dominates $\eta^{\rho}(\cdot-t/2)$ inside the trapezoid at all times (for simplicity and by monotonicity, we can suppose assume that $X^{\rho+\varepsilon}_{s_{3}}-X^{\rho}_{s_{3}}$ is exactly equal to $t$, as pictured in Figure 2). This step is quite technical, and formulated precisely as Condition (C.2) in Section 3.1. It requires that different couplings can be performed on disjoint contiguous intervals and glued in a ‘coherent’ fashion, whence the name surgery coupling. Again, we will prove this specifically for the SEP in a dedicated section, see Lemma 6.9. Since this second coupling operates on a much longer time interval, we can now ensure its success with overwhelming probability, say $1-O(L^{-100})$.

We still need to address the difficulty described in the second bullet point above, which corresponds to the situation where the small coupling described at 1) fails. As argued, will happen a number of times within [0,L] because $t$ is small. If this occurs, we lose control of the domination of the environments seen by the two walkers. This leads to:

1. 3)

   (Parachute coupling). If the coupling in 1) is unsuccessful, instead of the surgery coupling, we perform a (parachute) coupling of $\eta^{\rho}$ and $\eta^{\rho+\varepsilon}$ in $[s_{3},T]$ so that $\eta^{\rho+\varepsilon}(\cdot-(T-s_{3}))$ covers $\eta^{\rho}(\cdot+(T-s_{3}))$, as we did in the first part of the proof. This again happens with extremely good probability $1-O(L^{-100})$. The price to pay is that on this event, the two walks may have drifted linearly in the wrong direction during $[s_{3},T]$ (see the dashed blue trajectories in Figure 2), but we ensure at least that $X^{\rho+\varepsilon}_{T}-X^{\rho}_{T}\geq-2(T-s_{3})$, and we have now re-coupled the environments seen from the walkers, hence we are ready to start a new such step during the interval $[T,2T]$, etc. This last point is crucial, as we need to iterate to ensure an expected gain $\mathbb{E}^{\rho+\varepsilon}[X_{L}]-\mathbb{E}^{\rho}[X_{L}]$ that is large enough, cf. (1.15).

We now sketch a back-of-the-envelope calculation to argue that this scheme indeed generates the necessary discrepancy between $\mathbb{E}^{\rho+\varepsilon}[X_{L}]$ and $\mathbb{E}^{\rho}[X_{L}]$. During an interval of the form $[iT,(i+1)T]$ for $i=0,1,\ldots,\lfloor L/T\rfloor-1$ (we take $i=0$ in the subsequent discussion for simplicity), with probability $\simeq 1-e^{-ct}$, we do not have the first separating event (around time $s_{1}$) between $X^{\rho+\varepsilon}$ and $X^{\rho}$, and we simply end up with $X^{\rho+\varepsilon}_{T}-X^{\rho}_{T}\geq 0$, which is a nonnegative expected gain.

Else, with probability $\simeq e^{-ct}$ we have the first separation happening during $[s_{1},s_{2}]$. On this event, the coupling in 1) (and the subsequent impermeability of the green trapezoid) succeeds with probability $\geq 1-e^{-t^{1/100}}$, and we end up with $X^{\rho+\varepsilon}_{T}-X^{\rho}_{T}\geq t$, resulting in an expected gain of order $\simeq t$. If the first coupling fails, and we resort to the parachute coupling, we have a (negative) expected gain $\simeq-2Te^{-t^{1/100}}$. Multiplying by the probability of the first separation, we obtain a net expected gain $\simeq e^{-ct}(t-2Te^{-t^{1/100}})$.

Finally, we evaluate the possibility that the surgery coupling (in step 2) above) or the parachute coupling 3) fails, preventing us to restore the domination of $\eta^{\rho}(\cdot+X^{\rho})$ by $\eta^{\rho+\varepsilon}(\cdot+X^{\rho+\varepsilon})$ at time $T$, and we do not control anymore the coupling of the walks. This has probability $O(L^{-100})$, and in the worst case we have deterministically $X^{\rho+\varepsilon}_{L}-X^{\rho}_{L}=-2L$, so the expected loss is $O(L^{-99})$.

