Refined Universality for Critical KCM: Upper Bounds

We study a general class of interacting particle systems called kinetically constrained models (KCM) in two dimensions. They are tightly linked to the monotone cellular automata called bootstrap percolation. Among the three classes of such models (Bollobás et al. in Combin Probab Comput 24(4):687–722, 2015), the critical ones are the most studied. Together with the companion paper by Marêché and the author (Hartarsky and Marêché in Combin Probab Comput 31(5):879–906, 2022), our work determines the logarithm of the infection time up to a constant factor for all critical KCM. This was previously known only up to logarithmic corrections (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). We establish that on this level of precision critical KCM have to be classified into seven categories. This refines the two classes present in bootstrap percolation (Bollobás et al. in Proc Lond Math Soc (3) 126(2):620–703, 2023) and the two in previous rougher results (Hartarsky et al. in Probab Theory Relat Fields 178(1):289–326, 2020, Ann Probab 49(5):2141–2174, 2021, Martinelli et al. in Commun Math Phys 369(2):761–809, 2019). In the present work we establish the upper bounds for the novel five categories and thus complete the universality program for equilibrium critical KCM. Our main innovations are the identification of the dominant relaxation mechanisms and a more sophisticated and robust version of techniques recently developed for the study of the Fredrickson-Andersen 2-spin facilitated model (Hartarsky et al. in Probab Theory Relat Fields 185(3):993–1037, 2023).


Introduction
Kinetically constrained models (KCM) are interacting particle systems featuring challenging obstacles, preventing the use of conventional mathematical tools in the field, including non-ergodicity, non-attractivity, hard constraints, cooperative dynamics and dramatically diverging time scales. They originated in physics in the 80s [8,9] as toy models for the liquid-glass transition, which is still a hot and largely open topic for physicists [2]. The idea behind them is that one can induce glassy behaviour without the intervention of static interactions, disordered or not, but rather with simple kinetic constraints. The latter translate the phenomenological observation that at high density particles in a super-cooled liquid become trapped by their neighbours and require a scarce bit of empty space in order to move at all. We direct the reader interested in the motivations of these models and their position in the landscape of glass transition theories to [2,10,31].
Bootstrap percolation is the natural monotone deterministic counterpart of KCM (see [30] for an overview). Nevertheless, the two subjects arose for different reasons and remained fairly independent until the late 2000s. That is when the very first rigorous results for KCM came to be [6], albeit much less satisfactory than their bootstrap percolation predecessors. The understanding of these two closely related fields did not truly unify until the recent series of works [14-17, 19, 26-28] elucidating the common points, as well as the serious additional difficulties in the non-monotone stochastic setting. It is the goal of this series that is accomplished by the present paper.

Models
Let us introduce the class of U-KCM introduced in [6]. In d ě 1 dimensions an update family is a nonempty finite collection of finite nonempty subsets of Z d zt0u called update rules. The U-KCM is a continuous time Markov chain with state space Ω " t0, 1u Z d . Given a configuration η P Ω, we write η x for the state of x P Z d in η and say that x is infected (in η) if η x " 0. We say that the constraint at x P Z d is satisfied if there exists an update rule U P U such that x`U :" tx`y : y P Uu is fully infected and denote the corresponding indicator by where η A denotes the restriction of η to A Ă Z d , 0 A is the completely infected configuration with A omitted when it is clear from the context as here.
The final parameter of the model is its equilibrium density of infections q P r0, 1s. We denote by µ the product measure such that µpη x " 0q " q for all x P Z d and by Var the corresponding variance. Given a finite set A Ă Z d and real function f : Ω Ñ R, we write µ A pf q for the average µpf pηq|η Z d zA q of f over the variables in A. We write Var A pf q for the corresponding conditional variance, which is thus also a function from Ω Z d zA to R, where Ω B " t0, 1u B for B Ă Z d .
With this notation the U-KCM can be formally defined via its generator Lpf qpηq " ÿ xPZ d c x pηq¨pµ x pf q´f q pηq and its Dirichlet form reads where µ x and Var x are shorthand for µ txu and Var txu . Alternatively, the process can be defined via a graphical representation as follows (see [24] for background). Each site x P Z d is endowed with a standard Poisson process called clock. Whenever the clock at x rings we assess whether its constraint is satisfied by the current configuration. If it is, we update η x to an independent Bernoulli variable with parameter 1´q and call this a legal update. If the constraint is not satisfied, the update is illegal, so we discard it without modifying the configuration. It is then clear that µ is a reversible measure for the process (there are others, e.g. the Dirac measure on the fully non-infected configuration 1).
Our regime of interest is q Ñ 0, corresponding to the low temperature limit relevant for glasses. A quantitative observable, measuring the speed of the dynamics, is the infection time of 0 τ 0 " inf tt ě 0 : η 0 ptq " 0u , where pηptqq tě0 denotes the U-KCM process. More specifically, we are interested in its expectation for the stationary process E µ rτ 0 s, namely the process with random initial condition distributed according to µ. This quantifies the equilibrium properties of the system and is closely related e.g. to the more analytic quantity called relaxation time (i.e. inverse of the spectral gap of the generator) that the reader may be familiar with.
U-bootstrap percolation is essentially the q " 1 case of U-KCM started out of equilibrium, from a product measure with q 0 Ñ 0 density of infections. More conventionally, it is defined as a synchronous cellular automaton, which updates all sites of Z d simultaneously at each discrete time step, by infecting sites whose constraint is satisfied and never removing infections. As the process is monotone, it may alternatively be viewed as a growing subset of the grid generated by its initial condition. We denote by rAs U the set of sites eventually infected by the U-bootstrap percolation process with initial condition A Ă Z d , that is, the sites which can become infected in the U-KCM in finite time starting from ηp0q " p½ xRA q xPZ d . Strictly speaking, other than this notation, bootstrap percolation does not feature in our proofs, but its intuition and techniques are omnipresent. On the other hand, some of our intermediate results can translate directly to recover some bootstrap percolation results of [4,5].

Universality setting
We direct the reader to the companion paper by Marêché and the author [15], as well as a monograph of Toninelli [33] and the author's PhD thesis [21], for comprehensive background on the universality results for two-dimensional KCM and their history. Instead, we provide a minimalist presentation of the notions we need. The definitions in this section were progressively accumulated in [4,5,11,15,19,27] and may differ in phrasing from the originals, but are usually equivalent thereto (see [15] for more details).
Henceforth, we restrict our attention to models in two dimensions. The Euclidean norm and scalar product are denoted by }¨} and x¨,¨y, and distances are w.r.t. }¨}. Let S 1 " tx P R 2 : }x} " 1u be the unit circle consisting of directions, which we occasionally identify with R{2πZ in the standard way.
We denote the open half plane with outer normal u P S 1 and offset l P R by H u plq " x P R 2 : xx, uy ă l ( and omit l if it is 0. We further denote its closure by H u plq, omitting zero offsets. We often refer to continuous sets such as H u , but whenever talking about infections or sites in them, we somewhat abusively identify them with their intersections with Z 2 without further notice. Fix an update family U. Definition 1.1 (Stability). A direction u P S 1 is unstable if there exists U P U such that U Ă H u and stable otherwise.
Clearly, unstable directions form a finite union of finite open intervals in S 1 . We say that a stable direction is semi-isolated (resp. isolated ) if it is the endpoint of a nontrivial (resp. trivial) interval of stable directions. • supercritical if there exists C P C such that all u P C are unstable; • subcritical if every semicircle contains infinitely many stable directions; • critical otherwise.
The following notion measures "how stable" a stable direction is. • 8 if u is stable, but not isolated; • mintn : DZ Ă Z 2 , |Z| " n, |rH u Y Zs U zH u | " 8u otherwise.
The difficulty of U is α " min CPC max uPC αpuq.
We say that a direction u P S 1 is hard if αpuq ą α.
See Fig. 1 for an example update family with α " 2 and its difficulties. It can be shown that a model is critical iff 0 ă α ă 8 and supercritical iff α " 0, so difficulty is tailored for critical models and refines Definition 1.2. More precisely, αpuq P r1, 8q for isolated stable directions. Furthermore, for supercritical models the notions of stable and hard direction coincide. • unrooted if it is not rooted; • unbalanced if there exist two opposite hard directions; • balanced if it is not unbalanced, that is, there exists a closed semicircle containing no hard direction.
We further partition balanced unrooted update families into • semi-directed if there is exactly one hard direction; • isotropic if there are no hard directions.
We further consider the distinction between models with finite and infinite number of stable directions. The latter being necessarily rooted, but possibly balanced or unbalanced, we end up with a partition of all (two-dimensional non-subcritical) families into the seven classes studied in detail below in the critical case. We invite the interested reader to consult [15, Fig. 1 (g) 1, 0, 0 Table 1: Classification of critical U-KCM with difficulty α. For each class E µ rτ 0 s " expˆΘp1q´1 q α¯β´l og 1 q¯γ´l og log 1 q¯δ˙a s q Ñ 0. The label of the class and the exponents β, γ, δ are indicated in that order.
representatives of each class with rules contained in the the lattice axes and reaching distance at most 2 from the origin. Naturally, many more examples have been considered in the literature (also see Fig. 1).
Let us remark that models in each class may have one axial symmetry, but non-subcritical models invariant under rotation by π are necessarily either isotropic or unbalanced unrooted (thus with finite number of stable directions), while invariance by rotation by π{2 implies isotropy.

Results
Our result, summarised in Table 1, together with the companion paper by Marêché and the author [15], is the following complete refined classification of two-dimensional critical KCM (for the classification of supercritical ones, which only features the rooted/unrooted distinction, see [25][26][27]).
Theorem 1 (Universality classes of two-dimensional critical KCM). Let U be a two-dimensional critical update family with difficulty α. We have the following exhaustive alternatives as q Ñ 0 for the expected infection time of the origin under the stationary U-KCM. If U is (a) unbalanced with infinite number of stable directions (so rooted), then 1 E µ rτ 0 s " exp˜Θ`p logp1{qqq 4q 2α¸; (b) balanced with infinite number of stable directions (so rooted), then E µ rτ 0 s " expˆΘ p1q q 2α˙; (c) unbalanced rooted with finite number of stable directions, then α¸; (d) unbalanced unrooted (so with finite number of stable directions), then α¸; (e) balanced rooted with finite number of stable directions, then (f ) semi-directed (so balanced unrooted with finite number of stable directions), then E µ rτ 0 s " expˆΘ plog logp1{qqq q α˙; (g) isotropic (so balanced unrooted with finite number of stable directions), then This theorem is the result of a tremendous amount of effort by a panel of authors. It would be utterly unfair to claim that it is due to the present paper and its companion [15] alone. Indeed, parts of the result (sharp upper or lower bounds for certain classes) were established by (subsets of) Marêché, Martinelli, Morris, Toninelli and the author [16,19,27,28]. Moreover, particularly for the lower bounds, the classification of two-dimensional critical U-bootstrap percolation models by Bollobás, Duminil-Copin, Morris and Smith [4] (featuring only the balanced/unbalanced distinction) are heavily used, while upper bounds additionally use prerequisites from [17,18]. Thus, a fully self-contained proof of Theorem 1 from common probabilistic background is currently contained only in all the above references combined and spans hundreds of pages. Our contribution is but the conclusive step.
More precisely, the lower bound for classes (d) and (g) was deduced from [4] in [28]; the lower bound for class (b) was established in [16], while the remaining four were proved in [15]. Turning to upper bounds, the one for class (a) was given in [27] and the one for class (c) is due to [19]. The remaining five upper bounds are new and those are the subject of our work. The most novel and difficult ones concern classes (e) and (f), the latter remaining quite mysterious prior to our work. Indeed, [19,Conjecture 6.2] predicted the above result with the exception of this class, whose behaviour was unclear. We should note that this conjecture itself rectified previous ones from [27,30], which were disproved by the unexpected result of [19], and was new to physicists, as well as mathematicians.
Remark 1.5. Let us note that for reasons of extremely technical nature, we do not provide a full proof of (the upper bound of) Theorem 1(e). More precisely, we prove it as stated for models with rules contained in the axes of the lattice. We also prove a fully general upper bound of expˆO plogp1{qqq log log logp1{qq q α˙.
Furthermore, with very minor modifications (see Remark 5.1), the error factor can be reduced from log log log to log˚, where log˚denotes the number of iterations of the logarithm before the result becomes negative (the inverse of the tower function). Unfortunately, removing this minuscule error term requires further work, which we omit for the sake of concision. Instead, we provide a sketch of how to achieve this at the end of Section 5.1.2.

Organisation
The paper is organised as follows. In Section 2 we begin by outlining all the relevant relaxation mechanisms used by critical KCM, providing detailed intuition for the proofs to come. This section is particularly intended for readers unfamiliar with the subject, as well as physicists, for whom it may be sufficiently convincing on its own. In Section 3 we gather various notation and simple preliminaries. As we explain in detail in Section 2, the proof of Theorem 1 is not performed class by class, but rather scale by scale and mechanism by mechanism. In Section 4 we formally state the two fundamental techniques we use to move from one scale to the next, namely East-extensions and CBSEPextensions, which import and generalise ideas of [17]. They will be used in various combinations throughout the rest of the paper. The proofs of the results about those extensions, including the microscopic dynamics are deferred to Appendix A, since they are quite technical and do not require new ideas. Sections 5 and 6 are the core of our work and establish estimates on the relaxation times of "super good droplets" from microscopic scales up to and past the "critical" scale by means of East-extensions and CBSEPextensions respectively. In Section 7 we recall, adapt and apply mechansims from [17,19], allowing us to go from "post-critical" to infinite scales. It is only at that point that we assemble Theorem 1 class by class from the general tools gathered in the previous sections. Finally, in Appendix B we establish bounds on conditional probabilities, which, although technical and not particularly conceptual, serve a key role in both Sections 5 and 6. They establish general analogues of [17,Appendix A] in an entirely new way and may be of independent interest for bootstrap percolation.
Familiarity with the companion paper [15] or bootstrap percolation [4] is not needed. Inversely, familiarity with [17,19] is strongly recommended for going beyond Section 2 and achieving a complete view of the proof of the upper bounds of Theorem 1. Nevertheless, we will systematically state the implications of intermediate results of those works for our setting in a self-contained fashion.

Mechanisms
In this section we attempt an heuristic explanation of Theorem 1 from the viewpoint of mechanisms, which is mostly related to upper bound proofs. Yet, let us say a few words about the lower bounds. The proof of the lower bounds in the companion paper [15] has the advantage and disadvantage of being unified for all seven classes. This is undeniably practical and spotlights the fact that all scaling behaviours can be viewed through the lens of the same bottleneck (few energetically costly configurations through which the dynamics has to go to infect the origin) on a class-dependent length scale. However, the downside is that it provides little insight on the particularities of each class, which turn out to be quite significant. To prove upper bounds we need a clear vision of an efficient mechanism for infecting the origin in each class. Since we work with the stationary process, efficient means that it should avoid configurations which are too unlikely w.r.t. µ. However, while lower bounds only identify what cannot be avoided, they do not tell us how to avoid everything else, nor indeed how to reach the unavoidable bottleneck.
Instead of outlining the mechanism used by each class, we focus on techniques which are somewhat generic and then apply combinations thereof to each class. In figurative terms, we will develop several computer hardware components (three processors, four RAMs, etc.), give a general scheme of how to compose a generic computer out of generic components and, finally, assemble seven concrete computer configurations, using the appropriate components for each, sometimes changing a single component from a machine to the other. Moreover, within each component type different instances will be strictly comparable, so, at the assembly stage, we might simply choose the best possible component fitting with the requirements of model at hand. The purpose is twofold. Firstly, this enables us to highlight the robust tools developed and refined recently, which correspond to the components and how they are manufactured, as well as give a clean universal proof scheme into which they are plugged. Secondly, on the technical level, the modular structure will allow us to create each component only once and force us to make it as multipurpose as possible. Indeed, as it is clear from Table 2b, proceeding component-wise (row by row) is much easier than model-wise (column by column). We hope that the reader will be able to navigate through this more efficient, albeit less straightforward, procedure.
Our different components are called the microscopic, internal, mesoscopic and global dynamics and correspond to progressively increasing length scales on which we are able to relax, given a suitable infection configuration. As the notion of "suitable," which we call super good (SG), depends on the class and lower scale mechanisms used, we will mostly use it as a black box input extended progressively over scales in a recursive fashion. When we say that a convex polygonal region, called droplet and systematically equipped with a SG event (which makes the KCM inside the droplet ergodic), "relaxes," we mean that in a certain "relaxation time" the dynamics restricted to the SG event and to this region "mixes." Formally, this translates to a constrained Poincaré inequality for the conditional measure, but this is unimportant for our discussion.
One should think of droplets as extremely unlikely objects, which are able to move within a (slightly) favourable environment. Indeed, at all stages of our treatment, we need to control the inverse probability of droplets (being SG) and their relaxation times, keeping them as small as feasible. Furthermore, due to their inductive definition, the favourable environment required for their movement should not be too costly. Indeed, that would result in the deterioration of the probability of larger scale droplets, as those incorporate the lower scale environment in their internal structure.

Scales
Microscopic dynamics is about modifying infections at the level of the lattice along the boundary of a droplet, while respecting the KCM con-

straint.
Internal dynamics is about relaxation on scales from the lattice level to the internal scale ℓ i " C 2 logp1{qq{q α , where C is a large constant depending on U. This is the most delicate and novel step. Up to ℓ i we account for the main contribution to the probability of droplets, which then saturates at a certain value ρ D . Thus, it is important to only very occasionally ask for more than α infections to appear close to each other (we call such groups of infections helping sets). This means that up to the internal scale hard directions are practically impenetrable.
Mesoscopic dynamics is about relaxation on scales from ℓ i to the mesoscopic scale ℓ m " 1{q C . As our droplets grow to the mesoscopic scale and past it, it becomes possible to require larger helping sets, which we call W -helping sets. These allow droplets to move also in hard directions of finite difficulty, while nonisolated stable directions are still blocking.
Global dynamics is about relaxation on scales from ℓ m to infinity. The extension to infinity being fairly standard (and not hard), one should rather focus on scales up to the global scale given by ℓ g " expp1{q 3α q, which is notably much larger than all time scales we are aiming for.
Roughly speaking, on each of the last three scales, one should decide how to move a droplet of the lower scale in a domain on the larger scale.
For simplicity, in the remainder of Section 2, we assume that all update rules are contained in the axes of the lattice. This allows considering rectangular droplets (see Section 3.3). We further assume that all directions in the left semicircle have difficulties at most α (under the above assumption only the four axis directions can be isolated or semi-isolated), while the down direction is hard, unless there are no hard directions (isotropic class).

Microscopic dynamics
The microscopic dynamics is the only place where we actually deal with the KCM directly and is the same, regardless of the size of the droplet and the universality class. Roughly speaking, from the outside of the droplet, we may think of it as fully infected, since it is able to relax and, therefore, bring infections where they are needed. Thus, it is as though we are working on a single line of lattice sites, say column, next to an infected region. For an isolated (or semi-isolated) stable direction this induces a supercritical 1dimensional KCM on the column. Hence, provided a few suitable helping sets close to the column, we can apply results on 1-dimensional supercritical KCM to establish that the column can relax in time exppOplogp1{qqq 2 q, as for the East model described in Section 2.3.2. Assuming we know how to relax on the droplet itself, this allows us to relax on a droplet with one column appended. However, applying this procedure recursively line by line is not efficient enough to be useful for extending droplets more significantly.

One-directional extensions
We next explain two fundamental techniques beyond the microscopic dynamics which we use to extend droplets on any scale in a single direction, say, left. Each of them can be viewed as a large scale version of a simple spin model, which we review first.
As mentioned above, our droplets have three aspects: geometry, SG event and relaxation mechanism bounding the relaxation time conditionally on the SG event. An extension takes as input a droplet with all its aspects and produces a larger (wider or taller), extended, one. While extending the geometry and SG event is a matter of definition and the relaxation mechanism is an heuristic image of the dynamics, bounds on the probability of the event and the relaxation time require a proof. This proof reflects the nature of the extension of the geometry and event, itself guided by the intuition of the underlying one-dimensional spin model and enabling the use of the proof technique for its relaxation time. We thus collectively refer to the procedure of extending the geometry, event and relaxation of a droplet as extension.

CBSEP-extension
In the one-dimensional spin version of CBSEP [17,18] we work on tÒ, Óu Z . At rate 1 we resample each couple of neighbouring spins, provided that at least one of them is Ò. Their state is resampled w.r.t. the reference product measure, which will be reversible, conditioned to still have a Ò in at least one of the two sites. In other words, Ò can perform coalescence, branching and symmetric simple exclusion moves, hence the name. The relaxation time of this model on volume V is roughly at most minpV, 1{qq 2 in one dimension and minpV, 1{qq in two and more dimensions, where q is the equilibrium density of Ò [17,18].
For us Ò will represent SG droplets, which we would like to move within a larger volume. However, as we would like them to be able to move possibly by an amount smaller than the size of the droplet, we need to generalise the model a bit. We equip each site of Z with a state space corresponding to the state of a column of the height of our droplet of interest in the original lattice Z 2 . Then the event "there is a SG droplet" may occur on a group of ℓ sites (columns). The long range generalised CBSEP (which is actually a generalisation of what is called generalised CBSEP in [17,18]), which we will call CBSEP by abuse, fixes some range R ą ℓ and resamples groups of R consecutive sites if they contain a SG droplet, preserving this feature. Thus, one move of this process essentially delocalises the droplet within the range.
It is important to note (and this was crucial in [17]) that CBSEP does not have to create a droplet in order to evolve. Indeed, conditionally on having a droplet within a certain domain, its position will be approximately uniform owing to the symmetric construction, so that, as long as it is able to move easily by one line both left and right, its position will quickly mix. It is for this initial step that we rely on the microscopic dynamics and helping sets. However, in order to achieve the displacement by one line we further need to be able to internally shuffle the SG event in an amoeba-like manner, so as to contract most of its internal structure in the direction we are moving to. Then, together with a suitable structure on the additional column granted by the microscopic dynamics, it becomes a droplet shifted by one step.
Below CBSEP-extension (Definition 4.4) refers to the procedure of extending a droplet's geometry, event and relaxation with CBSEP as underlying toy model. Geometry is simply extended both left and right, while the extended SG event requires the presence of the original SG droplet inside the extended one, in addition to helping sets throughout the rest of the extended droplet sufficient to catalyse the motion of the original droplet in both directions. The relaxation of the extended droplet via this mechanism is very swift. Indeed, the time needed to move the droplet is roughly a power of the volume times the inverse rate of the microscopic dynamics, which is itself fast, and the inverse rate of contraction, which is small, as we will discuss later. However, CBSEP-extensions can only be used for sufficiently symmetric update families. That is, the droplet needs to be able to move indifferently in both directions and its position should not be biased in one direction or the other.

East-extension
East [22] is the one-dimensional KCM with U " tt1uu. That is, we are only allowed to resample the left neighbour of an infection. An efficient recursive mechanism for its relaxation is the following [29]. Assume we start with an infection at 0. In order to bring an infection to´2 n`1 , using at most n infections at a time (excluding 0), we first bring one to´2 n´1`1 , using n´1 infections. We then place an infection at´2 n´1 and reverse the procedure to remove all infections except 0 and´2 n´1 . Finally, start over with n´1 infections, viewing´2 n´1 as the new origin, thus reaching´2 n`1 . It is not hard to check that this is as far as one can get with n infections. Thus, a number of infections logarithmic in the desired distance is needed. This is to be contrasted with CBSEP, for which only one infection is ever needed, as it can be moved indefinitely by SEP moves. The relaxation time of East on a segment of length L is q´O plog minpL,1{qqq [1,6,7], where q is the equilibrium density of infections. This corresponds to the cost of n infections when 2 n " minpL, 1{qq is the typical distance to the nearest infection.
Its long range generalised version is defined similarly to the one of CBSEP. The only difference is that now R ą ℓ consecutive columns are resampled together if there is a SG droplet on their extreme right. It is clear that this does not allow moving the droplet, but rather forces us to recreate a new droplet at a shifted position before we can progress. The associated Eastextension (Definition 4.2) of a droplet corresponds to extending its geometry to the left, while the extended SG event requires that the original SG droplet is present in the rightmost position and helping sets are available in the rest of the extended droplet to allow its (long range generalised) East evolution.
The generalised East process goes back to [27], while the long range version is implicitly used in [19]. However, both works used a brutal strategy consisting of creating the new droplet from scratch. Instead, in this work we will have to be much more careful, particularly in view of semi-directed models. Indeed, take ℓ large and R " ℓ`5. Then it is intuitively clear that the presence of the original rightmost droplet overlaps greatly with the occurrence of the shifted SG one we would like to craft. Hence, the idea is to take advantage of this and only pay the conditional probability of the droplet we are creating, given the presence of the original one. This is not as easy as it sounds for several reasons. Firstly, we should make the SG structure soft enough (in contrast with what was done e.g. in [19,27]) so that small shifts do not change it much. Secondly, we need to actually have a quantitative estimate of the conditional probability of a complicated multi-scale event, given its translated version, which necessarily does not quite respect the same multi-scale geometry. To make matters worse, we do not have at our disposal a very sharp estimate of the probability of SG events (contrary to what was the case in [17]), so directly computing the ratio of two rough estimates would yield a very poor bound on the conditional probability. In fact, this problem is also present when contracting an amoeba in the CBSEP-extension-we need to evaluate the probability of a contracted version of the amoeba conditionally on the original amoeba being present.
We deal with these issues in Appendix B. We establish that, as intuition may suggest, to create a droplet shifted by R´ℓ, given the original one, we roughly only need to pay the probability of a droplet on scale R´ℓ rather than ℓ, which provides a substantial gain. Hence, the time necessary for an East-extension of a droplet to relax is essentially the product of the inverse probabilities of a droplet on scales of the form 2 m up to the extension length.

Internal dynamics
The internal dynamics is where most of our work will go. This is not surprising, as the probability of SG events will saturate at its final value ρ D (exppΘp1q{q α q for balanced models and exppΘplog 2 p1{qqq{q α q for unbalanced ones) at the internal scale, as it is the case in bootstrap percolation.

Unbalanced internal dynamics
Let us begin with the simplest case of unbalanced models. If U has unbalanced with infinite number of stable directions (class (a)), droplets in [27] on the internal scale consist of several infected consecutive columns, so that no relaxation is needed (the SG event is a singleton). The columns have size ℓ i , which justifies the value of ρ D .
If U is unbalanced with finite number of stable direction (classes (c) and (d)), droplets on the internal scale are fully infected square frames of thickness Op1q and size ℓ i , which gives a similar value of ρ D . In order to relax inside the frame, one can infect several columns next to the frame (inside it) and move them throughout the area enclosed in the frame with the help of the frame. This can be done similarly to a CBSEP-extension, by infecting the next column and removing the previous one (see [19,Fig. 8]). This was already done in [19] and the time necessary for this relaxation is easily seen to be ρ Op1q D (the cost for creating the infected columns).

CBSEP internal dynamics
If U is isotropic (class (g)), up to the conditioning problems of Appendix B described above, we need only minor adaptations of the strategy of [17] for the paradigmatic isotropic model called FA-2f. Droplets on the internal scale will have an internal structure as obtained by iterating Fig. 5a (see also [17,Fig. 2]). Our droplets will be extended little by little alternating between the horizontal and vertical directions, so that their size is multiplied essentially by a constant at each extension. Thus, roughly logp1{qq extensions are required to reach ℓ i . As isotropic models do not have any hard directions, we can move in all directions and thus the symmetry required for CBSEP-extensions is granted. Hence, this mechanism leads to a very fast relaxation of droplets in time exppq´o p1q q. 2 Remark 2.1. The vigilant reader may have noticed that CBSEP requires an actual symmetry, while for a general isotropic model we only know that there are no hard directions. We circumvent this issue by artificially symmetrising our droplets and events, asking for helping sets in directions which do not need any and asking for the symmetric of the helping set of the opposite direction. Although these are totally useless for the dynamics, they are important to ensure that the positions of droplets are indeed uniform rather than suffering from a drift towards an "easier" non-hard direction.

East internal dynamics
The most challenging case is the balanced non-isotropic one (classes (b), (e) and (f)). Indeed, the hard direction prevents us from using CBSEPextensions. To be precise, for semi-directed models (class (f)) it is possible to perform CBSEP-extensions horizontally, but the gain is insignificant, so we treat all balanced non-isotropic models identically up to the internal scale.
We still extend droplets, starting from a microscopic one, by a constant factor alternating between the horizontal and vertical directions. However, in contrast with the isotropic case, extensions are done in an oriented fashion, so that the original microscopic droplet remains anchored at the corner of larger ones (see Fig. 3b). Thus, we may apply East-extensions on each step and obtain that the cost is given by the product of conditional probabilities from Section 2.3.2 over all scales and shifts of the form 2 n . In total, a droplet of size 2 n needs to be paid once per scale larger than 2 n . A careful computation shows that only droplets larger than q´α provide the dominant contribution and those all have probability essentially ρ D . Thus, the total cost would be ρ Θplog logp1{qqq 2 {q α D , since there are log logp1{qq scales from q´α to ℓ i , as they increase exponentially. This is unfortunately a bit too rough for the semi-directed class. However, the solution is simple-it suffices to introduce scales growing double-exponentially above q´α, so that the product becomes dominated by its last term, giving the final cost ρ Θplog logp1{qq{q α D .

Mesoscopic dynamics
For the mesoscopic dynamics we are given as input a SG event for droplets on scale ℓ i and a bound on their relaxation time.

CBSEP mesoscopic dynamics
If U is unrooted (classes (d), (f) and (g)), recall that the hard directions (if any) are vertical. Then we can perform a horizontal CBSEP-extension directly from ℓ i to ℓ m , since ℓ i " logp1{qq{q α makes it likely for helping sets to appear along all segments of length ℓ i until we reach scale ℓ m " q´C. The resulting droplet is very wide, but short (see Fig. 6). However, this is enough for us to be able to perform a vertical CBSEP-extension, requiring W -helping sets, since they are now likely to be found. Again, CBSEP dynamics being very efficient, its cost is negligible.

East mesoscopic dynamics
If U is rooted (classes (a)-(c) and (e)), CBSEP-extensions are still inaccessible. We may instead East-extend horizontally from ℓ i to ℓ m in a single step. If the model is rooted balanced (classes (b) and (e)), we may proceed similarly in the vertical direction, reaching a droplet of size ℓ m in time ρ Another way of viewing this is simply as extending the procedure used for the internal dynamics all the way up to the mesoscopic scale.

Stair mesoscopic dynamics
For unbalanced families with infinite number of stable directions (class (a)) the following stair mesoscopic dynamics was introduced in [27]. Recall that for unbalanced models the internal droplet is simply a fully infected frame. While moving the droplet left via an East motion, we pick up W -helping sets above or below the droplet. These sets allow us to make all droplets to their left shifted up or down by one row. Hence, we manage to create a copy of the droplet far to its left but also slightly shifted up or down (see [27,Fig. 6]). Repeating this (with many steps in our staircase) in a two-dimensional East-like motion, we can now relax on a mesoscopic droplet with horizontal dimension much larger than ℓ m but still polynomial in 1{q and vertical dimension ℓ m in time ρ Θplogp1{qqq D .

Snail mesoscopic dynamics
For unbalanced rooted models with finitely many stable directions (class (c)) the snail mesoscopic dynamics of [19] is as follows. As in the stair case, it is possible to find W -helping sets above and below the droplet moving in an East way to its left. However, at present it is possible to reverse the East motion, pulling the W -helping set back to the original position of the droplet, thus effectively moving up or down (see [19,Fig. 2]). Note that the vertical directions are hard, so such an indirect approach is necessary. Thus, we can relax on a droplet with sides Θpℓ m q in time ρ Θplogp1{qqq D (see [19, Fig. 4]).

Global dynamics
The global dynamics receives as input a SG event for a droplet on scale ℓ m with probability roughly ρ D and a bound on its relaxation time, as provided by the mesoscopic dynamics. Its goal is to move such a droplet efficiently to the origin from its typical initial position at distance roughly ρ´1 {2 D .

CBSEP global dynamics
If U has a finite number of stable directions (classes (c)-(g)), since the mesoscopic droplet is large enough, it can perform a CBSEP motion in a typical environment. Therefore, the cost of this mechanism is given by the relaxation time of CBSEP on large volumes with density of Ò given by ρ D . Performing this strategy carefully and using the 2-dimensional CBSEP, this yields roughly 1{ρ D . Prefering a 2-dimensional over a 1-dimensional CBSEP strategy is not of particular importance for our result, since we only know log ρ D up to a constant factor. However, this was crucial in [17] in order to determine the sharp asymptotics of log E µ rτ 0 s for the FA-2f model. Moreover, it was announced in [17] that in the present work we would prove similar results for a subclass of isotropic models. Unfortunately, this is not possible, since we have uncovered an error invalidating the result of the bootstrap percolation preprint those were based on. We are, therefore, forced to postpone more general sharp results based on this strategy until the necessary bootstrap percolation prerequisites are enlisted.

East global dynamics
If U has infinite number of stable directions (classes (a) and (b)), the strategy is identical to the CBSEP global dynamics, but employs an East dynamics. Now the cost becomes the relaxation time of an East model with density of infections ρ D , which yields exppOplogp1{ρ D qq 2 q.

Assembling the components
To conclude, in Table 2a we provide a summary of the mechanisms for each scale and their cost to the relaxation time. The results are expressed in terms of the probability of a droplet ρ D , which equals expp´Oplogp1{qqq 2 {q α q for unbalanced models and expp´Op1q{q α q for balanced ones. The final bound The relaxation time cost associated to each choice of dynamics mechanism on each scale in terms of the probability of a droplet ρ D .
The fastest mechanism available to each class of Theorem 1 on each scale. The * indicates a leading contribution for the class (column). on E µ rτ 0 s for each class then corresponds to the product of the costs of the mechanism employed at each scale. To complement this, in Table 2b we indicate the fastest mechanism available for each class on each scale and further indicate which one gives the dominant contribution to the final result appearing in Theorem 1, once the bill is footed. Finally, let us avert the reader that, for the sake of concision, the proof below does not systematically implement the optimal strategy for each class as indicated in Table 2b if that does not deteriorate the final result. Similarly, when that is unimportant, we may give weaker bounds than the ones in Table 2a. When importing results from [19], we tacitly use a global mechanism not listed above, which is a weaker precursor of the CBSEP one.

Correlation inequalities
Let us recall two well-known correlation inequalities due to Harris [13] and van den Berg-Kesten [34]. The Harris inequality will be used throughout and we state some particular formulations that will be useful for us. The BK inequality is not natural to use for an upper bound in our setting and has not been employed to this purpose until now. Nevertheless, it will prove crucial in Appendix B to estimate certain conditional probabilities.
For Section 3.1 we fix a finite Λ Ă Z 2 . We say that an event A Ă Ω Λ is decreasing if adding infections does not destroy its occurrence. We collectively refer to this proposition and the following corollaries as Harris inequality.

Directions
Throughout this work we fix a critical update family U with difficulty α. We call a direction u P S 1 rational if uR X Z 2 ‰ ∅. By the definition of α there exists a semicircle with rational midpoint u 0 such that all directions in the semicircle have difficulty at most α. We may assume without loss of generality that the direction u 0`π {2 is hard unless U is isotropic. It is not difficult to show (see e.g. [5,Lemma 5.3]) that one can find a set S 1 of rational directions such that: • all isolated and semi-isolated stable directions are in S 1 ; We further consider the set p S " S 1`t 0, π{2, π, 3π{2u obtained by making S 1 invariant by rotation by π{2. We will refer to the elements of p S as quasistable directions or simply directions, as they are the only ones of interest to us. We label the elements of p S " pu i q iPr4ks clockwise and consider their indices modulo 4k (we write rns for t0, . . . , n´1u), so that u i`2k "´u i is perpendicular to u i`k . In figures we take p S " π 4 Z and u 0 " p´1, 0q. Further observe that if all U P U are contained in the axes of Z 2 , then p S " π 2 Z. For i P r4ks we introduce ρ i " mintρ ą 0 : Dx P Z 2 , xx, u i y " ρu and λ i " mintλ ą 0 : λu i P Z 2 u, which are both well-defined, as the directions are rational (in fact ρ i λ i " 1, but we will use both notations for transparency).

Droplets
We next define the geometry of the droplets we will use.
for r with positive coordinates (see the black regions in Fig. 2). We say that a droplet is symmetric if it is of the form x`Λprq with 2x P Z 2 and r i " r i`2k for all i P r2ks. For a set of radii r we define the side lengths s " ps i q iPr4ks with s i the length of the side of Λprq with outer normal u i .
Note that if all U P U are contained in the axes of Z 2 , then droplets are simply rectangles with sides parallel to the axes.
We write pe i q iPr4ks for the canonical basis of R 4k and we write 1 " ř iPr4ks e i , so that Λpr1q is a polygon with inscribed circle of radius r and sides perpendicular to p S. It will often be more convenient to parametrise dimensions of droplets differently. For i P r4ks we set This way Λpr`v i q is obtained from Λprq by extending the two sides parallel to u i by 1 in direction u i and leaving all other sides unchanged. Note that if Λprq is symmetric, then so is Λpr`λ i v i q for i P r4ks.
Definition 3.5 (Tube). Given i P r4ks, r and a multiple l of λ i , we define the tube of length l, direction i and radii r (see the thickened regions in Fig. 2) T pr, l, iq " Λpr`lv i qzΛprq.
We will often need to consider boundary conditions for our events on droplets and tubes. Given two disjoint finite regions A, B Ă Z 2 and two configurations η P Ω A and ω P Ω B , we define η¨ω P Ω AYB as

Scales
Throughout the work we consider the positive integer constants Each one is assumed to be large enough depending on U and, therefore, p S and α (e.g. W ą α), and much larger than the next. These constants are not allowed to depend on q. Whenever asymptotic notation is used, its implicit constants are not allowed to depend on the above ones, but only on U.
The following are our main scales corresponding to the mesoscopic and internal dynamics.

Helping sets
We next introduce various constant-sized sets of infections sufficient to induce growth. As the definitions are quite technical in general, in Fig. 1 we introduce a deliberately complicated example, on which to illustrate them. Let us fix a direction u i P p S with αpu i q ă 8. Let S be a nonempty discrete segment perpendicular to u i . We will assume that S is of the form x P Z 2 : xx, u i y " 0, }x} ď r ( for some r ě W , but all definitions are extended by translation. Definition 3.6 (W -helping set). A W -helping set for S is any set of W consecutive infected sites in S, that is a set of the form x`rW sλ i`k u i`k for some x P S. We denote by H W d pSq the event that there is a W -helping set for S at distance at least d from its endpoints.
The relevance of W -helping sets is that, since W is large enough, together with a suitable neighbourhood of S in H u i they fully infect S by expanding the infected interval one site at a time. We next define some smaller sets which are sufficient to induce such growth but have the annoying feature that they are not necessarily contained in S and do not necessarily induce growth in a simple sequential way like W -helping sets. Let us note that except in Appendix A.1 the reader will not lose anything conceptual by thinking that u i and α-helping sets defined below are simply single infected sites in S.
Tubes traversable in the horizontal and vertical directions respectively. Note that for non-rectangular geometry, i.e. k ą 1, a tube's shape uniquely determines the direction (see Fig. 2). Without the highlighted infection traversability is lost in both directions. Only the left tube is ST (ST " T for u 0 and u 2 ). The right one is T ω in direction u 3 , but not T " T 1 . Boundary conditions are irrelevant in other directions. The C 2`d offset (see Fig. 2) of Definition 4.1 is omitted.
for some integers k j . Moreover, we choose x i so that the period is independent of i and sufficiently large so that for all i P r4ks with αpu i q ď α, the diameter of t0u Y Z i is much smaller than Q. We may choose Q " Op1q and all Z i within distance Op1q from the origin.
In the example of Fig. 1 only the u 3 direction has Z 3 such that rZ 3 YH u 3 s U only contains every second site of the line H u i zH u i . This is indeed necessary, since at least 4 sites are needed to infect the full line. For this model we might take Q " 2, corresponding to the fact that we need two translates of Z 3 with suitable residues modulo 2, in order to infect the entire line. • If αpu i q ď α and αpu i`2k q ą α, then a α-helping set is a u i -helping set.
• If αpu i q ď α and αpu i`2k q ď α, then a α-helping set for S is a set of the form H Y H 1 with H a u i -helping set and´H 1 " t´h : h P H 1 u a u i`2k -helping set.
If αpu i q ă 8, any set which is either a W -helping set or a α-helping set is called helping set. If αpu i q " 8, there are no helping sets.
In the example of Fig. 1 u 0 and u 2 are both of difficulty α " 2, so α-helping sets correspond to a pair of consecutive infections and a pair of infections at distance 2. These two pairs may be distant from each other within S. The consecutive infections are not u 0 -helping sets, but we include them in α-helping sets in order for α-helping sets in direction u 0 to be the symmetric ones of those in direction u 2 . Definition 3.9. Let ω P Ω Z 2 zS be a boundary condition. The event H ω d pSq occurs if S has a helping set such that the vectors by which the sets Z i (and, possibly,´Z i`2k ) are translated in Eq. (2) are contained in S and are at distance at least d from the endpoints of S. It will be convenient, given a domain Λ Ą S and a boundary condition ω P Ω Z 2 zΛ to define H ω d pSq " tη P Ω Λ : η S P H ω¨η ΛzS d pSqu by abuse. We write simply HpSq if d " 0 and the domain Λ is such that the boundary condition ω is irrelevant.
The following observation will be used systematically in probability estimates. It follows easily from the definitions above (see e.g. [4,Lemma 4.2]).

Super good events
Throughout the paper we will refer to SG events for various droplets but we will usually not need to know exactly how they are constructed. However, we will systematically assume that for any sequence of radii r and boundary condition ω it holds that • for x P Z 2 we define SG ω px`Λprqq by translating SG ωp¨´xq pΛprqq by x; • any SG event is nonempty and decreasing both in the configuration and in the boundary condition; • µpSG ω pΛprqqq ď q´O pW q µpSGpΛprqqq, systematically writing SG for SG 1 .

Constrained Poincaré inequalities
Finally, we define (constrained) the Poincaré constants of various regions. Henceforth we will use the shorthand notation µ Λ p¨|SG ω q " µ Λ p¨|SG ω pΛqq and similarly for traversable tubes (see Definition 4.1 below), as well as for conditional variances. Given a finite Λ Ă Z 2 such that SGpΛq is defined, let γpΛq be the smallest constant γ ě 1 such that the inequality holds for all f : Ω Ñ R, where c Λ,ω x pηq " c x pη Λ¨ωZ 2 zΛ q (recall Eq. (1)) and we omit Λ, when c Λ,ω x is inside µ Λ like we do for SG, etc.
Remark 3.11. It is important to take note of the absence of conditioning on SG in the r.h.s. of Eq. (3). This definition follows [19] and differs from the one in [17]. Although this nuance is not important most of the time, this choice is crucial for the proof of Theorem 5.8 below. Unfortunately, this enforces some minor adaptations when importing intermediate results from [17], as dealt with by the following observation.
Typically, in [17] we rather transform terms of the form µ B p½ A µ A pf |Aq|Bq into µ B pf |Bq relying on additional information about the relative structure of A and B, which will not always be available to us in the present work. Instead, in [19] we would simply disregard the numerator in Observation 3.12, which is too rough for our purposes. Therefore, corresponding amendments are needed for adapting arguments from the latter work as well, although the definition Eq. (3) is unchanged.

One-directional extensions
We first need the following traversability T and symmetric traversability ST events, for which we make the same conventions as for SG events (see Section 3.6). The definition is illustrated in Figs. 1 and 2.
Definition 4.1 (Traversability). Fix i P r4ks, r and l multiple of ρ i . Assume that αpu j q ă 8 for all j P pi´k, i`kq. For m ě 0 and j P pi´k, i`kq write S j,m " Λpr`mv i`ρ j e j qzΛpr`mv i q and implicitly always assume that the indices are such that S j,m Ă T pr, l, iq ": T . For ω P Ω Z 2 zΛpr`lv i q we say that T is pω, dq-traversable if for all m and j the event H ω C 2`d pS j,m q occurs. We denote by T ω d pT q the event that the tube is pω, dq-traversable and omit ω if it is 1 and d if it is 0.
We define ST ω d pT q (the tube is pω, dq-symmetrically traversable) identically to T ω d pT q, except that we replace H ω C 2`d pS j,m q by H W d pS j,m q for all j such that maxpαpu j q, αpu j`2k qq ą α. In particular, if no such j exist, ST " T .
We will use two different ways to enlarge droplets to larger scales based on the East and CBSEP-extensions from Definitions 4.2 and 4.4. Both share the following setting.  Let r " q´O pCq , i P r4ks and l P p0, ℓ m`s be a multiple of λ i . Following [17], define d m " λ i tp3{2q m u for m P r1, Mq and M " mintm : λ i p3{2q m ě lu. Let d M " l, Λ m " Λpr`d m v i q and s m´1 " d m´dm´1 for m P r2, Ms.

East-extension
Definition 4.2 (East-extension). Fix i P r4ks, r and l multiple of λ i . Assume that SGpΛprqq is defined 4 and that αpu j q ă 8 for all j P pi´k, i`kq. We say that we East-extend Λprq by l in direction u i (see Fig. 2a) if for all s P p0, ls divisible by λ i and ω P Ω Z 2 zΛpr`sv i q we have η P SG ω pΛpr`sv i qq iff η Λprq P SGpΛprqq, η T pr,s,iq P T ω pT pr, s, iqq.
Recall γ from Section 3.7. The following is proved in Appendix A.3.

Proposition 4.3.
Assume that we East-extend Λprq by l in direction u i . Then

CBSEP-extension
Definition 4.4 (CBSEP-extension). Fix i P r4ks, r and l divisible by λ i . Assume that SGpΛprqq is defined 4 and that U has a finite number of stable directions. We say that we CBSEP-extend Λprq by l in direction u i (see Fig. 2b) if for all s P p0, ls divisible by λ i and ω P Ω Z 2 zΛpr`sv i q we have SG ω pΛpr`sv i qq " Ť x SG ω x pΛpr`sv i qq and for offsets x P r0, ss divisible by λ i we define η P SG ω x pΛpr`sv i qq iff the following all hold: The following is proved in Appendix A.3 based on [17].

East-type dynamics
In this section we treat various East-type dynamics on all scales. This is the most novel and central part of our work, albeit the most technical.

Internal East dynamics
For this section we assume that U is balanced and without loss of generality that we may fix i P p0, 2kq such that αpu j q ď α for all j P p´k, i`kq. Let r p0q " pr p0q j q jPr4ks be a symmetric sequence of radii such that r j " Θp1{εq is a (a) Case case i " 2k´1 of Section 5.1.1. For n P N droplets are symmetric and homothetic to the black Λ p0q . Intermediate ones Λ p1`1{4q , Λ p1`2{4q and Λ p1`3{4q obtained by East-extensions (see Fig. 2a) in directions u 0 , u 1 and u 2 respectively are drawn in progressive shades of grey.
The black, grey and white droplets are Λ p0q , Λ p1q and Λ p2q respectively. In this case no fractional scales are introduced. Figure 3: Geometry of the nested droplets Λ pnq used in Section 5.1 for k " 2.
multiple of λ j for all j, the vertices of Λpr p0q q are in λ i u i Z`λ 0 u 0 Z and the corresponding side lengths s p0q are also Θp1{εq. We define and s pnq k and s pnq i`k as required to be the sides of a droplet, where N c " mintn : W n ě q´αu We denote Λ pnq " Λpr pnq q, where r pnq is the sequence of radii corresponding to s pnq such that r pnq k " r p0q k and r pnq i`k " r p0q i`k (see Fig. 3a).
Remark 5.1. Note that despite the extremely fast divergence of ℓ pnq q α , for n P pN c , N i s it holds that W ď ℓ pn`1q {ℓ pnq ă pℓ pnq q α q 2 ă log 4 p1{qq. The sharp divergence will ensure that some error terms below sum to the largest one, so as to avoid additional factors of the order of N i´N c in the final answer, particularly for the semi-directed class (f). This technique was introduced in [20, Eq. (16)], while the geometrically increasing scale choice relevant for small n originates from [12]. It should be noted that this divergence can be further amplified up to a tower of exponentials of height linear in n´N c . In that case log log logp1{qq error term in Theorem 5.8 becomes log˚p1{qq, but is, alas, still divergent.

Semi-directed models
In Section 5.1.1 we assume that U is semi-directed. In that case we may set i " 2k´1. This is the only feature of semi-directed models used in this section. Hence, the reasoning applies equally well to all balanced models with rules contained in the axes of the lattice, since then k " 1 and we can always set i " 1 for balanced models. Observe that, since i " 2k´1, we may obtain Λ pn`1q from Λ pnq by 2k successive extensions in directions u 0 through u i (see Fig. 3a). We denote the droplets obtained this way by Λ pn`j{p2kqq for j P p0, 2kq and denote their radii and side lengths by r pn`j{p2kqq and s pn`j{p2kqq respectively. We write l pn`j{p2kqq " s pn`1q j`k´s pnq j`k " Θpℓ pn`1q {εq for the length l such that r pn`pj`1q{p2kqq " r pn`j{p2kqq`l v j .
Definition 5.2 (Semi-directed internal SG). Let U be semi-directed. Let SGpΛ p0q q be the event that Λ p0q is fully infected. Recursively, for n P rN i s and j P r2ks, we define SGpΛ pn`pj`1q{p2kqq q by East-extending Λ pn`j{p2kqq in direction u j by l pn`j{p2kqq (recall Definition 4.2). Theorem 5.3. Let U be semi-directed. Then Proof of Theorem 5.3. For n P 1{p2kqN, j P r2ks and m ě 2, such that n ă N i and pn´j{p2kqq P N set For the sake of simplifying expressions we will abusively assume that for all l pnq are of the form λ j tp3{2q m u with integer m. Without this assumption, one would need to treat the term corresponding to m " M´1 in Proposition 4.3 separately, but identically.

From Proposition 4.3 we get
where the product is over n P 1{p2kqN and M pnq " log l pnq { logp3{2q`Op1q. Indeed, by the Harris inequality a m in Eq. (4) for r " r pnq is at most To evaluate the r.h.s. of Eq. (9) we will need the following lemma.
Lemma 5.4. Let n P 1{p2kqN be such that n ď N i and m ě 1. Then Moreover, if the following improvements holds Let us finish the proof of Theorem 5.3 before proving Lemma 5.4. Using the trivial bound a pnq m ď expp1{pε 2 q α qq from Eq. (10) we get which is the main contribution. Note that n ă N c´1 {ε implies M pnq ă logp1{q α q{ logp3{2q, so the above product exhausts the terms in Eq. (9) with large m.
Next, using the first bound on a pnq m from Eq. (10), we obtaiń Finally, we use Eq. (12) to show Plugging the last result and Eqs. (13) to (15) in Eq. (9) and recalling Eq. (10), we conclude the proof of Theorem 5.3.
Proof of Lemma 5.4. Let us fix m and n as in the statement for Eq. (10).
We next turn to proving Eq. (12), so we fix n and m satisfying the corresponding hypotheses of Lemma 5.4. Denote s m " ptp3{2q m`1 u´tp3{2q m uqλ j u j for j " jpnq as in Eq. (8), so that a pnq m " µ´1`SG`Λ pnq`s m˘ˇS G`Λ pnq˘˘.
Our goal is then to bound the last factor, using Corollary B.4, which quantifies the fact that "small perturbations s m do not modify traversability much." Let us fix p as above, denote T " T pr ppq , l ppq , jppqq and T 1 " T`s m . From Eq. (11) it is not hard to check that the hypotheses of Corollary B.4 are satisfied, so that Finally, we can plug Eqs. (10), (20) and (21) in Eq. (19) to obtain the following bounds. If p3{2q m ď q´α`1 {2´op1q , then the terms corresponding to µ´1 Λ pnmq pSGq and to values of p in the intervals rn m , N c s, pN c , N c`∆ s and pN c`∆ , N i q respectively. Indeed, in the last term for small m we used Eq. (20), while for large m, we directly applied Eq. (21). Observing that the product of the second case for the first term, the second term and the third term can be bounded by we obtain the desired Eq. (12).

Balanced rooted models
We now assume that U is balanced rooted (the rooted character is assumed only in order not to override definitions from Sections 5.1.1 and 6.1, but is not needed otherwise). We set i " 1 in this case, which is always valid, since the model is balanced (recall i from the beginning of Section 5.1). We need to define a two-directional East-extension which is morally the concatenation of one in direction u 0 and one in direction u 1 , but whose actual definition is much more technical, so as to respect the homothetic relation between the Λ pnq and yet maintain a product structure.
We begin with some geometric preparations. Fix n P rN i s (since the definitions of Section 5.1.1 no longer apply, but only the ones from the beginning of Section 5.1 do, n is an integer here). Observe that we can cover Λ pn`1q with droplets pD κ q κPrKs so that the following conditions hold (see Fig. 4).
• For all κ P rKs, D κ Ă Λ pn`1q ; • any segment of length ℓ pnq {pCεq perpendicular to u j for some j P p´k, ks intersects at most Op1q of the D κ ; • droplets are assigned a generation g P t0, 1, 2u, so that only D 0 :" Λ pnq is of generation g " 0, only D 1 :" Λpr pnq`l 1 v 1 q is of generation g " 1, where l 1 " r pn`1q k´r pnq k xu 1 , u k y , so that D 1 spans the u k`1 -side of Λ pn`1q ; • if κ ě 2, then D κ is of generation g " 2, and is of the form D κ " y κ u 1`Λ pr pnq`l κ v 0 q for certain l κ ě 0 and y κ P r0, l 1 s multiple of λ 1 .
To construct the D κ of generation 2, it essentially suffices to increment y κ by Θpℓ pnq {εq and define l κ to be the largest possible, so that D κ Ă Λ pn`1q . Finally, we add to our collection of droplets the ones with y κ corresponding to a corner of Λ pn`1q and again take l κ maximal (see Fig. 4).
The droplets D κ corresponding to corners of Λ pn`1q . The generation 0 droplet is given in black, while the one of generation 1 is shaded.
(b) All droplets D κ . In the second generation, for visibility, droplets alternate between shaded, thickened and hatched.
for some family I of subsets of rKs. We say that R is n-traversable (T n pRq occurs) if for all j P p´k, kq and every segment S Ă R perpendicular to u j of length at least δℓ pnq {ε at distance • at least W from the boundary of all D κ , the event HpSq occurs; • at most W from a side of a D κ parallel to S for some κ P rKs, but S does not intersect any non-parallel side of any D κ 1 , the event H W pSq occurs.
We say that we East-extend Λ pnq to Λ pn`1q if the event SGpD 1 q is defined by East-extending Λ pnq by l 1 in drection u 1 and SGpΛ pn`1q q is defined as SGpD 1 q X T n pΛ pn`1q zD 1 q. Indeed, Remark 5.6. Note that these n-traversability events are product over the disjoint regions into which all the boundaries of pD κ q κPrKs partition Λ pn`1q .
Armed with this notion, we are ready to define our SG events up to the internal scale for our models of interest.
Definition 5.7 (Balanced rooted internal SG). Let U be balanced rooted. Let SGpΛ p0q q be the event that Λ p0q is fully infected. We define SGpΛ pnq q for n P rN i s by successively East-extending Λ ppq to Λ pp`1q .
Proof of Theorem 5.8. For m ě 1 and n P rN i s denote (23) For the sake of simplifying expressions we will abusively assume that for all κ P rKs the length l κ is of the form λ 0 tp3{2q m u with integer m. Without this assumption, one would need to treat the term corresponding to m " M´1 in Proposition 4.3 separately, but identically. We next deduce Theorem 5.8 from the following two lemmas. Lemma 5.9. For n ă N i we have γ`Λ pn`1q˘ď γpΛ pnq qe OpC 2 q log 2 p1{qq pµ Λ pn`1q pSGqµ Λ pn`1q pT n qq Op1q where M pnq " r1{εs`rlog ℓ pn`1q { logp3{2qs.
Proof of Lemma 5.9. Let us start with a general observation. Consider two regions A, B Ă Z 2 and a measure ν on Ω AYB . The law of total variance reads Var ν AYB pf q " ν BzA pVar ν A pf qq`Var ν BzA pν A pf qq .
The latter term can be bounded from above by ν AzB`νAXB pVar ν BzA pf qq˘ď ν AzB pVar ν B pf qq , using the convexity of the variance and the law of total variance. Hence, Fix n P rN i s. Applying the above inequality several times, we obtain that (26) and turn to bounding a generic summand. We East-extend Λ pnq in direction u 0 by an arbitrarily large amount, which defines SGpD κ q for all κ ě 2 (it was already defined in Definition 5.5 for D 1 by East-extending in direction u 1 ). Observe that by the Harris inequality and the product structure of Definition 4.2, as in Eq. (9), for any κ P r2, Kq Proposition 4.3 gives with Mpκq " mintm : λ 0 p3{2q m ě l κ u and the same holds for D 1 with Mp1q " mintm : λ 1 p3{2q m ě l 1 u. Without loss of generality fix κ " 2, since all droplets of generation 2 are treated identically. Our goal is to show y Var y pf q˘, (28) from which Lemma 5.9 clearly follows in view of Eq. (26). Let V " D 1 Y D 2 (that is a %-shaped region in Fig. 4) and SGpV q :" SGpD 1 q X T n pD 2 zD 1 q.
By the product structure of traversability (see Definition 5.5), it is clear that so that by convexity of the variance Further using a two-block dynamics (see e.g. Lemma A.3), we have where E " SGpΛ pnq`y 2 u 1 q X T n pD 1 X D 2 q. By convexity of the variance and the fact that E X T n pD 2 zD 1 q Ă SGpΛ pnq`y 2 u 1 q X T pD 2 zpΛ pnq`y 2 u 1 qq " SGpD 2 q (31) (recall Definitions 4.2 and 5.5 and the fact that each segment of length ℓ pnq {pεCq " δℓ pnq {ε intersects at most Op1q droplets), we have Indeed, in the last line we recalled the definitions of SGpD 2 q and SGpV q, while in the second one we took into account that for any events A Ă B with µpAq ą 0 it holds that Varpf |Bq (33) and Eq. (31). Plugging Eq. (32) in Eq. (30) and noting again that by the Harris inequality µ V pE|SGq ě µpEq ě µ Λ pnq pSGqµ Λ pn`1q pT n q, to get Applying Eq. (27), we obtain since M pnq ě max κPrKs Mpκq. Inserting this in Eq. (29), we complete the proof of Eq. (28) and of Lemma 5.9.
Proof of Lemma 5.10. The first inequality in Eq. (24) follows from the Harris inequality, while the second one is trivial, so we turn to the last one and fix n P rN i s. Note that by Definition 5.5 We will therefore proceed by induction starting with Moreover, from Definition 5.5, to ensure the occurrence of T n pΛ pn`1q q, it suffices to have OpW Kℓ pn`1q q{pℓ pnq δq well-placed W -helping sets, as well as Oppℓ pn`1q q 2 q{pℓ pnq δεq helping sets for segments of length δℓ pnq {p3εq. Indeed, we may split lines perpendicular to each u j for j P p´k, kq into successive disjoint segments of length δℓ pnq {p3εq with a possible smaller leftover and place W -helping sets or helping sets depending on whether the segment under consideration is close to a parallel boundary of one of the D κ . Recalling that 1{ε " 1{δ " W " 1, ℓ pN c q " W Op1q q α , K " Opℓ pn`1q {ℓ pnq q, the explicit expressions Eq. (7), the Harris inequality and Observation 3.10, we obtain µ Λ pn`1q pT n q ě q OpW 2 Kℓ pn`1q q{pℓ pnq δq´1´p 1´q α q δℓ pnq {Opεq¯O ppℓ pn`1q q 2 {pℓ pnq δεqq ě e´l og Op1q p1{qq´1´e´q α δℓ pnq {Opεq¯O ppℓ pn`1q q 2 {pℓ pnq δεqq (37) Essentially the same computation leads to the same bound for µ D 1 zD 0 pT q, the only difference being that only Op1q W -helping sets and Opℓ pn`1q {εq helping sets are needed. Further recalling Eqs. Removing the surplus factor To conclude, let us briefly sketch how to remove the log log logp1{qq factor appearing in Theorem 5.8, which would also propagate to pollute Theorem 1(e).
Theorem 5.11. Let U be balanced rooted. Instead of Definition 5.7, it is possible to define SGpΛ pN i q q in such a way that Sketch proof of Theorem 5.11. To prove this, one should combine the techniques of Sections 5.1.1 and 5.1.2. More precisely, a less crude bound on a pnq m than Eq. (24) should be established along the lines of Eq. (12). As in Eq. (19), we may further decompose a pnq m into a product over scales p ď n. The relevant values of the parameters correspond to p3{2q m ď 1{plog W p1{qqq α q, say, and p P rN c , ns, as other cases can be dealt with using the crude bound Eq. (24). Further, as in Eq. (21), we can also discard p ě N c`∆ . Hence, we need to focus for the remaining values of m and p on lower bounding nd µpT ppD 1 zD 0 q`s m q|T pD 1 zD 0 qq, the latter being treated exactly like µpT pT 1 q|T pT qq in Eq. (20). Turning to the former conditional probability, it can be further decomposed as a product over elementary regions delimited by the boundaries of the pD κ q κPrKs . Unfortunately, for such (non-convex) polygonal regions R, bounding µ p T p pR`s m q| T p pRqq is no easy feat. Indeed, Corollary B.4 only treats tubes and, more importantly deals with helping sets for one direction only in each part of the tube (recall Fig. 2a), while T p pRq requires helping sets in various directions, which are all dependent. To make matters worse, for certain families U it may happen that a single set of α infections is simultaneously a helping set for different directions and this would create complex and heavy dependency among different directions, which could, a priori, make boundary regions attract such sets.
To deal with this issue, one could further elaborate Definition 5.5. Indeed, we may split Λ pp`1q zD 1 into disjoint horizontal strips (recall Fig. 4b) of width ℓ ppq {pW εq. Each strip is assigned a direction u j , j P p´k, kq and we will only ask for helping sets for this direction to be present. These requirements are again cut essentially along the boundaries of all D κ into parallelograms as in Lemma B.3, placing W -helping sets on segments close to the boundaries. Naturally, some leftover regions remain without helping sets as in Definition 5.5, but they are unimportant as in Section 5.1.2.
By doing this, we make the event T n pRq the intersection of traversability events of parallelograms in the sense of Lemma B.3, so that its result can be applied as in the proof of Corollary B.4, leading to a calculation similar to the one in Theorem 5.3. The only significant change is that now there are OpW ℓ pp`1q {ℓ ppq q parallelograms instead of a constant number. This is not really a problem, but, if one wishes to avoid careful computations, given that we are interested in the range p P pN c , N c`∆ q, we can brutally bound this by its maximum, which is log Op1q logp1{qq by the definition of ∆.

Mesoscopic East dynamics
We next treat the East mesoscopic dynamics, which is essentially an extension of the internal one. Although they actually apply to all balanced models, the results of this section will only be used for balanced models with infinite stable directions, so the definitions will only apply to that class. As for that class there is a lot of margin, our reasoning will be far from tight for the sake of simplicity.
Extending the notation from Section 5.1.2, for n ą N i , we set ℓ pnq " W n´N i ℓ pN i q and define s pnq , r pnq , Λ pnq as in Section 5.1.2. Further let N m " inftn : ℓ pnq {ε ě ℓ m " q´Cu " ΘpC logp1{qq{ log W q and assume for simplicity that ℓ pN m q " q´Cε. We will only be interested in n ď N m and Definitions 5.5 and 5.7 remain unchanged for such n.
Proof. The proof is essentially identical to the one of Theorem 5.8, so we will only indicate the necessary changes. To start with, Lemma 5.9 applies without change for n P rN i , N m q. Also, the Harris inequality still implies that a pnq m ď µ´1 Λ pnq pSGq ď µ´1 Λ pN m q pSGq. Therefore, Recalling the bound of γpΛ pN i q q established in Theorem 5.8, together with the fact that N m ď C logp1{qq and M pN m´1 q ď OpC logp1{qqq, it suffices to prove that in order to conclude the proof of Theorem 5.12.
Once again, the proof of Eq. (38) proceeds similarly to the one of Eq. (24) in Lemma 5.10. Indeed, the same computation as Eq. (37) in the present setting gives that for n P rN i , N m q we have From Eq. (35) it follows that Plugging Eqs. (24) and (39) in the r.h.s., this yields Eq. (38) as desired.

CBSEP-type dynamics
In this section we establish results involving CBSEP-type dynamics. It is relevant only for isotropic models up to the internal scale and for unrooted models on mesoscopic level.

Isotropic internal and mesoscopic dynamics
Let U be isotropic. We follow and generalise [17]. Let r p0q be a symmetric sequence of radii with r p0q i " r p0q i`2k for all i P r2ks, such that r p0q i " Θp1{εq is a multiple of λ i for all i P r4ks and the corresponding side lengths s p0q i are also Θp1{εq. For any i P r2ks and n " 2km`r with r P r2ks we define and Λ pnq " Λpr pnq q with r pnq the sequence of radii associated to s pnq satisfying r pnq i " r pnq i`2k for all i P r2ks. Further set N m`" 2krlogpεℓ m`q { log 2s (recall ℓ m`f rom Section 3.4, where m is not a variable and stands for 'mesoscopic').
Note that Λ pnq are nested symmetric droplets extended in one direction at each step satisfying Λ p2kmq " 2 m Λ p0q . Recall Definition 4.4 and Fig. 2b. (a) A generic realisation of SGpΛ pnq q depicting the SG translates of Λ pnq , . . . , Λ pn´2kq involved in progressive shades of grey. Each extension is as in Fig. 2b. is thickened, while the droplets Λpr i q, 0 ď i ď 2k are in progressive shades of grey, starting from the black Λpr 0 q " Λ pn´2kq . Figure 5: Geometry of isotropic SG and SG events. Definition 6.1 (Isotropic SG). Let U be isotropic. We say that Λ p0q is SG if it is fully infected. We then recursively define SGpΛ pn`1q q for n ě 0 by CBSEP-extending Λ pnq in direction u n by l pnq :" s pnq n`k " Θp2 n{2k {εq (recall that indices of directions and sequences are considered modulo 4k as needed and see Fig. 5a). Theorem 6.2. Let U be isotropic. Then for all n ď N mγ´Λ Proof of Theorem 6.2. We seek to apply Proposition 4.5, in order to recursively upper bound γpΛ pnq q for all n ď N m`. To that end, we need the following definition. Definition 6.3. Fix 2km`r " n ď N m`w ith r P r2ks and let Λ pnq 1 " T pr pnq , λ r , r`2kq, Λ pnq 2 " Λpr pnq´λ r v r q and Λ pnq 3 " T pr pnq , λ r , rq´λ r u r (as in Proposition 4.5 with r " r pnq , l " l pnq and i " r). If n ă 2k, we define SGpΛ For n ě 2k, we define ST pΛ pnq 1 q to be the event that for every segment S Ă Λ pnq 1 perpendicular to some u j with j ‰ r˘k of length 2 m {pW εq the event H W pSq occurs. ST pΛ pnq 3 q is defined analogously and also does not depends on the configuration outside of Λ pnq 3 . Finally, for n ě 2k, we define SGpΛ pnq 2 q as the intersection of the following events (see Fig. 5b).
• SGpΛpr 0 qq, where r 0 " r pn´2kq ; • ST W pT pr 0 , l pn´2kq {2´λ r , rqq X ST W pT pr 0 , l pn´2kq {2´λ r , r`2kqq; • for all i P p0, 2kq • for every i P rn´2k, ns, j P r4ks and segment S Ă Λ pnq 2 perpendicular to u j of length 2 m {pW εq at distance at most W from the u j -side (parallel to S) of Λpr piq q, the event H W pSq holds.
In words, SGpΛ pnq 2 q is close to being the event that the central copy of Λ pn´2kq in Λ pnq 2 is SG, the two tubes of equal length around it corresponding to a CBSEP-extension by l pn´2kq in direction u r and so on until we reach Λ pnq after 2k extensions. However, we have modified this event in the following ways. Firstly, the first extension is shortened by 2λ r , so that the final result after the 2k extensions fits inside Λ pnq 2 and actually only its u r`k and u r´k -sides are shorter than those of Λ pnq 2 by λ r . Secondly, the traversability events for tubes are required to occur with segments shortened by W (recall Definition 4.1) on each side. Finally, we require W helping sets for the last OpW q lines of each tube, as well as the first OpW q outside the tube.
Taking this into account, we claim that Eq. (5) is verified. Indeed, if SGpΛ pnq 2 q and ST pΛ pnq 3 q occur, then the droplets Λ piq for i P rn´2k, ns are all SG. To see this, proceed by induction and observe that each traversability appearing in Definition 6.3 together with the W -helping sets implies the corresponding traversability for the droplets Λ piq , since the droplets are perturbed (shifted position and modified side lengths) by Op1q, which is less than the amount, W , by which we the segments required in ST W are contracted compared to those in ST . In total, for each segment appearing implicitly in the ST events defining SGpΛ piq q (via Definition 4.4), we have asked either for a W -helping set or a helping set in a slightly shorter segment.
Hence, we may apply Proposition 4.5 to get pST |ST 0 q (40) for n ě 2k and γpΛ pnq q ď e OpC 2 q log 2 p1{qq for n ă 2k. By the Harris inequality and the fact that both ST pΛ pnq 1 q and ST pΛ pnq 3 q can be guaranteed by OpW 2 q W -helping sets, we have Indeed, applying Lemma B.1 2k times gives that, conditionally on SGpΛ pnq 1 Y Λ pnq 2 q, with probability q OpCq we have SGpΛ pn´2kq q and the traversability events corresponding to symmetrically extending it in 2k steps to Λ Hence, Theorem 6.2 follows immediately from Lemma 6.4 below.
Lemma 6.4. The following bounds hold for n P r2k, N m`s and m " tn{p2kqu.

Unbalanced internal dynamics
For unbalanced U with finite number of stable directions the internal dynamics is essentially trivial and so is the SG event up to the internal scale.
Definition 6.5 (Unbalanced internal SG). If U is unbalanced with finite number of stable directions, we say that Λ p0q :" Λpr p0q q, defined by the side lengths s p0q j " λ j rℓ i {λ j s, is super good if all sites in Λ p0q zΛpr p0q´W 1q are infected.
The following straightforward result was proved in [19,Lemma 4.10].

CBSEP mesoscopic dynamics
Let U be unbalanced unrooted with finite number of stable directions (the unbalanced character is only assumed, so as not to override definitions for semi-directed and isotropic models). W.l.o.g. let αpu j q ď α for all j ‰˘k and minpαpu k q, αpu´kqq ą α. We will only use 4k scales for the mesoscopic dynamics. Recall Section 3.4. For i P r0, 2ks let Λ piq " Λpr piq q be centered at 0 with r piq defined by For i P p2k, 4ks, we define Λ piq similarly by These droplets are exactly as in Fig. 5a, except that the extensions are much longer. More precisely, we have Λ pi`1q " Λpr piq`lpiq pv i`vi`2k q{2q with l piq " p1´op1qqℓ m´i f i P r2ks and l piq " p1´Opδqqℓ m`i f i P r2k, 4kq.
Theorem 6.7. Let U be unbalanced unrooted with finite number of stable directions. Then Proof. We will proceed similarly to Theorem 6.2, but the first two steps are more special (see Fig. 6). For i P r4ks, as in Proposition 4.5, let  1 v 1 , l p0q {2, 0qq occurs, • for all j ‰˘k and segment S Ă Λ p1q 2 perpendicular to u j at distance at most W from the u j -side of Λ  • SGpΛ pi´2q q occurs; • for each m P t0, 2ku the following occurs • for all j P r4ks, m P ti´1, iu and segment S Ă Λ piq 2 perpendicular to u j of length s pmq j {W at distance at most W from the u j -side of Λ pmq the event H W pSq holds.
With these definitions it is again not hard to verify Eq. (5) (see Fig. 6), so that Proposition 4.5, Proposition 6.6 and the Harris inequality give It is not hard to check from Definition 6.8 that each SG, SG and ST event in Eq. (51) requires at most Cℓ i fixed infections, W Op1q W -helping sets and Op1q W -symmetrically traversable tubes. We claim that the probability of each tube being W -symmetrically traversable is q OpW q , which allows us to conclude, given Eq. (51). Indeed, as in Eq. (47), we have e.g.
µ T pr p0q´λ 1 v 1 ,l p0q ,0q pST W q ě q OpW q exp`´Opℓ m´q exp`´q α ℓ i {W˘˘ě q OpW q by the choice of scales in Section 3.4. Traversability for i ą 1 is slightly more subtle, since some of the parallelograms (recall Fig. 2b) require W -helping sets, since αpu k q ą α and αpu´kq ą α. However, the u k -side of Λ piq for i ą 0 has length Ωpℓ m´q , which is much larger than q´W , so we can still conclude the proof of our claim in the same way, using Observation 3.10.

Semi-directed mesoscopic dynamics
Let U be semi-directed and w.l.o.g. αpu i q ď α for all i ‰´k. Recall from Section 5.1 that we defined Λ pN i q , a symmetric droplet with side lengths s pN i q equal to Θpℓ pN i q {εq, as well as SGpΛ pN i q q in Definition 5.2. As in Section 6.2.2, for i P rN i`1 , N i`2 ks we define i´N i´k ď j ă k, λ j rℓ m´{ λ j s´k ď j ă i´N i´k , while for i P pN i`2 k, N i`4 ks, s piq is given by Eq. (49). We then define Λ pN i`i q " Λpr pN i`i q q with r pN i`i q the sequence of radii associated to s pN i`i q satisfying , which is p1´op1qqℓ m´f or i P r1, 2ks and p1´Opδqqℓ m`f or i P p2k, 4ks. Theorem 6.9. Let U be semi-directed. Then Proof. The proof proceeds exactly like Theorem 6.7, except that the first two steps are much more delicate and require taking into account the internal structure of SGpΛ pN i q q on all scales down to 0, which is, alas, rather complex (recall Fig. 3a) and also not symmetric w.r.t. Λ 1 and Λ 3 . This is not unexpected and is to some extent the crux of semi-directed models. As previously, for i P rN i , N i`4 kq we define Λ at distance at most ? W {ε from the origin are infected. For n P r0, N i s such that 2kn P N let Λ 1pnq " Λpr pnq´λ 0 pv 0`v2k qq and define SG 1 pΛ 1pnq q recursively exactly like SGpΛ pnq q in Definition 5.2 with all droplets replaced by their contracted versions Λ 1 and all traversability events required in East-extensions replaced by the corresponding W -traversability events (T W , see Definition 4.1). Let W 1 be the event that for every n P r0, N i s, j P r4ks and segment S Ă Λ   q be the event that for all j P r4ks, m P tN i , N i`1 u and every segment S Ă Λ pN i`1 q 3 perpendicular to u j of length s pmq j {W at distance at most W from the u j -side of Λ pmq the event H W pSq occurs.
For n P r0, N i s such that 2kn P N let nd define SG 2 pΛ 2pnq q like SG 1 pΛ 1pnq q in Definition 6.10. Further let Let W 2 (resp. I 2 ) be defined like W 1 (resp. I 1 ) in Definition 6.10 with Λ 1 replaced by Λ 2 and N i replaced by N i`1 . Finally, we set Furthermore, since Definition 6.8 for i P r2, 4kq does not inspect the internal structure of SGpΛ p0q q (see Fig. 6b), we may use the exact same definition for ST pΛ It therefore remains to bound each of the terms in the denominator by expp´1{pε Op1q q α qq in order to conclude the proof of Theorem 6.9.
Notice that a total of ε´O p1q fixed infections and W Op1q N i " q op1q Whelping sets are required in all the events in Eq. (52), which amounts to a negligible factor. The probability of SG 1 pΛ 1pN i q q and SG 2 pΛ 2pN i q q can be bounded exactly like SGpΛ pN i q q in Lemma 5.4, yielding a contribution of expp´1{pε Op1q q α qq. Finally, the remaining bounded number of ST W events are treated as in Theorem 6.7 to give a negligible q´O pW q factor.

Global dynamics and assembly of Theorem 1
In this section we recall and adapt global dynamics mechanisms from [17,19] and assemble the pieces to prove our main result Theorem 1 for each refined universality class. As already noted, all lower bounds are known from [15] and the upper ones for classes (a) and (c) were proved in [27] and [19] respectively, so we only need to establish the upper bounds for classes (b) and (d)-(g).

Global CBSEP dynamics
Let us recall the global CBSEP mechanism introduced in [17].
Let Λ m´a nd Λ m`b e droplets with sides Θpℓ m´q and Θpℓ m`q respectively. Consider a tiling of R 2 with square boxes Q i,j " r0, ℓ m qˆr0, ℓ m q`ℓ m pi, jq. We say that the box Q i,j is good if for every segment S Ă Q i,j perpendicular to some u P p S of length at least εℓ m´, H W pSq occurs and denote the corresponding event by G i,j . We say that it super good if Q i,j contains a super good translate of Λ m´a nd denote the corresponding event by SG i,j . Proposition 7.1. Let T " expp´log 4 p1{qq{q α q. Assume that SGpΛ m`q and SGpΛ m´q are defined so that p1´µpSGpΛ m´q qq T T 4 " op1q and for all x P Z 2 such that x`Λ m´Ă Λ m`w e have SGpx`Λ m´q XGpΛ m`q Ă SGpΛ m`q , where GpΛ m`q stands for the event that for every segment S Ă Λ m`p erpendicular to some u P p S of length at least 3εℓ m´t he event H W pSq occurs. Then E µ rτ 0 s ď γ`Λ m`˘l ogp1{µpSGpΛ m´q qq q OpCq .
We omit the proof, which is identical to [17,Section 5], 5 and turn to the proof of Theorem 1 for classes (d), (f) and (g).
Proof of Theorem 1(d). Let U be unbalanced unrooted. Recall Section 6.2.2 and let Λ m`" Λ p4kq and Λ m´" Λ p2kq . By concluding the proof.

Global FA-1f dynamics
We next import the global FA-1f dynamics together with much of the mesoscopic multi-directional East one simultaneously from [19].
Proposition 7.2. Let U have a finite number of stable directions, T " expp´log 4 p1{qq{q α q and r i be such that the associated side lengths satisfy C ď s i j ď Opℓ i q for all j P r4ks. Assume that for all l " Θpℓ m q multiple of λ 0 the event SGpΛpr i`l v 0 qq is defined so that p1´µpSGpΛpr i`l v 0 qqqq T T W " op1q. Then, E µ rτ 0 s ď max l"Θpℓ m q γpΛpr i`l v 0 qq pq 1{δ min l"Θpℓ m q µpSGpΛpr i`l v 0 qqqq logp1{qq{δ .
The proof is as in [19], the only difference being that one needs to replace the base of the snail there by Λ m :" Λpr i`λ 0 rℓ m {λ 0 sv 0 q, which has a similar shape by hypothesis; the corresponding event that the base is super good by SGpΛ m q; and [19,Lemma 4.9] by the definition Eq. (3) of γpΛ m q. As this proposition is essentially the entire content of [19] (see particularly Proposition 4.12 and Remark 4.8), we refer the reader to that work for the details.
Proof of Theorem 1(e). Let U be balanced rooted with finite number of stable directions. Recall Λ pN i q " Λpr pN i q q and r pN i q ": r i from Section 5.1. Fix l " Θpℓ m q multiple of λ 0 and East-extend Λ pN i q by l in direction u 0 . It is easy to check from Definition 4.2 and Observation 3.10 that Then, by Proposition 4.3, Theorem 5.11 and the Harris inequality, we obtain γ`Λpr i`l v 0 q˘ď expˆl ogp1{qq ε 3 q α˙, µ Λpr i`l v 0 q pSGq ě expˆ´2 ε 2 q α˙.

Global East dynamics
Finally, for class (b) we will need a simpler version of the procedure of [19,Section 5] with East dynamics instead of FA-1f.
Proof of Theorem 1(b). Let U be balanced with infinite number of stable directions, T " expp1{q 3α q and s m " s pN m q , r m " r pN m q and Λ m " Λ pN m q with the notation of Section 5.2. In particular, s m j " Θpℓ m q for j P r´k, k`1s and s m j " Opℓ m q for j P rk`2, 3k´1s. We East-extend Λ m by 2l " 2pλ 0`r m 0`r m 2k q in direction u 0 to obtain Λ " Λpr m`2 lv 0 q. Proposition 4.3, Theorem 5.12, the Harris inequality and the simple fact that µpT pT pr m , 2l, 0qqq " q OpW q (recall Observation 3.10) give A similar argument to the rest of the proof was already discussed thoroughly in [19,Section 5] and then in [17,Section 5], so we will only provide a sketch. The adapted approach of [19,Section 5] proceeds as follows.
(2) By finite speed of propagation we may work with the U-KCM on a large the discrete torus of size T " t˚.
(3) We partition the torus into strips and the strips into translates of the box Q " H u 0 pλ 0`r m 0 qXH u k pρ k`r m k qXH u´k pr ḿ k qXH u 2k pr m 2k q as shown in Fig. 8. We say Q is good (GpQq occurs) if for each segment S Ă Q perpendicular to some u P p S of length εℓ m the event H W pSq occurs. Further define SGpQq to occur if the (only integer) translate of Λ m contained in Q is SG. We say that the environment is good (E occurs) if all boxes are and in each strip at least one box is super good. The sizes are chosen so that it is sufficiently likely for this event to always occur up to t˚. Indeed, we have p1´µpSGpΛ m qqq T T W " op1q by Theorem 5.12 and p1´µ Q pGqqT W " op1q by Observation 3.10.
(4) By a standard variational technique it then suffices to prove a Poincaré inequality, bounding the variance of a function conditionally on E by the Dirichlet form on the torus. Moreover, since µ and E are product w.r.t. the partition of Fig. 8, it suffices to prove this inequality on a single strip.
(5) Finally, we prove such a bound, using an auxiliary East dynamics for the boxes and the definition of γ to reproduce the resampling of the state of a box by moves of the original U-KCM.
Let us explain the last step above in more detail, as it is the only one that genuinely differs from [19]. Let Q i " Q`ilu 0 and T " Ť iPrT s Q i be our strip of interest (indices are considered modulo T , since the strip is on the torus). As explained above, our goal is to prove that for all f : Ω T Ñ R it holds that where c T,1 x takes into account the circular geometry of T. By [27,Proposition 3.4] on the generalised East chain we have since Theorem 5.12 gives µ Q pSGq ě expp´2{pε 2 q α qq. 6 Next observe that Λ i Ą Q i , where Λ i :" Λ`pi´1qlu 0 (see Fig. 8). Hence, by convexity of the variance and the fact that µpEq " 1´op1q and we have writing GpΛ i zQ i q Ă GpQ i`1 q X GpQ i´1 q for the event that H W pSq holds for all segments S Ă Λ i zQ i of length 2εℓ m perpendicular to some u P p S and using Eq. (33) and SGpQ i´1 q X GpQ i q X GpΛ i zQ i q Ă SGpΛ i q (recall Definition 4.2) for the second inequality. Finally, recalling Eqs. (3), (53) and (55), we obtain Eq. (54) as desired.
at pieces of u i -helping sets for the last few lines of the droplet and add to Λ the sites which each piece can infect. The reason for introducing this is that helping sets may need to infect a few sites outside Λ before creating their periodic infections on the corresponding line and it is those sites that we wish to include in Λ ω i . We set Λ ω I " Ť iPI Λ ω i for I Ă r4ks.

A.1 Microscopic dynamics
Let i P r4ks be such that αpu j q ă 8 for all j P I " ti´k`1, . . . , i`k´1u.
Fix Λ " Λprq with sides at least Ωp1{δq and at most q´O pCq . Let l P r0, Op1qs, ω P Ω Z 2 zΛpr`lv i q , Λ`" pΛpr`lv i qq ω I and T " T pr, l, iq. Our goal is to provide a relaxation mechanism for an East-extension of bounded length.
Lemma A.1. In the above setting we have The proof is both standard and messy, so we only provide a sketch.
Sketch proof. By convexity of the variance, it is enough to upper bound Var Λ`zΛ pf |T ω pT qq. We will use the canonical path technique (see e.g. [32, Theorem 4.2.1]), so we need to define for any two configurations η, η 1 P Ω Λ`zΛ X T ω pT q ": A a sequence Γpη, η 1 q of configurations in A differing by single legal updates of the U-KCM with boundary condition 0 Λ¨ωZ 2 zΛ`, leading from η to η 1 . We call such sequences canonical paths.
Recalling the notation of Definition 3.7, for j P I such that αpu j q ď α let X j denote the intersection of rZ j Y H u j s U with a sufficiently long segment of H u j zH u j such that for all x P X j we have that ? W ď xx, x j y ď 2 ? W . It is easy to see that if H u j is fully infected, then X j can infect X j`Q x j only modifying states in H u j zH u j . Moreover, if Z j Y H u j is infected, then X j can be infected in at most Op ? W q steps. Finally, observe that a W -helping set in H u j zH u j can move freely in both directions along the line.
Let us first describe the path in the case when T consists of a single line perpendicular to, say, u j . Our paths will proceed in four stages. First, starting from η, we infect Op ? W q sites until we infect translates of X j with all possible residues modulo Q along the line. We next infect the last W sites of the line. Then we change η with that W -helping set to η 1 with the same W -helping set. Finally, we reach η 1 , which can be done as in the reverse of the first two stages, so we will only describe the first three stages. Note that if αpu j q ą α, the first two stages are not needed, as W -helping sets are guaranteed by T ω (which does not depend on ω in that case).
In the first stage we simply add the infections of rZ j Y H u j s U zH u j translated appropriately one by one until we infect the translate of X j in Op ? W q steps. Naturally, we do this for Q different translates, so as to obtain each residue.
In the second stage we perform an East motion of translates of X j , starting from the ones we infected in the first stage. As noted above, thinking of H u j as infected, using X j we can infect X j`Q x j (which may intersect X j or other infected sites of η, in which case we only infect the additional sites), then use X j`Q x j to infect X j`2 Qx j , then use X j`Q x j to remove any infections in X j`Q x j which are not in X j , X j`2 Q j or η. We continue similarly, as described for the East model in Section 2.3.2 (also see [1,Fig. 2]). Doing this, we can eventually infect the last W {Q translates of X j of the form X j`k Qx j in T . Repeating this for all Q translates with different residues modulo Q, we obtain the desired last W infections in the line. We finally remove all other auxiliary infections by reversing the same path. Note that by our choice of directions p S, it is indifferent whether H u j is infected or only Λ.
In the third stage we move the W -helping set to the other extremity of the segment, leaving behind η 1 . More precisely, if the W infections are currently at position x, we infect x´λ j`k u j`k and then remove the infection at x`pW´1qλ j`k u j`k (the last site of the W -helping set) if and only if it is not present in η 1 . Moreover, with a finite number of infections we can also infect any site Λ ω j zΛ at distance Op1q from the W -helping set, but at large enough distance from its extremities. Thus, as we move the W -helping set, we can ensure that all sites in Λ ω j zΛ on one side of its midpoint have the state η 1 and the others have state η. Finally, when η is completely replaced by η 1 , we move the W infections back to the end of the segment, still leaving η 1 behind.
To prove the lemma in the case of a single line, it suffices to bound the length of these paths, as well as where we neglected |T | (which is only polynomial in q) and used that by the Harris inequality and Observation 3.10 µpηq " q OpW q µpη|Aq. The length of the paths is polynomial in |T |, as for the East process (see [1]). Hence, we turn to upper bounding (57). Observe that in the first and second stages the configurations η 2 only differ from η in at most |X j |Oplog |T |q sites (this is the fundamental property of East paths), so µpηq{µpη 2 q ď q´O plog |T |q and there are at most pOp|T |qq Oplog |T |q possible choices for the discrepancies, hence the contribution to Eq. (57) is at most exppOpC 2 q log 2 p1{qqq. In the third stage at all times there are at most OpW q sites on which the configuration differs from both η and η 1 and Op|T |q possible positions of the W -helping set (we know that on one of its sides we have η and η 1 on the other). Recalling that µ is product, we obtain a contribution of only q OpCq to Eq. (57), which concludes the proof. In order to treat an arbitrary number of lines and take into account that the tube T is composed of segments in different directions, we proceed iteratively. Firstly, the structure of T ω (recall Fig. 2) makes different directions independent, and they do not overlap, so we may proceed one direction at a time. To treat several consecutive lines in the same direction, we first produce the W -helping set for the first line as above, then use it to act as an infected boundary condition for the second line, which we may place next to the site we want to update. This way we can also create a W -helping set for the second line and so on. Eventually, we have W -helping sets for all lines and we may perform the third stage, moving all of them simultaneously to change η into η 1 . We finally remove these W -helping sets by reversing the path from η 1 that would create them. The computation of the congestion of the path identical to the simpler one-line case.

A.2 Auxiliary three-block chain
We next prove a non-product variant of the standard two-block technique for the purposes of the proof of the East-extension Proposition 4.3. Let pΩ i , π i q 3 i"1 be finite positive probability spaces, pΩ, πq denote the associated product space and ν " πp¨|Hq for some event H Ă Ω. For ω P Ω we write ω i P Ω i for its i th coordinate. Consider an event F Ă Ω 1ˆΩ2 and set Dpf q " ν pVar ν pf |ω 3 q`½ F Var ν pf |ω 1 , ω 2 qq for any f : H Ñ R. Observe that D is the Dirichlet form of the continuous time Markov chain on H in which the couple pω 1 , ω 2 q is resampled at rate one from νp¨|ω 3 q and, if pω 1 , ω 2 q P F , then ω 3 is resampled with rate one from νp¨|ω 1 , ω 2 q. This chain is reversible w.r.t. ν.
Proof. We follow [17,Proposition 4.5]. Consider the Markov chain pωptqq tě0 described above. Given two arbitrary initial conditions ωp0q an ω 1 p0q we will construct a coupling of the two chains such that with probability Ωp1q we have ωptq " ω 1 ptq for t ą T :" max ω 3 PΩ 3 ν´1pF |ω 3 q. Standard arguments [23] then prove that the mixing time of the chain is OpT q and the lemma follows.
To construct our coupling, we use the following representation of the Markov chain. We are given two independent Poisson clocks with rate one and the chain transitions occur only at the clock rings. When the first clock rings, a Bernoulli variable ξ with probability of success νpF |ω 3 q is sampled. If ξ " 1, then the couple pω 1 , ω 2 q is resampled w.r.t. the measure πp¨|F q " νp¨|F , ω 3 q, while if ξ " 0, then pω 1 , ω 2 q is resampled w.r.t. the measure νp¨|F c , ω 3 q. Clearly, in doing so the couple pω 1 , ω 2 q is resampled w.r.t. νp¨|ω 3 q. If the second clock rings, we resample ω 3 from π 3 if ω P F and ignore the ring otherwise.
When the second clock rings, the two chins attempt to update to two maximally coupled couples of configurations with the corresponding distributions.
Suppose now that two consecutive rings occur at times t 1 ă t 2 at the first and second clocks respectively and the Bernoulli variables at time t 1 are both 1. Then the two configurations are clearly identical at t 2 . To conclude the proof, observe that for any time interval ∆ of length one the probability that there exist t 1 ă t 2 in ∆ as above is at least 1{p4T q.

A.3 Proofs of the one-directional extensions
We will require a more technical version of Eq. (3) accounting for a boundary condition. Let γ ω I pΛq be the smallest constant γ ě 1 such that for all f : For the rest of the section we assume the setting of Section 4 and set I " ti´k`1, . . . , i`k´1u. Proof. We will loosely follow [17,Eq. (3.11)]. Proceeding by induction it suffices to prove that for any m P r1, Mq and ω P Ω Z 2 zΛ m`1 γ ω I`Λ m`1˘ď max ω 1 PΩ Z 2 zΛ m γ ω 1 I pΛ m q a m q OpW q .
Lemma A.5. Assume we CBSEP-extend Λprq by l in direction u i . Then γ pΛpr`lv i qq ď max ω γ ω J pΛpr`λ i v i qq µ Λpr`λ i v i q pSGq µ Λpr`lv i q pSGq e OpC 2 q log 2 p1{qq .
Proof. As in [17,Eq. (3.11)] (with minor amendments as in Lemma A.4), we have Proof of Proposition 4.5. By Lemma A.5 it suffices to relate γ ω J pΛ 1 q and γ pΛprqq. This is done exactly as in [17,Lemma 3.12] (see particularly Eq. Lemma B.1. Let U have a finite number of stable directions. Fix i P r4ks and a symmetric droplet Λ " Λpr`lv i q obtained by CBSEP-extension by l in direction u i . Assume that l ď ℓ m`i s divisible by λ i . Then for all s P r0, ls divisible by λ i and ω, ω 1 P Ω Z 2 zΛ µ´SG ω s pΛq| SG ω 1 pΛq¯ě q OpCq .
Proof. We will prove that for all s, s 1 P r0, ls divisible by λ i and ω, ω 1 P Ω Z 2 zΛ we have µ Λ pSG ω s q µ Λ pSG ω 1 s 1 q " q OpW q .
To prove Eq. (62), let us first observe that by the symmetry of Definitions 3.8 and 4.4, µ Λ pSG ω s q µ Λ pSG ω 1 0 q " µ Ts pST ωs qµ T l´s pST ω l´s q µ T l pST ω l q , where T x " T pr, x, iq and the ω x are certain boundary conditions that can be expressed in terms of ω, ω 1 . Further note that for any x and ω 2 µ Tx´S T ω 2¯" q OpW q µ Tx pST q, by the Harris inequality, since it suffices to add W -helping sets on the last Op1q lines of the tube. Finally, observing that T s Y pT l´s`s u i q " T l , we get µ Ts pST qµ T l´s pST q ď µ T l pST q ď µ Ts pST 0 qµ T l´s pST q.
We next treat certain perturbations of traversability events, building them progressively from segments and parallelograms in the next lemmas.
Proof. Let us note that a stronger version of this result can be proved more easily by counting circular shifts of the configuration in a Op1q neighbourhood of S such that a given helping set remains at distance at least some constant from S 2 . We prefer to give the proof below as a preparation for Lemma B.3. For concreteness, let us assume that αpu i`2k q ą α, other cases being treated similarly. Thus, helping sets are just u i -helping sets or W -helping sets. Recall from Definition 3.7 that a u i -helping set is composed of Q translates of the set Z i . Further let S Ă H u i zH u i . For r P rQs we denote by H r pSq the event that S has a translate of Z i by a vector of the form pr`k r Qqλ i`k u i`k with k r P Z and similarly define H r pS 1 q. In words, we look for the part of the helping set with a specified reminder r modulo Q.
Since |S| " q´α`o p1q , the probability that there are α`1 infected sites at distance Op1q from each other and from S is q 1´op1q . Furthermore, if this does not happen, but HpSq occurs, then all H r pSq for r P rQs occur disjointly. Therefore, by the BK inequality Proposition 3.3, µpHpSqq ď q 1´op1q`ź rPrQs µpH r pSqq ď`1`q 1´op1q˘ź rPrQs µpH r pSqq, since, as in Observation 3.10, we have µpH r pSqq ě 1´p1´q α q Ωp|S|q ě q op1q .
Using Eq. (66) and applying the Harris inequality, we get µpHpS 1 qq µpHpSqq ě`1´q 1´op1q˘ź rPrQs µpH r pS 1 qq µpH r pSqq ě`1´q 1´op1q˘ˆ| S 1 |´Op1q |S|˙Q , where in the last inequality we used that H r pSq and H r pS 1 q can be expressed in terms of the i.i.d. (and therefore exchangeable) Bernoulli variables corresponding to each translate of the helping set being infected. Recalling that |S| ě W |S 2 |, this concludes the proof.
Lemma B.3. Let i, j P r4ks be such that αpu i q ď α and j R ti, i`2ku. Consider the parallelogram R " Rpl, hq " H u i plq X H u j phq X H u j`2k p0q X H u i`2k p0q for l P rρ i , e q´o p1q s and h " q´α`o p1q . We say that R is traversable in direction u i (T pRq occurs), if for each nonempty segment of the form S " R X H u i ph 1 qzH u i ph 1 q the event H 1 Z 2 zRpl`W,hq C 2 pSq occurs. Let R 1 " Rpl, h 1 q with 1 ą h 1 {h ě 1´1{W . Then µ pT pR 1 q|T pRqq ěˆ1´W 1{2ˆ1´h 1 h˙´q 1´op1q˙O plq Proof. Let us write simply H m for H 1 Z 2 zR C 2 pR X Hpmρ i qzH u i pmρ i qq and similarly define H 1 m for R 1 . Let m take its values in rMs for some integer M. Separate R into its lower and upper halves R 1 and R 2 , consisting of tM{2u and rM{2s segments perpendicular to u i respectively. If T pRq occurs, then one of the following occurs.
• There is a set of α`1 infections at distance Op1q from each other and from both R 1 and R 2 , and the rectangles, formed by removing in each of R 1 and R 2 the Op1q lines closest to their common boundary, are both traversable.
• The rectangles R 1 and R 2 are disjointly traversable.
The last fraction can be bounded, using Lemma B.2, to obtain µ pT pR 1 q|T pRqq ěˆ1´W 1{3ˆ1´h 1 h˙´q 1´op1q˙M .
As a result, we are able to prove the following result, which vastly generalises [17,Lemma A.5].
Corollary B.4 (Perturbation cost). Let T " T pr, l, iq be a tube with i P r4ks such that αpu j q ď α for all j P pi´k, i`kq. Denote the side lengths of Λprq by s as usual. Assume that l P rΩp1q, e q´o p1q s, s :" min i´kăjăi`k s j " q´α`o p1q and max i´kăjăi`k s j " q´α`o p1q . For some ∆ P p0, s{ ? W s, let r 1 and l 1 be such that 0 ď s j´s 1 j ď Op∆q for all j P ri´k, i`ks and 0 ď l´l 1 ď Op∆q.
Further let x P R 2 be such that }x} " Op∆q and d, d 1 P r0, Op∆qs with d ď d 1 . Denoting T 1 " T pr 1 , l 1 , iq`x, for any boundary conditions ω, P Ω Z 2 zT and ω 1 P Ω Z 2 zT 1 , we have Proof. Recalling Definition 4.1, it is clear that T ω d pT q is the intersection of 2k´1 independent traversability events for parallelograms of length l in the sense of Lemma B.3. Let us denote them by pR j q i`k´1 j"i´k`1 and, similarly, pR 1 j q i`k´1 j"i´k`1 for T 1 with R j and R 1 j having sides perpendicular to u j (see Fig. 9). Finally, set R 2 j " R j X R 1 j " Rpl´Op∆q, s j´O p∆`C 2 qq for j P pi´k, i`kq and observe that R 1 j zR 2 j consists of two disjoint parallelograms R 1 j " RpOp∆q, s j´O p∆`C 2 qq and R 2 j " Rpl´Op∆q, Op∆qq with the notation of Lemma B.3 (up to translation).
Observe that that T ω 1 d 1 pT 1 q is implied by the presence of W -helping sets on the last Op1q lines of each R 1 j and R 1 j and the traversability of all R 1 j and R 2 j . Then by the Harris inequality and the independence of T pR j q we have that µ´T ω 1 d 1 pT 1 qˇˇT ω d pT q¯ě q OpW q ź j µ`T`R 1 j˘˘µ`T`R 2 j˘ˇT pR j q˘.
We may then conclude, using Lemma B.3 and that by Observation 3.10 µ`T`R 1 j˘˘ě`1´p 1´q α q Ωps j q˘O p∆q .