Putting it all together, we get, for $L$ large enough by choosing e.g. $t=\sqrt{\log L}$ and $T=(\log L)^{100}$ (it turns out that our couplings work with this choice of scales), that

As long as the surgery coupling or the parachute succeed, we can repeat this coupling for up to $\simeq L/T$ times during $[0,L]$, thus obtaining

which establishes (1.15), with a sizeable margin.

In Section 2, we give rigorous definitions of general environments with a minimal set of properties (P.1)-(2.2), and of the random walk $X$. In Section 3, we impose the mild coupling conditions (C.1)-(C.3) on the environments, state our general result, Theorem 3.1, and deduce Theorems 1.1 and 1.2 from it. We then define the finite-range models and give a skeleton of the proof. In Section 4, we proceed to the first part of the proof, and in Section 5, we proceed to the second part (cf. Section 1.3 reagrding the two parts). In Section 6, we define rigorously the SEP and show that it satisfies all the conditions mentioned above. The (short) Appendix A contains a few tail estimates in use throughout this article. In Appendix B, we introduce the PCRW environment and show that it equally satisfies the above conditions, which entails that our results also apply to this environment. Throughout this article all quantities may implicitly depend on the two parameters $p_{\bullet},p_{\circ}\in(0,1)$ that are assumed to satisfy $p_{\bullet}>p_{\circ}$, cf. above (1.1) (and also Section 2.3).

We write $\mathbb{Z}_{+}$, resp. $\mathbb{R}_{+}$, for the set of nonnegative integers, resp. real numbers. We use the letter $z\in\mathbb{R}\times\mathbb{R}_{+}$ exclusively to denote space-time points $z=(x,t)$, and typically $x,y,x^{\prime},y^{\prime}\dots$ for spatial coordinates and $s,t,s^{\prime},t^{\prime},\dots$ for time coordinates. We usually use $m,n,\dots$ for non-negative integers. With a slight abuse of notation, for two integers $a\leq b$, we will denote by $[a,b]$ the set of integers $\{a,\ldots,b\}$ and declare that the length of $[a,b]$ is $|\{a,\ldots,b\}|=b-a+1$. Throughout, $c,c^{\prime},\dots$ and $C,C^{\prime},\dots$ denote generic constants in $(0,\infty)$ which are purely numerical and can change from place to place. Numbered constants are fixed upon first appearance.

We will consider stationary Markov (jump) processes with values in $\Sigma=(\mathbb{Z}_{+})^{\mathbb{Z}}$. The state space $\Sigma$ carries a natural partial order: for two configurations $\eta,\eta^{\prime}\in\Sigma$, we write $\eta\preccurlyeq\eta^{\prime}$ (or $\eta^{\prime}\succcurlyeq\eta$) if, for all $x\in\mathbb{Z}$, $\eta(x)\leq\eta^{\prime}(x)$. More generally, for $I\subset\mathbb{Z}$, $\eta|_{I}\preccurlyeq\eta^{\prime}|_{I}$ (or $\eta^{\prime}|_{I}\succcurlyeq\eta|_{I}$) means that $\eta(x)\leq\eta^{\prime}(x)$ for all $x\in I$. For a finite subset $I\subset\mathbb{Z}$, we use the notation $\eta(I)=\sum_{x\in I}\eta(x)$. We will denote $\geq_{\text{st.}}$ and $\leq_{\text{st.}}$ the usual stochastic dominations for probability measures. Let $J$ be a (fixed) non-empty open interval of $\mathbb{R}^{+}$. An environment is specified in terms of two families of probability measures $(\mathbf{P}^{\eta_{0}}:\eta_{0}\in\Sigma)$ governing the process $(\eta_{t})_{t\geq 0}$ and $({\mu_{\rho}}:\rho\in J)$, where $\mu_{\rho}$ is a measure on $\Sigma$, which are required satisfy the following conditions: