# Refined universality for critical KCM:
upper bounds

Ivailo Hartarsky<EMAIL_ADDRESS>
###### Abstract

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 [Bollobas15], the critical ones are the most studied.

Together with the companion paper by Marêché and the author
[Hartarsky22univlower], 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 [Martinelli19a, Hartarsky21a, Hartarsky20].
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 [Bollobas14] and the two in previous rougher results
[Martinelli19a, Hartarsky21a, Hartarsky20]. 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 [Hartarsky20FA].

MSC2020: Primary 60K35; Secondary 82C22, 60J27, 60C05
Keywords: Kinetically constrained models, interacting particle systems,
universality classes, Glauber dynamics, Poincaré inequality

###### Contents

  1. 1 Introduction
    1. 1.1 Models
    2. 1.2 Universality setting
    3. 1.3 Results
    4. 1.4 Organisation
  2. 2 Mechanisms
    1. 2.1 Scales
    2. 2.2 Microscopic dynamics
    3. 2.3 One-directional extensions
    4. 2.4 Internal dynamics
    5. 2.5 Mesoscopic dynamics
    6. 2.6 Global dynamics
    7. 2.7 Assembling the components
  3. 3 Preliminaries
    1. 3.1 Harris inequality
    2. 3.2 Directions
    3. 3.3 Droplets
    4. 3.4 Scales
    5. 3.5 Helping sets
    6. 3.6 Constrained Poincaré inequalities
    7. 3.7 Boundary conditions, translation invariance, monotonicity
  4. 4 One-directional extensions
    1. 4.1 Traversability
    2. 4.2 East-extension
    3. 4.3 CBSEP-extension
    4. 4.4 Conditional probability tools
  5. 5 Isotropic models
    1. 5.1 Isotropic internal and mesoscopic dynamics
    2. 5.2 CBSEP global dynamics
  6. 6 Unbalanced unrooted models
    1. 6.1 Unbalanced internal dynamics
    2. 6.2 CBSEP mesoscopic dynamics
    3. 6.3 CBSEP global dynamics
  7. 7 Semi-directed models
    1. 7.1 East internal dynamics
    2. 7.2 CBSEP mesoscopic dynamics
    3. 7.3 Global CBSEP dynamics
  8. 8 Balanced rooted models with finite number of stable directions
    1. 8.1 East internal dynamics
    2. 8.2 FA-1f global dynamics
  9. 9 Balanced models with infinite number of stable directions
    1. 9.1 East mesoscopic dynamics
    2. 9.2 East global dynamics
  10. A Extensions
    1. A.1 Auxiliary two-block chain
    2. A.2 Microscopic dynamics
    3. A.3 Proofs of the one-directional extensions
  11. B Conditional probabilities
  12. C The surplus factor for balanced rooted models with finite number of stable directions.

##  1 Introduction

Kinetically constrained models (KCM) are interacting particle systems. They
have challenging features including non-ergodicity, non-attractiveness, hard
constraints, cooperative dynamics and dramatically diverging time scales. This
prevents the use of conventional mathematical tools in the field.

KCM originated in physics in the 1980s [Fredrickson84, Fredrickson85] as toy
models for the liquid-glass transition, which is still a hot and largely open
topic for physicists [Arceri21]. 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 [Arceri21, Garrahan11, Ritort03].

Bootstrap percolation is the natural monotone deterministic counterpart of KCM
(see [Morris17] 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 [Cancrini08],
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 [Martinelli19, Martinelli19a,
Mareche20Duarte, Hartarsky20, Hartarsky22univlower, Hartarsky21a,
Hartarsky20FA] 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.

###  1.1 Models

Let us introduce the class of $\mathcal{U}$-KCM introduced in [Cancrini08]. In
$d\geqslant 1$ dimensions an _update family_ is a nonempty finite collection
of finite nonempty subsets of ${\mathbb{Z}}^{d}\setminus\\{0\\}$ called
_update rules_. The $\mathcal{U}$-KCM is a continuous time Markov chain with
state space ${\Omega}=\\{0,1\\}^{{\mathbb{Z}}^{d}}$. Given a configuration
${\eta}\in{\Omega}$, we write ${\eta}_{x}$ for the state of
$x\in{\mathbb{Z}}^{d}$ in ${\eta}$ and say that _$x$ is infected_ (in
${\eta}$) if ${\eta}_{x}=0$. We write ${\eta}_{A}$ for the restriction of
${\eta}$ to $A\subset{\mathbb{Z}}^{d}$ and $\mathbf{0}_{A}$ for the completely
infected configuration with $A$ omitted when it is clear from the context. We
say that _the constraint at $x\in{\mathbb{Z}}^{d}$ is satisfied_ if there
exists an update rule $U\in\mathcal{U}$ such that $x+U:=\\{x+y:y\in U\\}$ is
fully infected. We denote the corresponding indicator by

$c_{x}({\eta})={\mathbbm{1}}_{\exists
U\in\mathcal{U},{\eta}_{x+U}=\mathbf{0}}.$ (1)

The final parameter of the model is its _equilibrium density of infections_
$q\in[0,1]$. We denote by ${\mu}$ the product measure such that
${\mu}({\eta}_{x}=0)=q$ for all $x\in{\mathbb{Z}}^{d}$ and by
$\operatorname{Var}$ the corresponding variance. Given a finite set
$A\subset{\mathbb{Z}}^{d}$ and real function
$f:{\Omega}\rightarrow{\mathbb{R}}$, we write ${\mu}_{A}(f)$ for the average
${\mu}(f({\eta})|{\eta}_{{\mathbb{Z}}^{d}\setminus A})$ of $f$ over the
variables in $A$. We write $\operatorname{Var}_{A}(f)$ for the corresponding
conditional variance, which is thus also a function from
${\Omega}_{{\mathbb{Z}}^{d}\setminus A}$ to ${\mathbb{R}}$, where
${\Omega}_{B}=\\{0,1\\}^{B}$ for $B\subset{\mathbb{Z}}^{d}$.

With this notation the $\mathcal{U}$-KCM can be formally defined via its
generator

$\mathcal{L}(f)({\eta})=\sum_{x\in{\mathbb{Z}}^{d}}c_{x}({\eta})\cdot\left({\mu}_{x}(f)-f\right)({\eta})$
(2)

and its Dirichlet form reads

$\mathcal{D}(f)=\sum_{x\in{\mathbb{Z}}^{d}}{\mu}\left(c_{x}\cdot\operatorname{Var}_{x}(f)\right),$

where ${\mu}_{x}$ and $\operatorname{Var}_{x}$ are shorthand for
${\mu}_{\\{x\\}}$ and $\operatorname{Var}_{\\{x\\}}$. Alternatively, the
process can be defined via a graphical representation as follows (see
[Liggett05] for background). Each site $x\in{\mathbb{Z}}^{d}$ 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 ${\eta}_{x}$ 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 ${\mu}$ is a reversible measure for the process (there are
others, e.g. the Dirac measure on the fully non-infected configuration
$\mathbf{1}$).

Our regime of interest is $q\rightarrow 0$, corresponding to the low
temperature limit relevant for glasses. A quantitative observable, measuring
the speed of the dynamics, is the infection time of $0$

${\tau}_{0}=\inf\left\\{t\geqslant 0:{\eta}_{0}(t)=0\right\\},$

where $({\eta}(t))_{t\geqslant 0}$ denotes the $\mathcal{U}$-KCM process. More
specifically, we are interested in its expectation for the stationary process
${\mathbb{E}}_{{\mu}}[{\tau}_{0}]$, namely the process with random initial
condition distributed according to ${\mu}$. 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.

$\mathcal{U}$-bootstrap percolation is essentially the $q=1$ case of
$\mathcal{U}$-KCM started out of equilibrium, from a product measure with
$q_{0}\rightarrow 0$ density of infections. More conventionally, it is defined
as a synchronous cellular automaton, which updates all sites of
${\mathbb{Z}}^{d}$ 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 $[A]_{\mathcal{U}}$ the
set of sites eventually infected by the $\mathcal{U}$-bootstrap percolation
process with initial condition $A\subset{\mathbb{Z}}^{d}$, that is, the sites
which can become infected in the $\mathcal{U}$-KCM in finite time starting
from ${\eta}(0)=({\mathbbm{1}}_{x\not\in A})_{x\in{\mathbb{Z}}^{d}}$. 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 [Bollobas14, Bollobas15].

###  1.2 Universality setting

We direct the reader to the companion paper by Marêché and the author
[Hartarsky22univlower], as well as a monograph of Toninelli [Toninelli21] and
the author’s PhD thesis [Hartarsky22phd]*Chap. 1, 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 [Gravner99,
Bollobas15, Bollobas14, Martinelli19a, Hartarsky21a, Hartarsky22univlower] and
may differ in phrasing from the originals, but are usually equivalent thereto
(see [Hartarsky22univlower] for more details).

Henceforth, we restrict our attention to models in two dimensions. The
Euclidean norm and scalar product are denoted by $\|\cdot\|$ and
$\langle\cdot,\cdot\rangle$, and distances are w.r.t. $\|\cdot\|$. Let
$S^{1}=\\{x\in{\mathbb{R}}^{2}:\|x\|=1\\}$ be the unit circle consisting of
_directions_ , which we occasionally identify with
${\mathbb{R}}/2\pi{\mathbb{Z}}$ in the standard way. We denote the open half
plane with outer normal $u\in S^{1}$ and offset $l\in{\mathbb{R}}$ by

${\mathbb{H}}_{u}(l)=\left\\{x\in{\mathbb{R}}^{2}:\langle
x,u\rangle<l\right\\}$

and omit $l$ if it is $0$. We further denote its closure by
$\overline{\mathbb{H}}_{u}(l)$, omitting zero offsets. We often refer to
continuous sets such as ${\mathbb{H}}_{u}$, but whenever talking about
infections or sites in them, we somewhat abusively identify them with their
intersections with ${\mathbb{Z}}^{2}$ without further notice.

Fix an update family $\mathcal{U}$.

######  Definition 1.1 (Stability).

A direction $u\in S^{1}$ is _unstable_ if there exists $U\in\mathcal{U}$ such
that $U\subset{\mathbb{H}}_{u}$ 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.

######  Definition 1.2 (Criticality).

Let $\mathcal{C}$ be the set of open semicircles of $S^{1}$. An update family
is

  * •

_supercritical_ if there exists $C\in\mathcal{C}$ such that all $u\in C$ are
unstable;

  * •

_subcritical_ if every semicircle contains infinitely many stable directions;

  * •

_critical_ otherwise.

The following notion measures “how stable” a stable direction is.

######  Definition 1.3 (Difficulty).

For $u\in S^{1}$ the _difficulty_ ${\alpha}(u)$ of $u$ is

  * •

$0$ if $u$ is unstable;

  * •

$\infty$ if $u$ is stable, but not isolated;

  * •

$\min\\{n:\exists Z\subset{\mathbb{Z}}^{2},|Z|=n,|[{\mathbb{H}}_{u}\cup
Z]_{\mathcal{U}}\setminus{\mathbb{H}}_{u}|=\infty\\}$ otherwise.

The _difficulty_ of $\mathcal{U}$ is

${\alpha}=\min_{C\in\mathcal{C}}\max_{u\in C}{\alpha}(u).$

We say that a direction $u\in S^{1}$ is _hard_ if ${\alpha}(u)>{\alpha}$.

See Fig. 2 for an example update family with ${\alpha}=3$ and its
difficulties. It can be shown that a model is critical iff $0<{\alpha}<\infty$
and supercritical iff ${\alpha}=0$, so difficulty is tailored for critical
models and refines Definition 1.2. Moreover, ${\alpha}(u)\in[1,\infty)$ for
isolated stable directions. Furthermore, for supercritical models the notions
of stable and hard direction coincide. Finally, observe that the definition
implies that for any critical or supercritical update family there exists an
open semicircle with no hard direction.

######  Definition 1.4 (Refined types).

A critical or supercritical update family is

  * •

_rooted_ if there exist two non-opposite hard directions;

  * •

_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
[Hartarsky22univlower]*Fig. 1 for simple 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. 2).

Let us remark that models in each class may have one axial symmetry, but non-
subcritical models invariant under rotation by $\pi$ are necessarily either
isotropic or unbalanced unrooted (thus with finite number of stable
directions), while invariance by rotation by $\pi/2$ implies isotropy.

###  1.3 Results

Our result, summarised in Table 1, together with the companion paper by
Marêché and the author [Hartarsky22univlower], 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 [Martinelli19a, Mareche20Duarte, Mareche20combi]).

| Infinite stable directions | Finite stable directions
---|---|---
| Rooted | Unrooted
Unbalanced |  (a) $2,4,0$ |  (c) $1,3,0$ |  (d) $1,2,0$
Balanced |  (b) $2,0,0$ |  (e) $1,1,0$ |  (f) $1,0,1$ S.-dir. Iso. (g) $1,0,0$
Table 1: Classification of critical $\mathcal{U}$-KCM with difficulty
$\alpha$. For each class
${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\Theta(1)\left(\frac{1}{q^{\alpha}}\right)^{\beta}\left(\log\frac{1}{q}\right)^{\gamma}\left(\log\log\frac{1}{q}\right)^{\delta}\right)$
as $q\rightarrow 0$. The label of the class and the exponents
$\beta,\gamma,\delta$ are indicated in that order.

######  Theorem 1 (Universality classification of two-dimensional critical
KCM).

.
Let $\mathcal{U}$ be a two-dimensional critical update family with difficulty
$\alpha$. We have the following exhaustive alternatives as $q\rightarrow 0$
for the expected infection time of the origin under the stationary
$\mathcal{U}$-KCM.111We write $f=\Theta(g)$ if there exist $c,C>0$ such that
$cg(q)<f(q)<Cg(q)$ for all $q$ small enough and use other standard asymptotic
notation (see e.g. [Hartarsky22univlower]*Section 1.2). If $\mathcal{U}$ is

  1. (a)

unbalanced with infinite number of stable directions (so rooted), then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta\left(\left(\log(1/q)\right)^{4}\right)}{q^{2\alpha}}\right);$

  2. (b)

balanced with infinite number of stable directions (so rooted), then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta(1)}{q^{2\alpha}}\right);$

  3. (c)

unbalanced rooted with finite number of stable directions, then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta\left(\left(\log(1/q)\right)^{3}\right)}{q^{\alpha}}\right);$

  4. (d)

unbalanced unrooted (so with finite number of stable directions), then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta\left(\left(\log(1/q)\right)^{2}\right)}{q^{\alpha}}\right);$

  5. (e)

balanced rooted with finite number of stable directions, then222See Remark
1.6.

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta\left(\log(1/q)\right)}{q^{\alpha}}\right);$

  6. (f)

semi-directed (so balanced unrooted with finite number of stable directions),
then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta\left(\log\log(1/q)\right)}{q^{\alpha}}\right);$

  7. (g)

isotropic (so balanced unrooted with finite number of stable directions), then

${\mathbb{E}}_{{\mu}}[{\tau}_{0}]=\exp\left(\frac{\Theta(1)}{q^{\alpha}}\right).$

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 [Hartarsky22univlower] 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
[Martinelli19, Martinelli19a, Hartarsky21a, Hartarsky20]. Moreover,
particularly for the lower bounds, the classification of two-dimensional
critical $\mathcal{U}$-bootstrap percolation models by Bollobás, Duminil-
Copin, Morris and Smith [Bollobas14] (featuring only the balanced/unbalanced
distinction) is heavily used, while upper bounds additionally use
prerequisites from [Hartarsky22CBSEP, Hartarsky20FA]. Thus, a fully self-
contained proof of 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
[Bollobas14] in [Martinelli19]; the lower bound for class (b) was established
in [Hartarsky20], while the remaining four were proved in
[Hartarsky22univlower]. Turning to upper bounds, the one for class (a) was
given in [Martinelli19a] and the one for class (c) is due to [Hartarsky21a].
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,
[Hartarsky21a]*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 [Martinelli19a, Morris17], which were
disproved by the unexpected result of [Hartarsky21a], and was new to
physicists, as well as mathematicians.

######  Remark 1.5.

It should be noted that universality results including 1 apply to KCM more
general than the ones defined in Section 1.1. Namely, we may replace $c_{x}$
in Eq. 2 by a fixed linear combination of the constraints $c_{x}$ associated
to any finite set of update families. For instance, we may update vertices at
rate proportional to their number of infected neighbours. This and other
models along these lines have been considered e.g. in [Fredrickson84,
Blanquicett21, Alves21a]. For such mixtures of families, the universality
class is determined by the family obtained as their union. Indeed, upper
bounds follow by direct comparison of the corresponding Dirichlet forms, while
lower bounds (e.g. [Hartarsky22univlower]) generally rely on deterministic
bottlenecks, which remain valid.

######  Remark 1.6.

Let us note that for reasons of extremely technical nature, we do not provide
a full proof of (the upper bound of) 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

$\exp\left(\frac{O(\log(1/q))\log\log\log(1/q)}{q^{\alpha}}\right).$ (3)

Furthermore, with very minor modifications (see Remark 7.1), 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 in Appendix C.

###  1.4 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.

In Section 4 we formally state the two fundamental techniques we use to move
from one scale to the next, namely East-extensions and CBSEP-extensions, which
import and generalise ideas of [Hartarsky20FA]. 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 treated by [Hartarsky21b]
are deferred to Appendix A, since they are quite technical and do not require
new ideas. The bounds arising from extensions feature certain conditional
expectations. We provide technical tools for estimating them in Section 4.4.
We leave the entirely new proofs of these general analogues of
[Hartarsky20FA]*Appendix A to Appendix B.

Sections 5, 6, 7, 8 and 9 are the core of our work and use the extensions
mentioned above to prove the upper bounds of 1 for classes (g), (d), (f), (e),
(b) respectively. As we will discuss in further detail (see Sections 2 and
2(b)), some parts of the proofs will be common to several of these classes,
making the sections interdependent. Thus, they are intended for linear
reading.

We conclude in Appendix C by explaining how to remove the corrective
$\log\log\log(1/q)$ factor discussed in Remark 1.6 to recover the result of
1(e) as stated in full generality. Due to their technical nature, we delegate
Appendices A, C and B to the arXiv version of the present work.

Familiarity with the companion paper [Hartarsky22univlower] or bootstrap
percolation [Bollobas14] is not needed. Inversely, familiarity with
[Hartarsky21a, Hartarsky20FA] is strongly recommended for going beyond Section
2 and achieving a complete view of the proof of the upper bounds of 1.
Nevertheless, we will systematically state the implications of intermediate
results of those works for our setting in a self-contained fashion, without
re-proving them.

##  2 Mechanisms

Global | Mesoscopic | Internal
---|---|---
CBSEP | East | CBSEP | East, Stair | CBSEP | East | Unbal.
${\rho}_{\mathrm{D}}^{-1+o(1)}$ | ${\rho}_{\mathrm{D}}^{-O(\log(1/{\rho}_{\mathrm{D}}))}$ | $e^{q^{-o(1)}}$ | ${\rho}_{\mathrm{D}}^{-O(\log(1/q))}$ | $e^{q^{-o(1)}}$ | ${\rho}_{\mathrm{D}}^{-O(\log\log(1/q))}$ | ${\rho}_{\mathrm{D}}^{-O(1)}$
(a) The relaxation time cost associated to each choice of dynamics mechanism
on each scale in terms of the probability of a droplet ${\rho}_{\mathrm{D}}$.

| (a) | (b) | (c) | (d) | (e) | (f) | (g)
---|---|---|---|---|---|---|---
Global | East* | East* | CBSEP | CBSEP* | CBSEP | CBSEP | CBSEP*
Mesoscopic | Stair | East | East* | CBSEP | East* | CBSEP | CBSEP
Internal | — | East | Unbal. | Unbal.* | East | East* | CBSEP
(b) The fastest mechanism available to each class of 1 on each scale. The *
indicates a leading contribution for the class (column).

Table 2: Summary of the mechanisms and their costs. The microscopic one common
to all classes and with negligible cost is not shown (see Section 2.2).

In this section we attempt a heuristic explanation of 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 [Hartarsky22univlower] 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. ${\mu}$. 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. 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.

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.

In order to guide the reader through Section 2 and beyond, in Table 2, we
summarise the optimal mechanisms for each universality class on each scale and
its cost. While its full meaning will only become clear in Section 2.7, the
reader may want to consult it regularly, as they progress through Section 2.

The SG events will concern certain convex polygonal geometric regions called
_droplets_. These events will be designed so as to satisfy several conditions
ensuring that the configuration of infections inside the droplet is sufficient
to infect the entire droplet. The SG events defined by extensions from smaller
to larger scales will require the presence of a lower scale droplet inside the
large one (see Fig. 3) in addition to well-chosen more sparse infections
called _helping sets_ in the remainder of the larger scale droplet (see Fig.
2). Helping sets allow the smaller one to move inside the bigger one.

We will say that a droplet _relaxes_ in a certain _relaxation time_ if the
dynamics restricted to the SG event and to this region “mixes” in this much
time. 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. Hence, we
seek a balance between asking for many infections to make the movement
efficient and asking for few in order to keep the probability of droplets high
enough.

###  2.1 Scales

##### Microscopic dynamics

refers to modifying infections at the level of the lattice _along the boundary
of a droplet_ , while respecting the KCM constraint.

##### Internal dynamics

refers to relaxation on scales from the lattice level to the _internal scale_
$\ell^{\mathrm{i}}=C^{2}\log(1/q)/q^{\alpha}$, where $C$ is a large constant
depending on $\mathcal{U}$. This is the most delicate and novel step. Up to
$\ell^{\mathrm{i}}$ we account for the main contribution to the probability of
droplets. That is, at all larger scales the probability of a droplet
essentially saturates at a certain value ${\rho}_{\mathrm{D}}$, because
finding helping sets becomes likely. Thus, on smaller scales, it is important
to only very occasionally ask for more than ${\alpha}$ infections to appear
close to each other in order to get the right probability
${\rho}_{\mathrm{D}}$. This means that up to the internal scale hard
directions are practically impenetrable, since they require helping sets of
more that ${\alpha}$ infections.

##### Mesoscopic dynamics

refers to relaxation on scales from $\ell^{\mathrm{i}}$ to the _mesoscopic
scale_ $\ell^{\mathrm{m}}=1/q^{C}$. 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

refers to relaxation on scales from $\ell^{\mathrm{m}}$ 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
$\ell^{\mathrm{g}}=\exp(1/q^{3{\alpha}})$, which is notably much larger than
all time scales we are aiming for, but otherwise rather arbitrary.

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 the only four
relevant directions are the axis ones so that droplets have rectangular shape
(see Section 3.3). We further assume that all directions in the left
semicircle have difficulties at most ${\alpha}$, while the down direction is
hard, unless there are no hard directions (isotropic class).

###  2.2 Microscopic dynamics

The microscopic dynamics (see Section A.2) 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, the outer boundary of
the droplet behaves like a 1-dimensional KCM with update family reflecting
that we view the droplet as infected. Hence, provided there are enough helping
sets at the boundary to infect it, we can apply results on 1-dimensional KCM
supplied for this purpose by the author [Hartarsky21b].

This way we establish that one additional column can relax in time
$\exp(O(\log(1/q))^{2})$, similarly to the East model described in Section
2.3.2 below. 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.

###  2.3 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 (see
Section 4).

As mentioned above, our droplets are polygonal regions with a SG event
(presence of a suitable arrangement of infections in the droplet). An
extension takes as input a droplet and produces another one. In terms of
geometry, it contains the original one and is obtained by extending it, say,
horizontally, either to the left or both left and right (see Fig. 3). The
extended droplet’s SG event requires that the smaller one is SG and,
additionally, certain helping sets appear in the remaining volume. The choice
of where we position the smaller droplet (at the right end of the bigger one,
or anywhere inside it) depends on the type of extension. The additional
helping sets are required in such a way that, with their help, the smaller
droplet can, in principle, completely infect the larger one and, therefore,
make it relax (resample its configuration within its SG event).

Thus, an _extension_ is a procedure for iteratively defining SG events on
larger and larger scales. For each of our two types of extensions we need to
provide corresponding iterative bounds on the probability of the SG event and
on the relaxation time of droplets on this event. The former is a matter of
careful computation. For the latter task we will intuitively use a large-scale
version of an underlying one-dimensional spin model, which we describe first.

####  2.3.1 CBSEP-extension

In the one-dimensional spin version of CBSEP [Hartarsky20FA, Hartarsky22CBSEP]
we work on $\\{\uparrow,\downarrow\\}^{\mathbb{Z}}$. At rate 1 we resample
each pair of neighbouring spins, provided that at least one of them is
$\uparrow$. Their state is resampled w.r.t. the reference product measure,
which will be reversible, conditioned to still have a $\uparrow$ in at least
one of the two sites. In other words, $\uparrow$ can perform coalescence,
branching and symmetric simple exclusion moves, hence the name. The relaxation
time of this model on volume $V$ is roughly $\min(V,1/q)^{2}$ in one dimension
and $\min(V,1/q)$ in two and more dimensions [Hartarsky22CBSEP,
Hartarsky20FA], where $q$ is the equilibrium density of $\uparrow$, which we
think of as being small.

For us $\uparrow$ 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 a finite interval of
${\mathbb{Z}}$ with a state space corresponding to the state of a column of
the height of our droplet of interest in the original lattice
${\mathbb{Z}}^{2}$. Then the event “there is a SG droplet” may occur on a
group of $\ell$ consecutive sites (columns). The long range generalised CBSEP,
which, abusing notation, we will call CBSEP, is defined as follows. We fix
some _range_ $R>\ell$ and resample at rate 1 each group of $R$ consecutive
sites, if they contain a SG droplet. The resampling is performed conditionally
on preserving the presence of a SG droplet in those $R$ sites. Thus, one move
of this process essentially delocalises the droplet within the range.

It is important to note (and this was crucial in [Hartarsky20FA]) that CBSEP
does not _have to_ create an additional droplet in order to evolve. Since SG
droplets are unlikely, it suffices to move an initially available SG droplet
through our domain in order to relax. Since infection needs to be able to
propagate both left and right from the SG droplets, we will define (see
Section 4.3 and particularly Definitions 4.7 and 3(b)) _CBSEP-extension_ by
extending our domain horizontally and asking for the SG droplet anywhere
inside with suitable “rightwards-pointing” helping sets on its right and
“leftwards-pointing” on its left.

While we now know that droplets will evolve according to CBSEP, it remains to
see how one can reproduce one CBSEP move via the original dynamics. This is
done inductively on $R$ by a bisection procedure, the trickiest part being the
case $R=\ell+1$. We then dispose with a droplet plus one column—exactly the
setting for microscopic dynamics. However, we not only want to resample the
state of the additional column, but also allow the droplet to move by one
lattice step. To achieve this, we have to look inside the structure of the SG
droplet and require for its infections (which have no rigid structure and may
therefore move around like the organelles of an amoeba) to be somewhat more on
the side we want to move towards (see e.g. Fig. 5 and also Definitions 5.3,
7.7, 7.8 and 6.5). Then, together with a suitable configuration on the
additional column provided by the microscopic dynamics, we easily recover our
SG event shifted by one step, since most of the structure was already provided
by the version of the SG event “contracted” towards the new column.

This amoeba-like motion (moving a droplet, by slightly rearranging its
internal structure) leads to a very small relaxation time of the dynamics.
Indeed, the time needed to move the droplet is the product of three
contributions: the relaxation time of the 1-dimensional spin model; the
relaxation time of the microscopic dynamics; the time needed to see a droplet
contracting as explained above (see Proposition 4.9). The first of these is a
power of the volume (number of sites); the second is
$\exp(O(\log(1/q)))^{2})$; the third is also small, as we will discuss in
Section 2.3.2.

However, CBSEP-extensions can only be used for sufficiently symmetric update
families. That is, the droplet needs to be able to move indifferently both
left and right and its position should not be biased in one direction or the
other. Specifically, if we are working on a scale that requires the use of
helping sets of size $\alpha$, these have to exist both for the left and right
directions, so the model needs to be unrooted (if instead we use larger
helping sets, having a finite number of stable directions suffices). The
reason is that otherwise the position of the SG droplet is biased in one
direction instead of being approximately uniform. This would break the analogy
with the original one-dimensional spin model, which is totally symmetric. When
symmetry is not available, we recourse to the East-extension presented next,
which may also be viewed as a totally asymmetric version of the CBSEP-
extension.

####  2.3.2 East-extension

The East model [Jackle91] is the one-dimensional KCM with
$\mathcal{U}=\\{\\{1\\}\\}$. That is, we are only allowed to resample the left
neighbour of an infection. An efficient recursive mechanism for its relaxation
is the following [Mauch99]. 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 $q^{-O(\log\min(L,1/q))}$ [Aldous02,
Cancrini08, Chleboun14], where $q$ is the equilibrium density of infections.
This corresponds to the cost of $n$ infections when $2^{n}\sim\min(L,1/q)$ is
the typical distance to the nearest infection.

The long-range generalised version of the East model is defined similarly to
that of CBSEP. The only difference is that now $R>\ell$ 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 _East-extension_ of a droplet corresponds to extending its geometry
to the left (see Section 4.2 and particularly Definitions 4.4 and 3(a)). The
extended SG event requires that the original SG droplet is present in the
rightmost position and “leftwards-pointing” helping sets are available in the
rest of the extended droplet.

The generalised East process goes back to [Martinelli19a], while the long
range version is implicitly used in [Hartarsky21a]. 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 for semi-
directed models. Indeed, take $\ell$ large and $R=\ell+5$. 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 e.g. [Martinelli19a,
Hartarsky21a]) 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 (unlike in [Hartarsky20FA]), 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 a droplet in the CBSEP-
extension—we need to evaluate the probability of a contracted version of the
droplet, conditionally on the original droplet being present.

We deal with these issues in Section 4.4 (see also Appendix B). We establish
that, as intuition may suggest, to create a droplet shifted by $R-\ell$, given
the original one, we roughly only need to pay the probability of a droplet on
scale $R-\ell$ rather than $\ell$, 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 (see Proposition 4.6).

###  2.4 Internal dynamics

The internal dynamics (see Sections 8.1, 7.1, 5.1 and 6.1) is where most of
our work will go. This is not surprising, as the probability of SG events will
saturate at its final value ${\rho}_{\mathrm{D}}$ at the internal scale. The
value of ${\rho}_{\mathrm{D}}$ is given by $\exp(-O(1)/q^{\alpha})$ for
balanced models and $\exp(-O(\log(1/q))^{2}/q^{\alpha})$ for unbalanced ones,
as in bootstrap percolation [Bollobas14]. However, relaxation times for some
classes will keep growing past the internal scale, so the internal dynamics
will not necessarily give the final answer in 1 (see Table 2(b)).

####  2.4.1 Unbalanced internal dynamics

Let us begin with the simplest case of unbalanced models. If $\mathcal{U}$ is
unbalanced with infinite number of stable directions (class (a)), droplets in
[Martinelli19a] 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 $\ell^{\mathrm{i}}$, which justifies the value of
${\rho}_{\mathrm{D}}=q^{-O(\ell^{\mathrm{i}})}=\exp(-O(\log(1/q))^{2}/q^{\alpha})$.

Assume $\mathcal{U}$ is unbalanced with finite number of stable directions
(classes (c) and (d), see Section 6.1). Then droplets on the internal scale
will be fully infected square frames of thickness $O(1)$ and size
$\ell^{\mathrm{i}}$ (see [Hartarsky21a]*Figs. 2-4 or Fig. 6 for more general
geometry), which gives a similar value of ${\rho}_{\mathrm{D}}$. In order to
relax inside the frame, one can divide its interior into groups of $O(1)$
consecutive columns (see [Hartarsky21a]*Fig. 8). We can then view them as
performing a CBSEP dynamics with $\uparrow$ corresponding to a fully infected
group of columns. This is possible, because with the help of the frame each
completely infected group is able to completely infect the neighbouring ones.
Here we are using that there are finitely many stable directions to ensure
both the left and right directions have finite difficulty, so finite-sized
helping sets, as provided by the frame, are sufficient to propagate our group
of columns. This was already done in [Hartarsky21a] and the time necessary for
this relaxation is easily seen to be ${\rho}_{\mathrm{D}}^{-O(1)}$ (the cost
for creating a group of infected columns)—see Proposition 6.2.

####  2.4.2 CBSEP internal dynamics

If $\mathcal{U}$ is isotropic (class (g), see Section 5.1), up to the
conditioning problems of Section 4.4 described above, we need only minor
adaptations of the strategy of [Hartarsky20FA] for the paradigmatic isotropic
model called FA-2f. Droplets on the internal scale will have an internal
structure as obtained by iterating Fig. 5(a) (see also [Hartarsky20FA]*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 $\log(1/q)$
extensions are required to reach $\ell^{\mathrm{i}}$. 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 $\exp(q^{-o(1)})$—see Theorem
5.2.333Note that in [Hartarsky20FA]*Proposition 4.7 a much larger internal
relaxation time of order $\exp(q^{-1/2+o(1)})$ was obtained, but the cost
${\rho}_{\mathrm{D}}^{-1}$ of SG droplets was much smaller than the one in the
present work, so our treatment here is by no means as sharp for FA-2f as
[Hartarsky20FA].

######  Remark 2.1.

Note that for CBSEP-extensions to be used, we need a very strong symmetry.
Namely, leftwards and rightwards pointing helping sets should be the same up
to rotation by $\pi$. Yet, for a general isotropic model we only know that
there are no hard directions, so helping sets have the same size (equal to the
difficulty $\alpha$ of the model), but not necessarily the same shape. We
circumvent this issue by artificially symmetrising our droplets and events.
Namely, whenever we require helping sets in one direction, we also require the
helping sets for the opposite direction rotated by $\pi$ (see Definitions 3.8,
4.1 and 4.7). 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 (see Lemma
4.10).

####  2.4.3 East internal dynamics

The most challenging case is the balanced non-isotropic one (classes (b), (e)
and (f)). It is treated in Sections 7.1 and 8.1, but for the purposes of the
present section only Section 7.1 is relevant. This is because we assume that
only the four axis directions are relevant and our droplets are rectangular.
The treatment of the general case for balanced rooted families is left to
Sections 8.1 and C (recall Remark 1.6).

For the internal dynamics the downwards hard direction prevents us from using
CBSEP-extensions. To be precise, for semi-directed models (class (f)) it is
possible to perform CBSEP-extensions horizontally (and not vertically), but
the gain is insignificant, so we treat all balanced non-isotropic models
identically up to the internal scale as follows.

We still extend droplets, starting from a microscopic one, by a constant
factor alternating between the horizontal and vertical directions (see Fig.
7). However, in contrast with the isotropic case (see Fig. 5(a)), extensions
are done in an oriented fashion, so that the original microscopic droplet
remains anchored at the corner of larger ones. 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 $2^{n}$:

$\prod_{n=1}^{\log_{2}(\ell^{\mathrm{i}})}\prod_{m=0}^{n}a_{m}^{(n)},$ (4)

where $a_{m}^{(n)}$ is the conditional probability of a SG droplet of size
$2^{n}$ being present position $2^{m}$, given that a SG droplet of size
$2^{n}$ is present at position 0. It is crucial that Eq. 4 is _not_ the
straightforward bound $\prod_{n}({\rho}_{\mathrm{D}}^{(n)})^{-n}$, with
${\rho}_{\mathrm{D}}^{(n)}$ the probability of a droplet of scale $n$, that
one would get by direct analogy with the East model (recall from Section 2.3.2
that the relaxation time of East on a small volume $L$ is $q^{-\log L}$), as
that would completely devastate all our results. Indeed, as mentioned in
Section 2.3.2, the term $a_{n}^{(m)}$ in Eq. 4 is approximately equal to
${\rho}_{\mathrm{D}}^{(m)}$, rather than ${\rho}_{\mathrm{D}}^{(n)}$. This is
perhaps one of the most important points to our treatment.

In total, a droplet of size $2^{n}$ needs to be paid once per scale larger
than $2^{n}$ (see Eq. 41). A careful computation shows that only droplets
larger than $q^{-{\alpha}}$ provide the dominant contribution and those all
have probability essentially ${\rho}_{\mathrm{D}}=\exp(-O(1)/q^{\alpha})$ (see
Eq. 42). Thus, the total cost would be

$\prod_{n=\log(1/q^{\alpha})}^{\log(\ell^{\mathrm{i}})}\prod_{m=\log(1/q^{\alpha})}^{n}{\rho}_{\mathrm{D}}={\rho}_{\mathrm{D}}^{-O(\log\log(1/q))^{2}}=\exp\left(O(\log\log(1/q))^{2}/q^{\alpha}\right),$
(5)

since there are $\log\log(1/q)$ scales from $q^{-{\alpha}}$ to
$\ell^{\mathrm{i}}$, as they increase exponentially.

Equation 5 is unfortunately a bit too rough for the semi-directed class,
overshooting 1(f). However, the solution is simple. It suffices to introduce
scales growing double-exponentially above $q^{-{\alpha}}$ instead of
exponentially (see Eq. 36), so that the product over scales $n$ in Eq. 5
becomes dominated by its last term, corresponding to droplet size
$\ell^{\mathrm{i}}$. This gives the optimal final cost

${\rho}_{\mathrm{D}}^{-\log(q^{\alpha}\ell^{\mathrm{i}})}={\rho}_{\mathrm{D}}^{-O(\log\log(1/q))}=\exp\left(O(\log\log(1/q))/q^{\alpha}\right)$

(see Theorem 7.3).

###  2.5 Mesoscopic dynamics

For the mesoscopic dynamics (see Sections 5.1, 9.1, 6.2 and 7.2) we are given
as input a SG event for droplets on scale
$\ell^{\mathrm{i}}=C^{2}\log(1/q)/q^{\alpha}$ and a bound on their relaxation
time and occurrence probability ${\rho}_{\mathrm{D}}$. We seek to output the
same on scale $\ell^{\mathrm{m}}=q^{-C}$. Taking $C\gg W$, once our droplets
have size $\ell^{\mathrm{m}}$, we will be able to find $W$-helping sets (sets
of $W$ consecutive infections, where $W$ is large enough).

####  2.5.1 CBSEP mesoscopic dynamics

If $\mathcal{U}$ is unrooted (classes (d), (f) and (g), see Sections 6.2 and
7.2), recall that the hard directions (if any) are vertical. Then we can
perform a horizontal CBSEP-extension directly from $\ell^{\mathrm{i}}$ to
$\ell^{\mathrm{m}}$, since $\ell^{\mathrm{i}}=C^{2}\log(1/q)/q^{\alpha}$ makes
it likely for helping sets (of size $\alpha$) to appear along all segments of
length $\ell^{\mathrm{i}}$ until we reach scale $\ell^{\mathrm{m}}=q^{-C}$.
The resulting droplet is very wide, but short (see Fig. 6(a)). However, this
is enough for us to be able to perform a vertical CBSEP-extension (see Fig.
6(b)), requiring $W$-helping sets, since they are now likely to be found.
Again, CBSEP dynamics being very efficient, its cost is negligible. Note that,
in order to perform the vertical extension, we are using that there are no
nonisolated stable directions, so that $W$ is larger than the difficulty of
the up and down directions, making $W$-helping sets sufficient to induce
growth in those directions. Thus, morally, there are no hard directions beyond
scale $\ell^{\mathrm{m}}$ for unrooted models.

####  2.5.2 East mesoscopic dynamics

If $\mathcal{U}$ is rooted (classes (a)-(c) and (e), see Section 9.1), CBSEP-
extensions are still inaccessible. We may instead East-extend horizontally
from $\ell^{\mathrm{i}}$ to $\ell^{\mathrm{m}}$ in a single step. If the model
is balanced or has a finite number of stable directions (classes (b), (c) and
(e)), we may proceed similarly in the vertical direction, reaching a droplet
of size $\ell^{\mathrm{m}}$ in time ${\rho}_{\mathrm{D}}^{-O(\log(1/q))}$
(here we use the basic bound $q^{-O(\log L)}$ for East dynamics recalled in
Section 2.3.2, which is fairly tight in this case, since droplets are small
compared to the volume:
$\log\ell^{\mathrm{m}}\approx\log(\ell^{\mathrm{m}}/\ell^{\mathrm{i}})$). For
the unbalanced case (class (c)) here we require $W$-helping sets along the
long side of the droplet like in Section 2.5.1. Another way of viewing this is
simply as extending the procedure used for the East internal dynamics all the
way up to the mesoscopic scale $\ell^{\mathrm{m}}$ (see Section 9.1).

It should be noted that a version of this mechanism, which coincides with the
above for models with rectangular droplets, but differs in general, was
introduced in [Hartarsky21a]. Though their _snail mesoscopic dynamics_ can be
replaced by our East one, for the sake of concision in Section 8.2 we will
directly import the results of [Hartarsky21a] based on the snail mechanism.

####  2.5.3 Stair mesoscopic dynamics

For unbalanced families with infinite number of stable directions (class (a))
the following _stair mesoscopic dynamics_ was introduced in [Martinelli19a].
Recall from Section 2.4.1 that for unbalanced models the internal droplet is
simply a fully infected frame or group of consecutive columns. 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
[Martinelli19a]*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 $\ell^{\mathrm{m}}$ but still
polynomial in $1/q$ and vertical dimension $\ell^{\mathrm{m}}$ in time
${\rho}_{\mathrm{D}}^{-O(\log(1/q))}$. Here, one should again intuitively
imagine we are using the bound $q^{-O(\log L)}$ but this time for the
relaxation time of the 2-dimensional East model.

###  2.6 Global dynamics

The global dynamics (see Sections 8.2, 5.2, 6.3, 7.3 and 9.2) receives as
input a SG event for a droplet on scale $\ell^{\mathrm{m}}$ with probability
roughly ${\rho}_{\mathrm{D}}$ 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
${\rho}_{\mathrm{D}}^{-1/2}$.

####  2.6.1 CBSEP global dynamics

If $\mathcal{U}$ has a finite number of stable directions (classes (c)-(g))
the mesoscopic droplet can perform a CBSEP motion in a typical environment.
Indeed, the droplet is large enough for CBSEP-extensions with $W$-helping sets
to be possible in all directions. Therefore, the cost of this mechanism is
given by the relaxation time of CBSEP on a box of size
$\ell^{\mathrm{g}}=\exp(1/q^{3\alpha})$ with density of $\uparrow$ given by
${\rho}_{\mathrm{D}}$. Performing this strategy carefully and using the
2-dimensional CBSEP, this yields a relaxation time
$\min((\ell^{\mathrm{g}})^{2},1/{\rho}_{\mathrm{D}})=1/{\rho}_{\mathrm{D}}$
(recall Section 2.3.1 and see Section 5.2).

####  2.6.2 East global dynamics

If $\mathcal{U}$ 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 ${\rho}_{\mathrm{D}}$, which yields a relaxation time of
${\rho}_{\mathrm{D}}^{-O(\log\min(\ell^{\mathrm{g}},1/{\rho}_{\mathrm{D}}))}={\rho}_{\mathrm{D}}^{-O(\log(1/{\rho}_{\mathrm{D}}))}$
(recall Section 2.3.2 and see Section 9.2).

###  2.7 Assembling the components

To conclude, let us return to the summary provided in Table 2. In Table 2(a)
we collect the mechanisms for each scale and their cost to the relaxation
time. The results are expressed in terms of the probability of a droplet
${\rho}_{\mathrm{D}}$, which equals $\exp(-O(\log(1/q))^{2}/q^{\alpha})$ for
unbalanced models and $\exp(-O(1)/q^{\alpha})$ for balanced ones. The final
bound on ${\mathbb{E}}_{{\mu}}[{\tau}_{0}]$ for each class then corresponds to
the product of the costs of the mechanism employed at each scale. To
complement this, in Table 2(b) we indicate the fastest mechanism available for
each class on each scale. We further indicate which one gives the dominant
contribution to the final result appearing in 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 2(b) if that does not deteriorate the final result.
Similarly, when that is unimportant, we may give weaker bounds than the ones
in Table 2(a). In Section 8.2 we tacitly import a weaker precursor of the
CBSEP global mechanism from [Hartarsky21a] not listed above.

##  3 Preliminaries

###  3.1 Harris inequality

Let us recall a well-known correlation inequality due to Harris [Harris60]. It
will be used throughout and we state some particular formulations that will be
useful to us.

For Section 3.1 we fix a finite ${\Lambda}\subset{\mathbb{Z}}^{2}$. We say
that an event $\mathcal{A}\subset{\Omega}_{\Lambda}$ is _decreasing_ if adding
infections does not destroy its occurrence.

######  Proposition 3.1 (Harris inequality).

Let $\mathcal{A},\mathcal{B}\subset{\Omega}_{\Lambda}$ be decreasing. Then

${\mu}(\mathcal{A}\cap\mathcal{B})\geqslant{\mu}(\mathcal{A}){\mu}(\mathcal{B}).$
(6)

######  Corollary 3.2.

Let $\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\subset{\Omega}_{\Lambda}$
be nonempty and decreasing events such that $\mathcal{B}$ and $\mathcal{D}$
are independent, then

$\displaystyle{\mu}(\mathcal{A}|\mathcal{B}\cap\mathcal{D})$
$\displaystyle{}\geqslant{\mu}(\mathcal{A}|\mathcal{B})\geqslant{\mu}(\mathcal{A}),$
(7)
$\displaystyle{\mu}(\mathcal{A}\cap\mathcal{C}|\mathcal{B}\cap\mathcal{D})$
$\displaystyle{}\geqslant{\mu}(\mathcal{A}|\mathcal{B}){\mu}(\mathcal{C}|\mathcal{D}).$
(8)

###### Proof.

The first inequality in Eq. 7 is Eq. 8 for $\mathcal{C}=\Omega_{\Lambda}$, the
second follows from Eq. 6 and
${\mu}(\mathcal{A}|\mathcal{B})={\mu}(\mathcal{A}\cap\mathcal{B})/{\mu}(\mathcal{B})$,
while Eq. 8 is

${\mu}(\mathcal{A}\cap\mathcal{C}|\mathcal{B}\cap\mathcal{D})=\frac{{\mu}(\mathcal{A}\cap\mathcal{C}\cap\mathcal{B}\cap\mathcal{D})}{{\mu}(\mathcal{B}\cap\mathcal{D})}\geqslant\frac{{\mu}(\mathcal{A}\cap\mathcal{B}){\mu}(\mathcal{C}\cap\mathcal{D})}{{\mu}(\mathcal{B}){\mu}(\mathcal{D})}={\mu}(\mathcal{A}|\mathcal{B}){\mu}(\mathcal{C}|\mathcal{D}),$

using Eq. 6 in the numerator and independence in the denominator. ∎

We collectively refer to Eqs. 6, 7 and 8 as _Harris inequality_.

###  3.2 Directions

Throughout this work we fix a critical update family $\mathcal{U}$ with
difficulty ${\alpha}$. We call a direction $u\in S^{1}$ _rational_ if
$u{\mathbb{R}}\cap{\mathbb{Z}}^{2}\neq\\{0\\}$. By Definition 1.3 there exists
an open semicircle with rational midpoint $u_{0}$ such that all directions in
the semicircle have difficulty at most ${\alpha}$. We may assume without loss
of generality that the direction $u_{0}+\pi/2$ is hard unless $\mathcal{U}$ is
isotropic. It is not difficult to show (see e.g. [Bollobas15]*Lemma 5.3) that
one can find a nonempty set $\mathcal{S}^{\prime}$ of rational directions such
that:

  * •

all isolated and semi-isolated stable directions are in
$\mathcal{S}^{\prime}$;

  * •

$u_{0}\in\mathcal{S}^{\prime}$;

  * •

for every two consecutive directions $u,v$ in $\mathcal{S}^{\prime}$ either
there exists a rule $X\in\mathcal{U}$ such that
$X\subset\overline{\mathbb{H}}_{u}\cap\overline{\mathbb{H}}_{v}$ or all
directions between $u$ and $v$ are stable.

We further consider the set
$\widehat{}\mathcal{S}=\mathcal{S}^{\prime}+\\{0,\pi/2,\pi,3\pi/2\\}$ obtained
by making $\mathcal{S}^{\prime}$ invariant by rotation by $\pi/2$. It is not
hard to verify that the three conditions above remain valid when we add
directions, so they are still valid for $\hat{}\mathcal{S}$ instead of
$\mathcal{S}^{\prime}$. We will refer to the elements of
$\widehat{}\mathcal{S}$ as _quasi-stable directions_ or simply _directions_ ,
as they are the only ones of interest to us. We label the elements of
$\widehat{}\mathcal{S}=(u_{i})_{i\in[4k]}$ clockwise and consider their
indices modulo $4k$ (we write $[n]$ for $\\{0,\dots,n-1\\}$), so that
$u_{i+2k}=-u_{i}$ (the inverse being taken in ${\mathbb{R}}^{2}$ and not
w.r.t. the angle) is perpendicular to $u_{i+k}$. In figures we take
$\widehat{}\mathcal{S}=\frac{\pi}{4}({\mathbb{Z}}/8{\mathbb{Z}})$ and
$u_{0}=(-1,0)$. Further observe that if all $U\in\mathcal{U}$ are contained in
the axes of ${\mathbb{Z}}^{2}$, then we may set
$\widehat{}\mathcal{S}=\frac{\pi}{2}({\mathbb{Z}}/4{\mathbb{Z}})$.

For $i\in[4k]$ we introduce ${\rho}_{i}=\min\\{{\rho}>0:\exists
x\in{\mathbb{Z}}^{2},\langle x,u_{i}\rangle={\rho}\\}$ and
${\lambda}_{i}=\min\\{{\lambda}>0:{\lambda}u_{i}\in{\mathbb{Z}}^{2}\\}$, which
are both well-defined, as the directions are rational (in fact
${\rho}_{i}{\lambda}_{i}=1$, but we will use both notations for transparency).

###  3.3 Droplets

We next define the geometry of the droplets we will use.

######  Definition 3.3 (Droplet).

A _droplet_ is a nonempty set of the form

${\Lambda}(\underline{r})=\bigcap_{i\in[4k]}\overline{\mathbb{H}}_{u_{i}}(r_{i})$

for some _radii_ $\underline{r}\in{\mathbb{R}}^{[4k]}$ (see the black regions
in Fig. 3). For a sequence of radii $\underline{r}$ we define the _side
lengths_ $\underline{s}=(s_{i})_{i\in[4k]}$ with $s_{i}$ the length of the
side of ${\Lambda}(\underline{r})$ with outer normal $u_{i}$.

We say that a droplet is _symmetric_ if it is of the form
$x+{\Lambda}(\underline{r})$ with $2x\in{\mathbb{Z}}^{2}$ and $r_{i}=r_{i+2k}$
for all $i\in[2k]$. If this is the case, we call $x$ the _center_ of the
droplet.

Note that if all $U\in\mathcal{U}$ are contained in the axes of
${\mathbb{Z}}^{2}$, then droplets are simply rectangles with sides parallel to
the axes.

We write $(\underline{e}_{i})_{i\in[4k]}$ for the canonical basis of
${\mathbb{R}}^{[4k]}$ and we write
$\underline{1}=\sum_{i\in[4k]}\underline{e}_{i}$, so that
${\Lambda}(r\underline{1})$ is a polygon with inscribed circle of radius $r$
and sides perpendicular to $\widehat{}\mathcal{S}$. It will often be more
convenient to parametrise dimensions of droplets differently. For $i\in[4k]$
we set

$\underline{v}_{i}=\sum_{j=i-k+1}^{i+k-1}\langle
u_{i},u_{j}\rangle\underline{e}_{j}.$ (9)

This way ${\Lambda}(\underline{r}+\underline{v}_{i})$ is obtained from
${\Lambda}(\underline{r})$ by extending the two sides parallel to $u_{i}$ by
$1$ in direction $u_{i}$ and leaving all other sides unchanged (see Fig.
3(a)). Note that if ${\Lambda}(\underline{r})$ is symmetric, then so is
${\Lambda}(\underline{r}+{\lambda}_{i}\underline{v}_{i})$ for $i\in[4k]$.

######  Definition 3.4 (Tube).

Given $i\in[4k]$, $\underline{r}$ and $l>0$, we define the _tube of length
$l$, direction $i$ and radii $\underline{r}$_ (see the thickened regions in
Fig. 3)

$T(\underline{r},l,i)={\Lambda}(\underline{r}+l\underline{v}_{i})\setminus{\Lambda}(\underline{r}).$

We will often need to consider boundary conditions for our events on droplets
and tubes. Given two disjoint finite regions $A,B\subset{\mathbb{Z}}^{2}$ and
two configurations ${\eta}\in{\Omega}_{A}$ and ${\omega}\in{\Omega}_{B}$, we
define ${\eta}\cdot{\omega}\in{\Omega}_{A\cup B}$ as

$({\eta}\cdot{\omega})_{x}=\begin{cases}{\eta}_{x}&x\in A,\\\
{\omega}_{x}&x\in B.\end{cases}$

###  3.4 Scales

Throughout the work we consider the positive integer constants

$1/{\varepsilon}\gg 1/{\delta}\gg C\gg W.$

Each one is assumed to be large enough depending on $\mathcal{U}$ and,
therefore, $\widehat{}\mathcal{S}$ and ${\alpha}$ (e.g. $W>{\alpha}$), and
much larger than any explicit function of the next (e.g. $e^{W}<C$). 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 $\mathcal{U}$.

The following are our main scales corresponding to the mesoscopic and internal
dynamics:

$\displaystyle\ell^{\mathrm{m}+}$ $\displaystyle{}=q^{-C}/\sqrt{{\delta}},$
$\displaystyle\ell^{\mathrm{m}}$ $\displaystyle{}=q^{-C},$
$\displaystyle\ell^{\mathrm{m}-}$
$\displaystyle{}=q^{-C}\cdot\sqrt{{\delta}},$ $\displaystyle\ell^{\mathrm{i}}$
$\displaystyle{}=C^{2}\log(1/q)/q^{\alpha}.$

###  3.5 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. 2 we
introduce a deliberately complicated example, on which to illustrate them.

(a) The five update rules $U\in\mathcal{U}$ given as dots. The cross marks the
origin.

$3$$3$$3$$3$$u_{2}$$u_{1}$$u_{0}$$u_{3}$ (b) The four stable directions, which
coincide with $\widehat{}\mathcal{S}$, and their difficulties.

$u_{0}$

$u_{1}$

$u_{2}$

$x_{3}$$u_{3}$$Z_{3}+k_{1}x_{3}+{\lambda}_{0}u_{0}$$Z_{3}$$\dots$

(c) Possible choice of $u_{i}$-helping sets. The hatched region represents
${\mathbb{H}}_{u_{i}}\cap{\mathbb{Z}}^{2}$.

Figure 2: An intricate isotropic example.

####  3.5.1 Helping sets for a line

######  Definition 3.5 ($W$-helping set).

Let $i\in[4k]$. A $W$-helping set is any set of $W$ consecutive infected sites
in $\overline{\mathbb{H}}_{u_{i}}\setminus{\mathbb{H}}_{u_{i}}$, that is, a
set of the form $x+[W]{\lambda}_{i+k}u_{i+k}$ for some
$x\in\overline{\mathbb{H}}_{u_{i}}\setminus{\mathbb{H}}_{u_{i}}$.

The relevance of $W$-helping sets is that, since $W$ is large enough,
$[Z\cup{\mathbb{H}}_{u_{i}}]_{\mathcal{U}}=\overline{\mathbb{H}}_{u_{i}}$ for
any direction $u_{i}$ such that $\alpha(u_{i})<\infty$ and $W$-helping set $Z$
(see [Bollobas15]*Lemma 5.2).

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 Section A.2 the reader will not
lose anything conceptual by thinking that the sets $Z_{i}$, $u_{i}$\- and
${\alpha}$-helping sets defined below are simply single infected sites in $S$
and the period $Q$ is 1.

######  Lemma 3.6.

Let $i\in[4k]$ be such that $0<{\alpha}(u_{i})\leqslant{\alpha}$. Then there
exists a set $Z_{i}\subset{\mathbb{Z}}^{2}\setminus{\mathbb{H}}_{u_{i}}$ and
$x_{i}\in{\mathbb{Z}}^{2}\setminus\\{0\\}$ such that $\langle
x_{i},u_{i}\rangle=0$, $|Z_{i}|={\alpha}$,
$|[Z_{i}\cup{\mathbb{H}}_{u_{i}}]_{\mathcal{U}}\setminus\overline{\mathbb{H}}_{u_{i}}|<\infty$
and $[Z_{i}\cup{\mathbb{H}}_{u_{i}}]_{\mathcal{U}}\supset x_{i}{\mathbb{N}}$,
where ${\mathbb{N}}=\\{0,1,\dots\\}$. That is, together with
${\mathbb{H}}_{u_{i}}$, $Z_{i}$ can infect a semi-sublattice of the first line
outside ${\mathbb{H}}_{u_{i}}$ and only a finite number of other sites.

###### Proof.

Definition 1.3 supplies a set
$Z\subset{\mathbb{Z}}^{2}\setminus{\mathbb{H}}_{u_{i}}$ such that
$\bar{Z}:=[{\mathbb{H}}_{u_{i}}\cup
Z]_{\mathcal{U}}\setminus{\mathbb{H}}_{u_{i}}$ is infinite and
$|Z|=\alpha(u_{i})$. Choose $Z$ to minimise $l:=\max\\{\langle
z,u_{i}\rangle:z\in Z\\}$. Yet, $u_{i}$ is stable, since $\alpha(u_{i})\neq
0$, so
$\bar{Z}\subset\overline{\mathbb{H}}_{u_{i}}(l)\setminus{\mathbb{H}}_{u_{i}}$,
which consists of a finite number of lattice lines perpendicular to $u_{i}$.
Then [Bollobas14]*Lemma 3.3 asserts that the intersection of $\bar{Z}$ with
the first line, $\overline{\mathbb{H}}_{u_{i}}\setminus{\mathbb{H}}_{u_{i}}$,
is either finite or contains a semi-sublattice. If it is infinite and
$\bar{Z}\setminus\overline{\mathbb{H}}_{u_{i}}$ is finite, we conclude by
setting $Z_{i}$ equal to $Z$ plus $\alpha-\alpha(u_{i})$ arbitrarily chosen
sites in $\bar{Z}\setminus Z$. Otherwise, we set
$Z^{\prime}=(Z-\rho_{i}u_{i})\setminus{\mathbb{H}}_{u}$ and observe that
$\bar{Z}^{\prime}$ is still infinite, so $|Z^{\prime}|=\alpha(u_{i})$ and we
obtain a contradiction with the minimality of $l$. ∎

In the example of Fig. 2 the $u_{3}$ direction admits a set $Z_{3}$ of
cardinality 3 such that $[Z_{3}\cup{\mathbb{H}}_{u_{3}}]_{\mathcal{U}}$ only
contains every second site of the line
$\overline{\mathbb{H}}_{u_{i}}\setminus{\mathbb{H}}_{u_{i}}$, while at least
$4$ sites are needed to infect the entire line. Thus, in order to efficiently
infect $\overline{\mathbb{H}}_{u_{3}}\setminus{\mathbb{H}}_{u_{3}}$, assuming
${\mathbb{H}}_{u_{3}}$ is infected, we may use two translates of $Z_{3}$ with
different parity. This technicality is reflected in the next definition.

######  Definition 3.7 ($u_{i}$-helping set).

For all $i\in[4k]$ such that $0<{\alpha}(u_{i})\leqslant{\alpha}$ fix a choice
of $Z_{i}$ and $x_{i}$ as in Lemma 3.6 in such a way that the _period_

$Q:=\frac{\|x_{i}\|}{{\lambda}_{i+k}}$

is independent of $i$ and sufficiently large so that the diameter of
$\\{0\\}\cup Z_{i}$ is much smaller than $Q$. A _$u_{i}$ -helping set_ is a
set of the form

$\bigcup_{j\in\left[Q\right]}\left(Z_{i}+j{\lambda}_{i+k}u_{i+k}+k_{j}x_{i}\right),$
(10)

for some integers $k_{j}$. For $i\in[4k]$ with ${\alpha}(u_{i})=0$, we define
$u_{i}$-helping sets to be empty. For $i\in[4k]$ with $\alpha(u_{i})>\alpha$
there are no $u_{i}$-helping sets.

Note that by Lemma 3.6 a $u_{i}$-helping set $Z$ is sufficient to infect a
half-line, but since that contains a $W$-helping set, we have
$[Z\cup{\mathbb{H}}_{u_{i}}]_{\mathcal{U}}\supset\overline{\mathbb{H}}_{u_{i}}$.

We further incorporate the artificial symmetrisation alluded to in Remark 2.1
in the next definition.

######  Definition 3.8 (${\alpha}$-helping set).

Let $i\in[4k]$.

  * •

If ${\alpha}(u_{i})\leqslant{\alpha}$ and
${\alpha}(u_{i+2k})\leqslant{\alpha}$, then a _${\alpha}$ -helping set_ is a
set of the form $H\cup H^{\prime}$ with $H$ a $u_{i}$-helping set and
$-H^{\prime}=\\{-h:h\in H^{\prime}\\}$ a $u_{i+2k}$-helping set.

  * •

If ${\alpha}(u_{i})\leqslant{\alpha}$ and ${\alpha}(u_{i+2k})>{\alpha}$, then
a _${\alpha}$ -helping set_ is a $u_{i}$-helping set.

  * •

If ${\alpha}<{\alpha}(u_{i})\leqslant\infty$, there are no _${\alpha}$
-helping sets_.

If ${\alpha}(u_{i})<\infty$, any set which is either a $W$-helping set or a
${\alpha}$-helping set is called _helping set_. If ${\alpha}(u_{i})=\infty$,
there are no _helping sets_.

In the example of Fig. 2 $u_{0}$ and $u_{2}$ are both of difficulty
${\alpha}=3$, so ${\alpha}$-helping sets in direction $u_{0}$ correspond to
the union of a translate by a vector in ${\mathbb{Z}}^{2}\cap
u_{1}{\mathbb{R}}$ of $\\{(0,0),(2,0),(3,0)\\}$ _and_ such a (possibly
different) translate of $\\{(0,0),(-2,1),(0,2)\\}$. The latter is not a
$u_{0}$-helping set, but we include it in ${\alpha}$-helping sets in direction
$u_{0}$, in order for ${\alpha}$-helping sets in direction $u_{0}$ and $u_{2}$
to be symmetric ($Z_{0}$ is a $\alpha$-helping set in direction $u_{0}$ if and
only if $-Z_{0}$ is a $\alpha$-helping set in direction $u_{2}$).

####  3.5.2 Helping sets for a segment

Let us fix a direction $u_{i}\in\widehat{}\mathcal{S}$ with
${\alpha}(u_{i})<\infty$. Let $S$ be a discrete segment perpendicular to
$u_{i}$. In this section we will assume that $S$ is of the form

$\left\\{x\in{\mathbb{Z}}^{2}:\langle x,u_{i}\rangle=0,\langle
x,u_{i+k}\rangle/{\lambda}_{i+k}\in[a,b]\right\\}$ (11)

for some integers $a,b$ with $b\geqslant W+a$, but all definitions are
extended by translation.444Note however, that a translate of a helping set for
a given line need not be a helping set for that line. For instance, in
Definition 3.10 if the helping set is a $W$-helping set, it has to be in $S$
and not on a translate in direction $u_{i}$ thereof. The direction $u_{i}$
will be kept implicit in the notation, so it may be useful to view $S$ as
having an orientation.

######  Definition 3.9 ($W$-helping set).

For $d\geqslant 0$, we denote by $\mathcal{H}^{W}_{d}(S)$ the event that there
is an infected $W$-helping set in $S$ at distance at least $d$ from its
endpoints. We write $\mathcal{H}^{W}(S):=\mathcal{H}^{W}_{0}(S)$.

For helping sets the definition is more technical, since they are not included
in $S$, so we require that they are close to $S$ and at some distance from its
endpoints.

######  Definition 3.10.

For $d\geqslant 0$, we denote by $\mathcal{H}_{d}(S)$ the event on $\Omega$
that there is an infected helping set $Z$ such that

$\langle z,u_{i}\rangle\in[0,Q]\quad\text{and}\quad\langle
z,u_{i+k}\rangle\in\left[a{\lambda}_{i+k}+d,b{\lambda}_{i+k}-d\right]$ (12)

for all $z\in Z$. Given a domain ${\Lambda}\supset S$ and a boundary condition
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}}$ we define
$\mathcal{H}^{{\omega}}_{d}(S)=\\{{\eta}\in{\Omega}_{{\Lambda}}:{\omega}\cdot{\eta}\in\mathcal{H}_{d}(S)\\}$.
We omit $d$ if it is $0$.

Note that if $\alpha(u_{i})<\infty$, then
$\mathcal{H}^{\omega}(S)\supset\mathcal{H}^{W}(S)$ for any ${\omega}$ with
equality if $\alpha(u_{i})>\alpha$. The next observation will be used
systematically in probability estimates. It follows easily from the
definitions above (see e.g. [Bollobas14]*Lemma 4.2).

######  Observation 3.11 (Helping set probability).

For any $\Lambda\supset S$ and
${\omega}\in\Omega_{{\mathbb{Z}}^{2}\setminus\Lambda}$ we have: if
${\alpha}(u_{i})<\infty$, then

${\mu}\left(\mathcal{H}^{\omega}(S)\right)\geqslant{\mu}\left(\mathcal{H}^{W}(S)\right)\geqslant
1-\left(1-q^{W}\right)^{\lfloor(|S|)/W\rfloor}\geqslant\max\left(q^{W},1-e^{-q^{2W}|S|}\right);$

if ${\alpha}(u_{i})\leqslant{\alpha}$, then

${\mu}(\mathcal{H}(S))\geqslant\left(1-(1-q^{\alpha})^{{\Omega}(|S|)}\right)^{O(1)}\geqslant\left(1-e^{-q^{\alpha}|S|/O(1)}\right)^{O(1)}.$

###  3.6 Constrained Poincaré inequalities

We next define the (constrained) Poincaré constants of various regions. For
${\Lambda}\subset{\mathbb{Z}}^{2}$, $\eta,\omega\in{\Omega}$ (or possibly
$\eta$ defined on a set including $\Lambda$ and $\omega$ on a set including
${\mathbb{Z}}^{2}\setminus\Lambda$) and $x\in{\mathbb{Z}}^{2}$, we set
$c_{x}^{{\Lambda},{\omega}}({\eta})=c_{x}({\eta}_{\Lambda}\cdot{\omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}})$
(recall Eq. 1). Given a finite ${\Lambda}\subset{\mathbb{Z}}^{2}$ and a
nonempty event
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda})\subset{\Omega}_{\Lambda}$, let
${\gamma}({\Lambda})$ be the smallest constant ${\gamma}\in[1,\infty]$ such
that the inequality

$\operatorname{Var}_{{\Lambda}}\left(f|\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda})\right)\leqslant{\gamma}\sum_{x\in{\Lambda}}{\mu}_{\Lambda}\left(c_{x}^{\Lambda,\mathbf{1}}\operatorname{Var}_{x}(f)\right)$
(13)

holds for all $f:{\Omega}\rightarrow{\mathbb{R}}$. Here we recall from Section
1.1 that ${\mu}$ denotes both the product Bernoulli probability distribution
with parameter $q$ and the expectation with respect to it. Moreover, for any
function $\phi:{\Omega}\rightarrow{\mathbb{R}}$,
${\mu}_{\Lambda}(\phi)={\mu}(\phi(\eta)|\eta_{{\mathbb{Z}}^{2}\setminus{\Lambda}})$
is the average on the configuration $\eta$ of law ${\mu}$ in ${\Lambda}$,
conditionally on its state in ${\mathbb{Z}}^{2}\setminus{\Lambda}$. Thus,
${\mu}_{\Lambda}(\phi)$ is a function on
${\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}}$. Similarly,
$\operatorname{Var}_{x}(f)={\mu}(f^{2}(\eta)|\eta_{{\mathbb{Z}}^{2}\setminus\\{x\\}})-{\mu}^{2}(f(\eta)|\eta_{{\mathbb{Z}}^{2}\setminus\\{x\\}})$
and

$\operatorname{Var}_{\Lambda}\left(f|\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda})\right)={\mu}\left(\left.f^{2}(\eta)\right|{\eta}_{\Lambda}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}(\Lambda),\eta_{{\mathbb{Z}}^{2}\setminus{\Lambda}}\right)\\\
-{\mu}^{2}\left(f(\eta)|{\eta}_{\Lambda}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}(\Lambda),\eta_{{\mathbb{Z}}^{2}\setminus{\Lambda}}\right).$

######  Remark 3.12.

It is important to note that in the r.h.s. of Eq. 13 we average w.r.t.
${\mu}_{\Lambda}$ and not
${\mu}_{\Lambda}(\cdot|\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}))$ (the
latter would correspond to the usual definition of Poincaré constant, from
which we deviate). In this respect Eq. 13 follows [Hartarsky21a]*Eq. (12) and
differs from [Hartarsky20FA]*Eq. (4.5). Although this nuance is not important
most of the time, this choice is crucial for the proof of Theorem 8.5 below.

###  3.7 Boundary conditions, translation invariance, monotonicity

Let us make a few conventions in order to lighten notation throughout the
paper. As we already witnessed in Section 3.5, it is often the case that much
of the boundary condition is actually irrelevant for the occurrence of the
event. For instance, in Definition 3.10, $\mathcal{H}^{\omega}(S)$ only
depends on the restriction of ${\omega}$ to a finite-range neighbourhood of
the segment $S$. Moreover, even the state in ${\omega}$ of sites close to $S$,
but in ${\mathbb{H}}_{u_{i}}$ is of no importance. Such occasions will arise
frequently, so, by abuse, we allow ourselves to specify a boundary condition
on any region containing the sites whose state actually matters for the
occurrence of the event.

We also need the following natural notion of translation invariance.

######  Definition 3.13 (Translation invariance).

Let $A\subset{\mathbb{R}}^{2}$. Consider a collection of events
$\mathcal{E}^{\omega}(A+x)$ for $x\in{\mathbb{Z}}^{2}$ and
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus(A+x)}$. We say that
$\mathcal{E}(A)$ is _translation invariant_ , if for all
$\eta\in{\Omega}_{A}$, ${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus A}$ and
$x\in{\mathbb{Z}}^{2}$ we have

$\eta\in\mathcal{E}^{\omega}(A)\Leftrightarrow\eta_{\cdot-x}\in\mathcal{E}^{{\omega}_{\cdot-x}}(A+x).$

Similarly, we say that $\mathcal{E}^{\omega}(A)$ is _translation invariant_ ,
if the above holds for a fixed ${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus
A}$.

It is clear from the definition that $\mathcal{H}(S)$ is translation invariant
for any segment $S$. The same will hold for $\mathcal{T}$ and
$\mathcal{S}\mathcal{T}$ events for tubes defined in Section 4.1 and
$\mathcal{S}\mathcal{G}$ events for droplets defined throughout the paper.
This will often be clear from the definitions, but when the definition itself
is stated for a specific region of space (as in Section 3.5), we
systematically extend it in a translation invariant way to all translates of
this region.

We also use the occasion to point out that, just like the event
$\mathcal{H}^{\omega}(S)$, all our $\mathcal{T}$, $\mathcal{S}\mathcal{T}$ and
$\mathcal{S}\mathcal{G}$ events will be decreasing in both the configuration
and the boundary condition, so that we will be able to apply Section 3.1 as
needed.

##  4 One-directional extensions

In this section we define our crucial one-directional CBSEP-extension and
East-extension techniques (recall Section 2.3).

###  4.1 Traversability

We first need the following traversability $\mathcal{T}$ and symmetric
traversability $\mathcal{S}\mathcal{T}$ events for tubes (recall Definition
3.4) requiring infected helping sets (recall Section 3.5.2) to appear for each
of the segments composing the tube. The definition is illustrated in Fig. 3.
Recall the constant $C$ from Section 3.4

######  Definition 4.1 (Traversability).

Fix a tube $T=T(\underline{r},l,i)$. Assume that $i\in[4k]$ is such that
${\alpha}(u_{j})<\infty$ for all $j\in(i-k,i+k)$. For $m\geqslant 0$ and
$j\in(i-k,i+k)$ write
$S_{j,m}={\mathbb{Z}}^{2}\cap{\Lambda}(\underline{r}+m\underline{v}_{i}+{\rho}_{j}\underline{e}_{j})\setminus{\Lambda}(\underline{r}+m\underline{v}_{i})$.
Note that $S_{j,m}$ is a discrete line segment perpendicular to $u_{j}$ of
length $s_{j}-O(1)$ (recall from Definition 3.3 that $\underline{s}$ is the
sequence of side lengths of $\Lambda(\underline{r})$). For
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r}+l\underline{v}_{i})}$
we denote by

$\mathcal{T}^{\omega}_{d}(T)=\bigcap_{j,m:\varnothing\neq S_{j,m}\subset
T}\mathcal{H}^{\omega}_{C^{2}+d}\left(S_{j,m}\right)$

the event that $T$ is $({\omega},d)$-_traversable_. We set
$\mathcal{T}^{\omega}(T)=\mathcal{T}^{\omega}_{0}(T)$.

If moreover $\alpha(u_{i})<\infty$ for all $i\in[4k]$, that is, $\mathcal{U}$
has a finite number of stable directions, we denote by

$\mathcal{S}\mathcal{T}^{\omega}_{d}(T)=\mathcal{T}_{d}^{\omega}(T)\cap\bigcap_{j:{\alpha}(u_{j})\leqslant\alpha<{\alpha}(u_{j+2k}))}\bigcap_{m:\varnothing\neq
S_{j,m}\subset T}\mathcal{H}^{W}_{C^{2}+d}\left(S_{j,m}\right)$

the event that $T$ is $({\omega},d)$-_symmetrically traversable_.

Thus, if all side lengths of ${\Lambda}(\underline{r})$ are larger than
$C^{2}+d$ by a large enough constant, the event
$\mathcal{T}^{\omega}_{d}(T(\underline{r},s,i))$ decomposes each of the
hatched parallelograms in Fig. 3(a) into line segments parallel to its side
that is not parallel to $u_{i}$. A helping set is required for each of these
segments for the direction perpendicular to them which has positive scalar
product with $u_{i}$. The last boundedly many segments may also use the
boundary condition ${\omega}$, but it is irrelevant for the remaining ones,
since it is far enough from them.

For symmetric traversability, we rather require $W$-helping sets for opposites
of hard directions (recall from Definition 3.8 that if the direction itself is
hard, helping sets are simply $W$-helping sets). In particular, if none of the
directions $u_{j}$ for $j\in[4k]\setminus\\{i+k,i-k\\}$ is hard (implying that
$\mathcal{U}$ is unrooted), we have
$\mathcal{S}\mathcal{T}^{\omega}_{d}(T(\underline{r},l,i))=\mathcal{T}^{\omega}_{d}(T(\underline{r},l,i))$.
The reason to call symmetric traversability so is that if $\mathcal{U}$ has a
finite number of stable directions and ${\Lambda}(\underline{r})$ is a
symmetric droplet (recall Section 3.3), then, for any $l$, $i$,
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T(\underline{r},l,i)}$,
${\eta}\in{\Omega}_{T(\underline{r},l,i)}$, we have

$\eta\in\mathcal{S}\mathcal{T}^{\omega}_{d}(T(\underline{r},l,i))\Leftrightarrow\eta^{\prime}\in\mathcal{S}\mathcal{T}^{{\omega}^{\prime}}_{d}(T(\underline{r},l,i+2k)),$
(14)

denoting by ${\omega}^{\prime}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus
T(\underline{r},l,i+2k)}$ the boundary condition obtained by rotating
${\omega}$ by $\pi$ around the center of ${\Lambda}(\underline{r})$ and
similarly for $\eta^{\prime}$. To see this, recall from Section 3.5 that
$\mathcal{H}^{\omega}(S)\supset\mathcal{H}^{W}(S)$ with equality when
$\alpha(u_{i})>\alpha$ and note that the same symmetry as in Eq. 14 holds at
the level of the segment $S_{j,m}$ and its symmetric one,
$S^{\prime}_{j+2k,m}={\mathbb{Z}}^{2}\cap{\Lambda}(\underline{r}+m\underline{v}_{i+2k}+{\rho}_{j+2k}\underline{e}_{j+2k})\setminus{\Lambda}(\underline{r}+m\underline{v}_{i+2k})$:

$\eta\in\begin{cases}\mathcal{H}^{\omega}_{C^{2}+d}(S_{j,m})&\alpha(u_{j+2k})\leqslant\alpha\\\
\mathcal{H}^{W}_{C^{2}+d}(S_{j,m})&\alpha(u_{j+2k})>\alpha\end{cases}\Leftrightarrow\eta^{\prime}\in\begin{cases}\mathcal{H}^{{\omega}^{\prime}}_{C^{2}+d}(S^{\prime}_{j+2k,m})&\alpha(u_{j})\leqslant\alpha\\\
\mathcal{H}^{W}_{C^{2}+d}(S^{\prime}_{j+2k,m})&\alpha(u_{j})>\alpha,\end{cases}$

all four cases following directly from Definitions 3.8, 3.10 and 3.9.

We next state a simple observation which will be used frequently to modify
boundary conditions as we like at little cost.

######  Lemma 4.2 (Changing boundary conditions).

Let ${\Lambda}(\underline{r})$ be a droplet, $l>0$ be a multiple of
${\lambda}_{i}$ and $i\in[4k]$ assume that its side lengths $s_{j}$ for
$j\in[4k]\setminus\\{i-k,i+k\\}$ satisfy $s_{j}\geqslant C^{3}$. Set
$T=T(\underline{r},l,i)$. Then there exists a decreasing event
$\mathcal{W}(T)\subset{\Omega}_{T}$ such that ${\mu}(\mathcal{W}(T))\geqslant
q^{O(W)}$ for any ${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T}$ and
$\eta\in\mathcal{W}(T)$ we have

${\eta}\in\mathcal{T}^{\omega}(T)\Leftrightarrow{\eta}\in\mathcal{T}^{\mathbf{1}}(T).$

Moreover,
${\mu}(\mathcal{T}^{\omega}(T))=q^{-O(W)}{\mu}(\mathcal{T}^{\mathbf{1}}(T))$
for all ${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T}$. The same holds
with $\mathcal{S}\mathcal{T}$ instead of $\mathcal{T}$.

###### Proof.

Recall the segments $S_{j,m}$ from Definition 4.1. Let $\mathcal{W}(T)$ be the
intersection of $\mathcal{H}^{W}_{C^{2}}(S_{j,m})$ for the largest
sufficiently large but fixed number of values of $m$ for each $j\in(i-k,i+k)$,
such that $\varnothing\neq S_{j,m}\subset T$. By Observation 3.11
${\mu}(\mathcal{W}(T))=q^{O(W)}$. Moreover, the boundary condition is not
important for the remaining segments, so $\mathcal{W}(T)$ is indeed as
desired. Finally, by Eq. 6 we have

$\displaystyle{\mu}\left(\mathcal{T}^{\mathbf{1}}(T)\right)$
$\displaystyle{}\leqslant{\mu}\left(\mathcal{T}^{\omega}(T)\right)\leqslant\frac{{\mu}(\mathcal{W}(T)\cap\mathcal{T}^{\omega}(T))}{{\mu}(\mathcal{W}(T))}$
$\displaystyle{}\leqslant
q^{-O(W)}{\mu}\left(\mathcal{W}(T)\cap\mathcal{T}^{\mathbf{1}}(T)\right)\leqslant
q^{-O(W)}{\mu}\left(\mathcal{T}^{\mathbf{1}}(T)\right).\qed$

Another convenient property allowing us to decompose a long tube into smaller
ones is the following.

######  Lemma 4.3 (Decomposing tubes).

Let $T=T(\underline{r},l,i)$ be a tube,
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T}$ be a boundary condition
and $s\in[0,l]$ be a multiple of ${\lambda}_{i}$. Set
$T_{1}=T(\underline{r},s,i)$ and $T_{2}=su_{i}+T(\underline{r},l-s,i)$. Then

${\eta}\in\mathcal{T}^{\omega}(T(\underline{r},l,i))\Leftrightarrow\left({\eta}_{T_{2}}\in\mathcal{T}^{\omega}(T_{2})\text{
and }{\eta}_{T_{1}}\in\mathcal{T}^{{\eta}_{T_{2}}\cdot{\omega}}(T_{1})\right)$

and the same holds for $\mathcal{S}\mathcal{T}$ instead of $\mathcal{T}$.

###### Proof.

This follows immediately from Definition 4.1, since for each of the segments
$S_{j,m}$ in Definition 4.1 either $S_{j,m}\subset T_{1}$ or $S_{j,m}\cap
T_{1}=\varnothing$ and similarly for $T_{2}$ (see Fig. 3(a)). ∎

${\omega}$$u_{i}$$s$${\Lambda}(\underline{r})$$T(\underline{r},s,i)$$u_{i+2k}$${\Lambda}(\underline{r})+xu_{i}$$T(\underline{r},s-x,i)+xu_{i}$$T(\underline{r},x,i+2k)+xu_{i}$
(a) East-extension. The thickened tube is traversable ($\mathcal{T}$).
${\omega}$${\omega}$$u_{i}$$u_{i+2k}$$s-x$$x$${\Lambda}(\underline{r})+xu_{i}$$T(\underline{r},s-x,i)+xu_{i}$$T(\underline{r},x,i+2k)+xu_{i}$
(b) CBSEP-extension. Thickened tubes are symmetrically traversable
($\mathcal{S}\mathcal{T}$).

Figure 3: One-directional extensions. The black droplet is SG. Helping sets
appear on each line of the hatched parallelograms as indicated by the hatching
direction. The white strips have width $\Theta(C^{2})$.

###  4.2 East-extension

We start with the East-extension (see Fig. 3(a)), which is simpler to state.

######  Definition 4.4 (East-extension).

Fix $i\in[4k]$, a droplet ${\Lambda}(\underline{r})$ and a multiple $l>0$ of
${\lambda}_{i}$. Let
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\subset{\Omega}_{{\Lambda}(\underline{r})}$
be an event and assume that ${\alpha}(u_{j})<\infty$ for all $j\in(i-k,i+k)$.
We use the expression “ _we East-extend ${\Lambda}(\underline{r})$ by $l$ in
direction $u_{i}$_” to state that, for all $s\in(0,l]$ multiple of
${\lambda}_{i}$ and
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r}+s\underline{v}_{i})}$,
we define the event
$\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+s\underline{v}_{i}))\subset{\Omega}_{{\Lambda}(\underline{r}+s\underline{v}_{i})}$
to occur for ${\eta}\in{\Omega}_{{\Lambda}(\underline{r}+s\underline{v}_{i})}$
if

${\eta}_{{\Lambda}(\underline{r})}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\quad\text{and}\quad{\eta}_{T(\underline{r},s,i)}\in\mathcal{T}^{{\omega}}(T(\underline{r},s,i)).$

In other words, given the event $\mathcal{S}\mathcal{G}^{\mathbf{1}}$ for the
droplet ${\Lambda}(\underline{r})$, we define the event
$\mathcal{S}\mathcal{G}^{\omega}$ (in particular for ${\omega}=\mathbf{1}$,
but not only) for the larger droplet
${\Lambda}(\underline{r}+l\underline{v}_{i})={\Lambda}(\underline{r})\sqcup
T(\underline{r},l,i)$. The event obtained on the larger droplet requires for
the smaller one to be $\mathbf{1}$-super good (SG) and for the remaining tube
to be ${\omega}$-traversable (recall Definition 4.1). Note that these two
events are independent. Further observe that if
$\mathcal{S}\mathcal{G}({\Lambda}(\underline{r}))$ is translation invariant,
then so is
$\mathcal{S}\mathcal{G}({\Lambda}(\underline{r}+s\underline{v}_{i}))$ for any
$s\in(0,l]$ multiple of ${\lambda}_{i}$, defined by East-extending
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$. To get a grasp on
Definition 4.4, let us note the following.

######  Lemma 4.5 (East-extension ergodicity).

Let $i\in[4k]$ be such that ${\alpha}(u_{j})<\infty$ for all $j\in(i-k,i+k)$.
Let ${\Lambda}(\underline{r})$ be a droplet and $l$ be a multiple of
${\lambda}_{i}$. Let
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\subset{\Omega}_{{\Lambda}(\underline{r})}$
and assume that ${\alpha}(u_{j})<\infty$ for all $j\in(i-k,i+k)$. Further
assume that
$\eta\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$ implies
that the $\mathcal{U}$-KCM with initial condition
${\eta}\cdot\mathbf{1}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r})}$
can entirely infect ${\Lambda}(\underline{r})$. If we East-extend
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$, then for any
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r}+l\underline{v}_{i})}$
and
$\eta\in\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+l\underline{v}_{i}))$
the $\mathcal{U}$-KCM with initial condition ${\omega}\cdot\eta$ can entirely
infect ${\Lambda}(\underline{r}+l\underline{v}_{i})$.

###### Proof.

The proof is rather standard and we will not use the lemma, so we only sketch
the reasoning. Let
$\eta\in\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+l\underline{v}_{i}))$.
Since
$\eta_{{\Lambda}(\underline{r})}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$
by Definition 4.4, by hypothesis we can completely infect
${\Lambda}(\underline{r})$, starting from ${\omega}\cdot{\eta}$. We next
proceed by induction on $s\in[0,l]$ to show that we can infect
${\Lambda}(\underline{r}+s\underline{v}_{i})$. When a new site in
${\mathbb{Z}}^{2}$ is added to this set, as we increase $s$, we actually add
to it an entire segment $S_{j,m}$ as in Definition 4.1 (at most one $m$ for
each $j\in(i-k,i+k)$). Since $T(\underline{r},l,i)$ is
$({\omega},0)$-traversable, by Definitions 3.10 and 4.1, this segment contains
a helping set. As noted in Section 3.5.1, in infinite volume helping sets
together with a half-plane infect the entire line on the boundary of the half-
plane. Since the helping set in our setting is only next to a finite fully
infected droplet ${\Lambda}(\underline{r}+s\underline{v}_{i})$, infection
spreads along its edge until it reaches a bounded distance from the corners
(see [Bollobas14]*Lemma 3.4). However, by our choice of $\hat{}\mathcal{S}$
(recall Section 3.2), for each $j\in[4k]$ there is a rule $X\in\mathcal{U}$
such that
$X\subset\overline{\mathbb{H}}_{u_{j}}\cap\overline{\mathbb{H}}_{u_{j+1}}$.
Using this rule, we can infect even the remaining sites to fill up the corner
between directions $u_{j}$ and $u_{j+1}$ of the droplet
${\Lambda}(\underline{r}+s^{\prime}\underline{v}_{i})$ with $s^{\prime}>s$
minimal such that
${\Lambda}(\underline{r}+s^{\prime}\underline{v}_{i})\setminus{\Lambda}(\underline{r}+s\underline{v}_{i})\neq\varnothing$
(see [Bollobas15]*Lemma 5.5 and Fig. 6). ∎

We next state a recursive bound on the Poincaré constant ${\gamma}$ from
Section 3.6 reflecting the recursive definition of SG events in an East-
extension. In rough terms, it states that in order to relax on the larger
volume, we need to be able to relax on the smaller one and additionally pay
the cost of creating logarithmically many copies of it shifted by
exponentially growing offsets, conditionally on the presence of the original
droplet. We further need to account for the cost of microscopic dynamics (see
the $e^{\log^{2}(1/q)}$ term below), but its contribution is unimportant.
Recall $\ell^{\mathrm{m}+}$ from Section 3.4.

######  Proposition 4.6 (East-extension relaxation).

Let $i\in[4k]$ be such that for all $j\in(i-k,i+k)$ we have
${\alpha}(u_{j})<\infty$. Let ${\Lambda}(\underline{r})$ be a droplet with
$\underline{r}=q^{-O(C)}$ and side lengths at least $C^{3}$. Let
$l\in(0,\ell^{\mathrm{m}+}]$ be a multiple of ${\lambda}_{i}$. Define
$d_{m}={\lambda}_{i}\lfloor(3/2)^{m}\rfloor$ for $m\in[1,M)$ and
$M=\min\\{m:{\lambda}_{i}(3/2)^{m}\geqslant l\\}$. Let $d_{M}=l$,
${\Lambda}^{m}={\Lambda}(\underline{r}+d_{m}\underline{v}_{i})$ and
$s_{m-1}=d_{m}-d_{m-1}$ for $m\in[2,M]$.

Let $\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$ be a
nonempty translation invariant decreasing event. Assume that we East-extend
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$. Then
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}+l\underline{v}_{i}))$
is also nonempty, translation invariant, decreasing and satisfies

${\gamma}({\Lambda}(\underline{r}+l\underline{v}_{i}))\leqslant\max\left({\gamma}({\Lambda}(\underline{r})),{\mu}^{-1}\left(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\right)\right)e^{O(C^{2})\log^{2}(1/q)}\prod_{m=1}^{M-1}a_{m},$

with

$a_{m}={\mu}^{-1}\left(\left.\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{m}+s_{m}u_{i}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{m})\right).$
(15)

The proof is left to Section A.3.

###  4.3 CBSEP-extension

We next turn our attention to CBSEP-extensions (see Fig. 3(b)). The definition
differs from Definition 4.4 in two ways. Firstly, we allow the SG event to
occur anywhere in the volume and ask for traversability on either side of it
in the direction away from the SG droplet (so that infection can spread,
starting from it). Secondly, we require our tubes to be symmetrically
traversable, instead of traversable, which will ultimately make the position
of the SG region roughly uniform.

######  Definition 4.7 (CBSEP-extension).

Assume that $\mathcal{U}$ has a finite number of stable directions
(equivalently, $\alpha(u_{j})<\infty$ for all $j\in[4k]$). Fix $i\in[4k]$, a
droplet ${\Lambda}(\underline{r})$ and a multiple $l$ of ${\lambda}_{i}$. Let
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$ be a
translation invariant event. We use the expression “ _we CBSEP-extend
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$_” to state that, for
all $s\in(0,l]$ multiple of ${\lambda}_{i}$ and
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r}+s\underline{v}_{i})}$,
we define the event
$\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+s\underline{v}_{i}))\subset{\Omega}_{{\Lambda}(\underline{r}+s\underline{v}_{i})}$
as follows.

For _offsets_ $x\in[0,s]$ divisible by ${\lambda}_{i}$ we define
${\eta}\in\mathcal{S}\mathcal{G}_{x}^{\omega}({\Lambda}(\underline{r}+s\underline{v}_{i}))$
if the following all hold:

$\displaystyle\eta_{T(\underline{r},s-x,i)+xu_{i}}$
$\displaystyle{}\in\mathcal{S}\mathcal{T}^{{\omega}}(T(\underline{r},s-x,i)+xu_{i});$
$\displaystyle{\eta}_{{\Lambda}(\underline{r})+xu_{i}}$
$\displaystyle{}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r})+xu_{i});$
$\displaystyle\eta_{T(\underline{r},x,i+2k)+xu_{i}}$
$\displaystyle{}\in\mathcal{S}\mathcal{T}^{{\omega}}(T(\underline{r},x,i+2k)+xu_{i}).$

We then set
$\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+s\underline{v}_{i}))=\bigcup_{x}\mathcal{S}\mathcal{G}^{\omega}_{x}({\Lambda}(\underline{r}+s\underline{v}_{i}))$.

Note that CBSEP-extending in direction $u_{i}$ gives the same result as CBSEP-
extending in direction $u_{i+2k}$. We further reassure the reader that in
applications Definitions 4.4 and 4.7 will not be used simultaneously for the
same droplet ${\Lambda}(\underline{r})$, so no ambiguity will arise as to
whether
$\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+l\underline{v}_{i}))$
is obtained by CBSEP- or East-extension. However, as it is clear from Table
2(b), it will sometimes be necessary to CBSEP-extend a droplet itself obtained
by East-extending an even smaller one. But for the time being, let us focus on
a single CBSEP-extension.

The following analogue of Lemma 4.5 holds for CBSEP-extension.

######  Lemma 4.8 (CBSEP-extension ergodicity).

Assume that $\mathcal{U}$ has a finite number of stable directions. Let
$i\in[4k]$, ${\Lambda}(\underline{r})$ be a droplet and $l$ be a multiple of
${\lambda}_{i}$. Let
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\subset{\Omega}_{{\Lambda}(\underline{r})}$.
Further assume that
$\eta\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$ implies
that the $\mathcal{U}$-KCM with initial condition
${\eta}\cdot\mathbf{1}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r})}$
can entirely infect ${\Lambda}(\underline{r})$. If we CBSEP-extend
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$, then for any
${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}(\underline{r}+l\underline{v}_{i})}$
and
$\eta\in\mathcal{S}\mathcal{G}^{\omega}({\Lambda}(\underline{r}+l\underline{v}_{i}))$
the $\mathcal{U}$-KCM with initial condition ${\omega}\cdot\eta$ can entirely
infect ${\Lambda}(\underline{r}+l\underline{v}_{i})$.

###### Proof.

By Definition 4.7, it suffices to prove that for each offset $x\in[0,s]$ the
conclusion holds for
$\eta\in\mathcal{S}\mathcal{G}^{\omega}_{x}({\Lambda}(\underline{r}+l\underline{v}_{i}))$.
By Definition 4.7, this implies that the events
$\mathcal{S}\mathcal{G}^{\mathbf{1}}(xu_{i}+{\Lambda}(\underline{r}))\cap\mathcal{S}\mathcal{T}^{\omega}(xu_{i}+T(\underline{r},s-x,i))$
and
$\mathcal{S}\mathcal{G}^{\mathbf{1}}(xu_{i}+{\Lambda}(\underline{r}))\cap\mathcal{S}\mathcal{T}^{\omega}(xu_{i}+T(\underline{r},x,i+2k))$
hold. Moreover, by Definition 4.1,
$\mathcal{S}\mathcal{T}^{{\omega}^{\prime}}(T)\subset\mathcal{T}^{{\omega}^{\prime}}(T)$
for any tube $T$ and boundary condition ${\omega}^{\prime}$. Therefore, we may
apply Lemma 4.5 to each of the droplets
${\Lambda}(\underline{r}+x\underline{v}_{i})$ and
$xu_{i}+{\Lambda}(\underline{r}+(s-x)\underline{v}_{i})$ (in direction
$u_{i+2k}$ for the former) to obtain the desired conclusion. ∎

We next state the CBSEP analogue of Proposition 4.6, which is more involved,
but also more efficient. Roughly speaking, we show that the time needed in
order to relax on a CBSEP-extended droplet, is the product of four
contributions: the Poincaré constant of the smaller droplet; the inverse
probability of the symmetric traversability events in Definition 4.7; the cost
of microscopic dynamics; the conditional probability of suitable contracted
versions of the super good and symmetric traversability events, given the
original ones (recall Section 2.3.1). The last two contributions will turn out
to be negligible, but the last one will require some care and make the
statement somewhat technical.

######  Proposition 4.9 (CBSEP-extension relaxation).

Assume that $\mathcal{U}$ has a finite number of stable directions. Let
$i\in[4k]$, ${\Lambda}(\underline{r})$ be a droplet with
$\underline{r}=q^{-O(C)}$ and side lengths at least $C^{3}$. Let
$l\in(0,\ell^{\mathrm{m}+}]$ be a multiple of ${\lambda}_{i}$. Define
$d_{m}={\lambda}_{i}\lfloor(3/2)^{m}\rfloor$ for $m\in[1,M)$ and
$M=\min\\{m:{\lambda}_{i}(3/2)^{m}\geqslant l\\}$. Let $d_{M}=l$,
${\Lambda}^{m}={\Lambda}(\underline{r}+d_{m}\underline{v}_{i})$ and
$s_{m-1}=d_{m}-d_{m-1}$ for $m\in[2,M]$.

Let $\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$ be a
nonempty translation invariant decreasing event. Assume that we CBSEP-extend
${\Lambda}(\underline{r})$ by $l$ in direction $u_{i}$. Then
$\mathcal{S}\mathcal{G}({\Lambda}(\underline{r}+l\underline{v}_{i}))$ is also
nonempty, translation invariant and decreasing.

Denote ${\Lambda}_{1}=T(\underline{r},{\lambda}_{i},i+2k)$,
${\Lambda}_{2}={\Lambda}(\underline{r}-{\lambda}_{i}\underline{v}_{i})$ and
${\Lambda}_{3}=T(\underline{r}-{\lambda}_{i}\underline{v}_{i},{\lambda}_{i},i)$,
so that
${\Lambda}(\underline{r}+{\lambda}_{i}\underline{v}_{i})-{\lambda}_{i}u_{i}={\Lambda}_{1}\sqcup{\Lambda}_{2}\sqcup{\Lambda}_{3}$
and
${\Lambda}_{2}\cup{\Lambda}_{3}={\Lambda}(\underline{r})=({\Lambda}_{1}\cup{\Lambda}_{2})+{\lambda}_{i}u_{i}$.
Consider some nonempty decreasing events555We will use a bar to denote
“contracted” versions of events (recall Section 2.3.1).
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2})\subset{\Omega}_{{\Lambda}_{2}}$,
$\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{2}}({\Lambda}_{1})\subset{\Omega}_{{\Lambda}_{1}}$
and
$\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{2}}({\Lambda}_{3})\subset{\Omega}_{{\Lambda}_{3}}$
for all ${\eta}_{2}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2})$.
Assume that

$\left\\{{\eta}:{\eta}_{{\Lambda}_{1}}\in\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{{\Lambda}_{2}}}({\Lambda}_{1}),{\eta}_{{\Lambda}_{2}}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}),{\eta}_{{\Lambda}_{3}}\in\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{{\Lambda}_{2}}}({\Lambda}_{3})\right\\}\\\
\subset\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{1}\cup{\Lambda}_{2})\cap\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{2}\cup{\Lambda}_{3}).$
(16)

Set
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{1}\cup{\Lambda}_{2})=\\{{\eta}:{\eta}_{{\Lambda}_{2}}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}),{\eta}_{{\Lambda}_{1}}\in\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{{\Lambda}_{2}}}({\Lambda}_{1})\\}$.
Then

${\gamma}({\Lambda}(\underline{r}+l\underline{v}_{i}))\leqslant\frac{{\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r})))}{{\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}+l\underline{v}_{i})))}\times\max\left({\mu}^{-1}\left(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))\right),{\gamma}({\Lambda}(\underline{r}))\right)\\\
\times\frac{e^{O(C^{2})\log^{2}(1/q)}}{{\mu}(\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{1}\cup{\Lambda}_{2})|\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{1}\cup{\Lambda}_{2}))\min_{{\eta}_{2}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2})}{\mu}(\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{2}}({\Lambda}_{3})|\mathcal{S}\mathcal{T}^{\mathbf{0}}({\Lambda}_{3}))}.$

Proposition 4.9 is proved in Section A.3 based on [Hartarsky20FA]. Equation 16
should be understood as follows. In the middle droplet ${\Lambda}_{2}$, which
has the shape of ${\Lambda}(\underline{r})$, but contracted in direction
$u_{i}$ by $O(1)$, we require an event
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2})$. This event will provide
simultaneously as much of the structure of
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{1}\cup{\Lambda}_{2})$ and
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{2}\cup{\Lambda}_{3})$ (these
regions both have the shape of ${\Lambda}(\underline{r})$), as one can hope
for, given that we are missing a tube of length $O(1)$ of these regions. Once
such a favourable configuration
${\eta}_{2}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2})$ is fixed, the
events
$\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{{\Lambda}_{2}}}({\Lambda}_{1})$
and
$\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{{\Lambda}_{2}}}({\Lambda}_{3})$
provide exactly the missing part of
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{1}\cup{\Lambda}_{2})$ and
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{2}\cup{\Lambda}_{3})$
respectively. In applications, these events will necessarily need to be
defined, taking into account the structure of
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}(\underline{r}))$, on which we
have made no assumptions at this point.

###  4.4 Conditional probability tools

In both Propositions 4.6 and 4.9 our bounds feature certain conditional
probabilities of SG events. We now provide two tools for bounding them.

The next result generalises [Hartarsky20FA]*Corollary A.3, which relied on
explicit computations unavailable in our setting. It shows that the offset of
the core of a CBSEP-extended droplet (see Fig. 3(b) and recall the notation
$\mathcal{S}\mathcal{G}_{x}^{\omega}$ from Definition 4.7) is roughly uniform.
This result is the reason for the somewhat artificial Definition 3.8 of
helping sets and Definition 4.1 of $\mathcal{S}\mathcal{T}$ (also see Remark
2.1).

######  Lemma 4.10 (Uniform core position).

Assume that $\mathcal{U}$ has a finite number of stable directions. Fix
$i\in[4k]$ and a symmetric droplet
${\Lambda}={\Lambda}(\underline{r}+l\underline{v}_{i})$ obtained by CBSEP-
extension by $l$ in direction $u_{i}$. Assume that
$l\leqslant\ell^{\mathrm{m}+}$ is divisible by ${\lambda}_{i}$ and that the
side lengths of ${\Lambda}(\underline{r})$ are at least $C^{3}$. Then for all
$s\in[0,l]$ divisible by ${\lambda}_{i}$ and
${\omega},{\omega}^{\prime}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus{\Lambda}}$

${\mu}\left(\left.\mathcal{S}\mathcal{G}^{\omega}_{s}({\Lambda})\right|\mathcal{S}\mathcal{G}^{{\omega}^{\prime}}({\Lambda})\right)\geqslant
q^{O(C)}.$

The proofs of Lemmas 4.10 and 4.11 are left to Appendix B. The latter vastly
generalises [Hartarsky20FA]*Lemma A.4 and is proved by different means. It is
illustrated in Fig. 4. In words, Lemma 4.11 states in a quantitative way that
the conditional probability of a tube of “critical” size
($q^{-{\alpha}+o(1)}$) being traversable, given that a slightly perturbed
version of it (shifted spatially, with different boundary condition, width of
the white strips in Fig. 3(a), radii and length) is traversable, is not very
low. We note that sizes other than the critical one are not important, so
cruder bounds will suffice.

######  Lemma 4.11 (Perturbing a tube).

Let $i\in[4k]$ such that ${\alpha}(u_{j})\leqslant{\alpha}$ for all
$j\in(i-k,i+k)$. Let ${\Lambda}(\underline{r})$ be a droplet with side lengths
$\underline{s}$ and let $T=T(\underline{r},l,i)$ be a tube. Assume that
$l\in[{\Omega}(1),e^{q^{-o(1)}}]$,
$s:=\min_{i-k<j<i+k}s_{j}=q^{-{\alpha}+o(1)}$ and
$\max_{i-k<j<i+k}s_{j}=q^{-{\alpha}+o(1)}$. For some
${\Delta}\in[C^{2},s/W^{2}]$, let $\underline{r}^{\prime}$ and $l^{\prime}$ be
such that $0\leqslant s_{j}-s_{j}^{\prime}\leqslant O({\Delta})$ for all
$j\in(i-k,i+k)$ and $0\leqslant l-l^{\prime}\leqslant O({\Delta})$, where
$\underline{s}^{\prime}$ are the side lengths of the droplet
${\Lambda}(\underline{r}^{\prime})$. Further let $x\in{\mathbb{R}}^{2}$ be
such that $\|x\|=O({\Delta})$ and $d,d^{\prime}\in[0,O({\Delta})]$ with
$d\leqslant d^{\prime}$. Denoting
$T^{\prime}=T(\underline{r}^{\prime},l^{\prime},i)+x$, for any boundary
conditions ${\omega}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T}$ and
${\omega}^{\prime}\in{\Omega}_{{\mathbb{Z}}^{2}\setminus T^{\prime}}$, we have

${\mu}\left(\left.\mathcal{T}^{{\omega}^{\prime}}_{d^{\prime}}(T^{\prime})\right|\mathcal{T}^{\omega}_{d}(T)\right)\geqslant
q^{O(W)}\left(1-(1-q^{\alpha})^{{\Omega}(s)}\right)^{O({\Delta})}\\\
\times\left(1-W{\Delta}/s-q^{1-o(1)}\right)^{O(l)}.$

$l$$l^{\prime}$$T^{\prime}$$T$$u_{i}$$u_{j}$ Figure 4: Illustration of the
perturbation of Lemma 4.11. The two thickened tubes are $T$ and $T^{\prime}$.
The regions concerned by their traversability are hatched in different
directions.

##  5 Isotropic models

For this section we assume $\mathcal{U}$ to be isotropic (class (g)). In this
case the reasoning closely follows and generalises [Hartarsky20FA]. We will
treat internal and mesoscopic dynamics simultaneously, since for this class
there is no difference between the two.

###  5.1 Isotropic internal and mesoscopic dynamics

We start by defining the geometry of our droplets and the corresponding length
scales. They will all be symmetric and every $2k$-th of them will be dilated
twice. Each such dilation is decomposed into $2k$ steps, so that their
geometry fits the setting of our extensions from Section 4 (see Fig. 5(a)).

Recall Section 3.3 and the constant ${\varepsilon}$ from Section 3.4. Let
$\underline{r}^{(0)}$ be a sequence of radii with $r_{i}^{(0)}=r_{i+2k}^{(0)}$
for all $i\in[2k]$, such that for all $i\in[4k]$,
$r_{i}^{(0)}=\Theta(1/{\varepsilon})$ and the corresponding side length
$s_{i}^{(0)}=\Theta(1/{\varepsilon})$ is a multiple of $2{\lambda}_{i+k}$. For
any integer $m\geqslant 0$, $i\in[2k]$ and $n=2km+r$ with $r\in[2k]$ we define

$s_{i}^{(n)}=s_{i+2k}^{(n)}=s^{(0)}_{i}2^{m}\times\begin{cases}2&k\leqslant
i<k+r\\\ 1&\text{otherwise}\end{cases}$ (17)

and ${\Lambda}^{(n)}={\Lambda}(\underline{r}^{(n)})$ with
$\underline{r}^{(n)}$ the sequence of radii associated to
$\underline{s}^{(n)}$ satisfying $r_{i}^{(n)}=r_{i+2k}^{(n)}$ for all
$i\in[2k]$. Further set
$N^{\mathrm{m}+}=2k\lceil\log({\varepsilon}\ell^{\mathrm{m}+})/\log 2\rceil$
(recall $\ell^{\mathrm{m}+}$ from Section 3.4, where $\mathrm{m}$ is not a
variable and stands for ‘mesoscopic’).

Note that, as claimed, ${\Lambda}^{(n)}$ are nested symmetric droplets
extended in one direction at each step satisfying
${\Lambda}^{(2km)}=2^{m}{\Lambda}^{(0)}$. Moreover, they are nested so that we
can define their SG events by extension (recall Definitions 4.7 and 3(b) for
CBSEP-extensions).

######  Definition 5.1 (Isotropic SG).

Let $\mathcal{U}$ be isotropic. We say that ${\Lambda}^{(0)}$ is _SG_
($\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(0)})$ occurs), if all sites
in ${\Lambda}^{(0)}$ are infected. We then recursively define
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n+1)})$ for
$n\in[N^{\mathrm{m}+}]$ by CBSEP-extending ${\Lambda}^{(n)}$ in direction
$u_{n}$ by $l^{(n)}:=s_{n+k}^{(n)}=\Theta(2^{n/2k}/{\varepsilon})$ (recall
from Section 3.2 that indices of directions and sequences are considered
modulo $4k$ as needed and see Fig. 5(a)).

(a) A generic realisation of
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n)})$ depicting the SG
translates of ${\Lambda}^{(n)},\dots,{\Lambda}^{(n-2k)}$ involved in
progressive shades of grey. Each extension is as in Fig. 3(b).
${\Lambda}_{1}^{(n)}$${\Lambda}_{2}^{(n)}$${\Lambda}_{3}^{(n)}$ (b) The
setting of Definition 5.3. The tubes ${\Lambda}_{1}^{(n)}$ and
${\Lambda}_{3}^{(n)}$ of length ${\lambda}_{r}$ are hatched,
${\Lambda}_{2}^{(n)}={\Lambda}^{(n)}\setminus{\Lambda}_{3}^{(n)}$ is
thickened, while the symmetrically traversable tubes are in progressive shades
of grey.

Figure 5: Geometry of isotropic $\mathcal{S}\mathcal{G}$ and
$\overline{\mathcal{S}\mathcal{G}}$ events.

Recall from Section 3.6 that once
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n)})$ is defined, so is
${\gamma}({\Lambda}^{(n)})$. We next prove a bound on this quantity.

######  Theorem 5.2.

Let $\mathcal{U}$ be isotropic (class (g)). Then for all $n\leqslant
N^{\mathrm{m}+}$

$\displaystyle{\gamma}\left({\Lambda}^{(N^{\mathrm{m}+})}\right)$
$\displaystyle{}\leqslant\frac{\exp(1/(\log^{C/2}(1/q)q^{\alpha}))}{{\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(N^{\mathrm{m}+})}))},$
$\displaystyle{\mu}\left(\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{(n)}\right)\right)$
$\displaystyle{}\geqslant\exp\left(1/\left(q^{\alpha}{\varepsilon}^{2}\right)\right).$

The bound on ${\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n)}))$ is
fairly standard in bootstrap percolation and could essentially be attributed
to [Bollobas14], but we will prove it in Lemma 5.6, since we also need some
better bounds on intermediate scales. Bounding
${\gamma}({\Lambda}^{(N^{\mathrm{m}+})})$ is more demanding and will be done
by iteratively applying Proposition 4.9, as suggested by Definition 5.1.

###### Proof of Theorem 5.2.

Note that ${\gamma}({\Lambda}^{(0)})=1$, since Eq. 13 is trivial, because
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(0)})$ is a singleton. We seek
to apply Proposition 4.9, in order to recursively upper bound
${\gamma}({\Lambda}^{(n)})$ for all $n\leqslant N^{\mathrm{m}+}$. To that end,
we need the following definition of contracted events. Since, in the language
of Proposition 4.9, the events
$\overline{\mathcal{S}\mathcal{T}}_{{\eta}_{2}}$ we define do not depend on
${\eta}_{2}$, we will directly omit it from the notation.

######  Definition 5.3 (Contracted isotropic events).

For $n=2km+r\in[N^{\mathrm{m}+}+1]$ with $r\in[2k]$, as in Proposition 4.9
with $\underline{r}=\underline{r}^{(n)}$, $l=l^{(n)}$ and $i=r$, let

$\displaystyle{\Lambda}^{(n)}_{1}$
$\displaystyle{}=T\left(\underline{r}^{(n)},{\lambda}_{r},n+2k\right)$
$\displaystyle{\Lambda}^{(n)}_{2}$
$\displaystyle{}={\Lambda}\left(\underline{r}^{(n)}-{\lambda}_{r}\underline{v}_{r}\right)$
(18) $\displaystyle{\Lambda}^{(n)}_{3}$
$\displaystyle{}=T\left(\underline{r}^{(n)}-{\lambda}_{r}\underline{v}_{r},{\lambda}_{r},r\right).$

If $n<2k$, we define $\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{1}^{(n)})$,
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})$ and
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$ to occur if
${\Lambda}_{1}^{(n)}$, ${\Lambda}_{2}^{(n)}$ and ${\Lambda}_{3}^{(n)}$ is
fully infected respectively.

For $n\geqslant 2k$, we define
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{1}^{(n)})\subset{\Omega}_{{\Lambda}_{1}^{(n)}}$
(resp.
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})\subset{\Omega}_{{\Lambda}_{3}^{(n)}}$)
to be the event that for every segment $S\subset{\Lambda}_{1}^{(n)}$ (resp.
${\Lambda}_{3}^{(n)}$) perpendicular to some $u_{j}$ with $j\neq r\pm k$ of
length $2^{m}/(W{\varepsilon})$ the event $\mathcal{H}^{W}(S)$ occurs (recall
Definition 3.9). Finally, for $n\geqslant 2k$, we define
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})$ as the intersection
of the following events (see Fig. 5(b)):666Recall from Definition 4.1 that
$\mathcal{S}\mathcal{T}_{W}$ refers to symmetric traversability with
parallelograms in Fig. 3(a) shrunken by $W$, but not necessarily requiring
$W$-helping sets. Further recall from Section 4.1 that for isotropic models
$\mathcal{T}$ and $\mathcal{S}\mathcal{T}$ events are the same.

  * •

$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n-2k)})$;

  * •

$\mathcal{S}\mathcal{T}^{\mathbf{1}}(T(\underline{r}^{(n-2k)},l^{(n-2k)}/2-{\lambda}_{r},r))\cap\mathcal{S}\mathcal{T}^{\mathbf{1}}(T(\underline{r}^{(n-2k)},l^{(n-2k)}/2-{\lambda}_{r},r+2k))$;

  * •

for all $i\in(0,2k)$

$\mathcal{S}\mathcal{T}^{\mathbf{1}}_{W}\left(T\left(\underline{r}^{(n-2k+i)}-{\lambda}_{r}(\underline{v}_{r}+\underline{v}_{r+2k}),l^{(n-2k+i)}/2,r+i\right)\right)\\\
\cap\mathcal{S}\mathcal{T}^{\mathbf{1}}_{W}\left(T\left(\underline{r}^{(n-2k+i)}-{\lambda}_{r}(\underline{v}_{r}+\underline{v}_{r+2k}),l^{(n-2k+i)}/2,r+i+2k\right)\right).$

  * •

for every $i\in[2k]$, $j\in[4k]$ and segment $S\subset{\Lambda}_{2}^{(n)}$,
perpendicular to $u_{j}$ of length $2^{m}/(W{\varepsilon})$ at distance at
most $W$ from the $u_{j}$-side (parallel to $S$) of ${\Lambda}^{(n-2k+i)}$,
the event $\mathcal{H}^{W}(S)$ holds.

In words, $\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})$ is close to
being the event that the central copy of ${\Lambda}^{(n-2k)}$ in
${\Lambda}_{2}^{(n)}$ is SG and several tubes are symmetrically traversable.
Namely, for each $i\in[2k]$, the two tubes of equal length around
${\Lambda}^{(n-2k+i)}$ corresponding to a CBSEP-extension by $l^{(n-2k+i)}$ in
direction $u_{r}$, finally reaching ${\Lambda}^{(n)}$ after $2k$ extensions.
However, we have modified this event in the following ways. Firstly, the first
extension is shortened by $2{\lambda}_{r}$, so that the final result after the
$2k$ extensions fits inside ${\Lambda}_{2}^{(n)}$ and actually only its
$u_{r+k}$ and $u_{r-k}$-sides are shorter than those of ${\Lambda}_{2}^{(n)}$
by ${\lambda}_{r}$ (see Fig. 5(b)). Secondly, the symmetric traversability
events for tubes are required to occur with segments shortened by $W$ (recall
Definition 4.1) on each side. Finally, we roughly require $W$ helping sets for
the last $O(W)$ lines of each tube, as well as the first $O(W)$ outside the
tube (without going out of ${\Lambda}^{(n)}_{2}$).

######  Lemma 5.4 (CBSEP-extension relaxation condition).

For all $n\in[N^{\mathrm{m}+}]$ we have
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})\times\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})\subset\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}_{2}^{(n)}\cup{\Lambda}_{3}^{(n)})$
and similarly for ${\Lambda}_{1}^{(n)}$ instead of ${\Lambda}_{3}^{(n)}$.

###### Proof.

If $n<2k$, this follows directly from Definition 5.3, since
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})\times\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$
is only the fully infected configuration and similarly for
${\Lambda}_{1}^{(n)}$. We therefore assume that $n\geqslant 2k$ and set
$n=2km+r$ with $r\in[2k]$.

We start with the first claim. Note that
${\Lambda}_{2}^{(n)}\cup{\Lambda}_{3}^{(n)}={\Lambda}^{(n)}$. Let
${\eta}\in\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})\times\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$.
We proceed by induction on $i$ to show that
${\eta}_{{\Lambda}^{(i)}}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(i)})$
for $i\in[n-2k,n]$.

The base is part of Definition 5.3. Assume
${\eta}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(i)})$ for some
$i\in[n-2k,n)$. Then by Definition 4.7, it suffices to check that

${\eta}\in\mathcal{S}\mathcal{T}^{\mathbf{1}}\left(T\left(\underline{r}^{(i)},l^{(i)}/2,i\right)\right)\cap\mathcal{S}\mathcal{T}^{\mathbf{1}}\left(T\left(\underline{r}^{(i)},l^{(i)}/2,i+2k\right)\right),$
(19)

since then
${\eta}\in\mathcal{S}\mathcal{G}^{\mathbf{1}}_{l^{(i)}/2}({\Lambda}^{(i+1)})\subset\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(i+1)})$.

Let us first consider the case $i=n-2k$ and assume for concreteness that $m$
is even (so that $u_{i}=u_{r}$). Then

$\eta\in\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\subset\mathcal{S}\mathcal{T}^{\mathbf{1}}\left(T\left(\underline{r}^{(i)},l^{(i)}/2-{\lambda}_{r},r\right)\right),$

so by Lemma 4.3 it suffices to check that
${\eta}\in\mathcal{S}\mathcal{T}^{\mathbf{1}}(u_{i}(l^{(i)}/2-{\lambda}_{r})+T(\underline{r}^{(i)},{\lambda}_{r},i))$,
in order for the first symmetric traversability event in Eq. 19 to occur. We
claim that this follows from
${\eta}\in\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$ and the
fourth condition in Definition 5.3. To see this, notice that for each
$j\in[4k]$ the $u_{j}$-side length of ${\Lambda}(\underline{r}^{(i)})$
satisfies $s_{j}^{(i)}=\Theta(s_{j}^{(0)}2^{m})\gg 2^{m}/(W{\varepsilon})$ by
Eq. 17. Further recall from Section 3.5 that
$\mathcal{H}^{W}(S)\subset\mathcal{H}^{\omega}(S)$ for any segment $S$ of
length at least $C$ and boundary condition ${\omega}$. Thus, for each of the
segments in Definition 4.1 for the tube
$u_{i}(l^{(i)}/2-{\lambda}_{r})+T(\underline{r}^{(i)},{\lambda}_{r},i)\subset{\Lambda}^{(n)}$,
we have supplied not only a helping set, but in fact several $W$-helping sets.
For directions $u_{j}$ with $j\in(r-k,r+k)\setminus\\{r\\}$, they are in
${\Lambda}^{(n)}_{2}$, while for $j=r$ they are found in
${\Lambda}^{(n)}_{3}$, if $k=1$ and $m$ is even, and in ${\Lambda}^{(n)}_{2}$
otherwise (see Fig. 5(b)). Hence, the claim is established. For the second
event in Eq. 19 the reasoning is the same except that when $k>1$ or $m$ is
even, the tube $T(\underline{r}^{(i)},l^{(i)}/2,i+2k)$ is entirely contained
in ${\Lambda}_{2}^{(n)}$, so only
$\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)})$ is needed.

We next turn to the case $i\in(n-2k,n)$, which is treated similarly. Indeed,

${\eta}\in\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\subset\mathcal{S}\mathcal{T}_{W}^{\mathbf{1}}\left(T\left(\underline{r}^{(i)}-{\lambda}_{r}(\underline{v}_{r}+\underline{v}_{r+2k}),l^{(i)}/2,i\right)\right).$

Comparing this tube to the desired one in Eq. 19,
$T(\underline{r}^{(i)},l^{(i)}/2,i)$, we notice that the lengths and positions
of their sides differ by $O(1)$ (see Fig. 4). However, recalling Definitions
4.1 and 3(a), decreasing the width of each parallelogram there by
${\Omega}(W)\gg O(1)$ (using the event
$\mathcal{S}\mathcal{T}_{W}^{\mathbf{1}}$ rather than
$\mathcal{S}\mathcal{T}^{\mathbf{1}}$) is enough to compensate for this
discrepancy (the shaded zones in Fig. 4 are empty in this case). It remains to
ensure that the first and last $O(1)$ segments in Definition 4.1 also have
helping sets. But this is guaranteed by the fourth condition in Definition 5.3
and (depending on the values of $k$, $i$ and $m$)
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$ exactly as in the
case $i=n-2k$.

Finally, the statement for ${\Lambda}_{1}^{(n)}$ is also proved analogously
(with the offset for $i=n-2k$ modified by ${\lambda}_{r}$ in Eq. 19), so the
proof is complete. ∎

By Lemma 5.4, Eq. 16 holds, so we may apply Proposition 4.9. This gives

$\displaystyle{\gamma}\left({\Lambda}^{(n+1)}\right)\leqslant{}$
$\displaystyle\max\left({\mu}^{-1}\left(\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{(n)}\right)\right),{\gamma}\left({\Lambda}^{(n)}\right)\right)e^{O(C^{2})\log^{2}(1/q)}$
$\displaystyle\times\frac{{\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n)}))}{{\mu}(\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n+1)}))}{\mu}^{-1}\left(\left.\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{3}^{(n)}\right)\right|\mathcal{S}\mathcal{T}^{\mathbf{0}}\left({\Lambda}_{3}^{(n)}\right)\right)$
(20)
$\displaystyle\times{\mu}^{-1}\left(\left.\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{1}^{(n)}\right)\cap\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right)$

for $n\geqslant 2k$ and ${\gamma}({\Lambda}^{(n)})\leqslant
e^{O(C^{2})\log^{2}(1/q)}$ for $n<2k$. We therefore assume that $n\geqslant
2k$. Recalling Definition 5.3, note that both
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{1}^{(n)})$ and
$\overline{\mathcal{S}\mathcal{T}}({\Lambda}_{3}^{(n)})$ can be guaranteed by
the presence of $O(W^{2})$ well chosen infected $W$-helping sets, since only
$O(W)$ disjoint segments of length $2^{m}/(W{\varepsilon})$ perpendicular to
$u_{j}$ for a given $j\in(r-k,r+k)$ can be fit in ${\Lambda}_{1}^{(n)}$ or
${\Lambda}^{(n)}_{3}$ (see Fig. 5(b)), so it suffices to have a $W$-helping
set at each end of those. This and the Harris inequality, Eqs. 7 and 8, give

$\displaystyle{\mu}\left(\left.\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{3}^{(n)}\right)\right|\mathcal{S}\mathcal{T}^{\mathbf{0}}\left({\Lambda}_{3}^{(n)}\right)\right)\geqslant{\mu}\left(\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{3}^{(n)}\right)\right)\geqslant
q^{W^{O(1)}},$ (21)
$\displaystyle\begin{multlined}{\mu}\left(\left.\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{1}^{(n)}\right)\cap\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right)\\\
\geqslant
q^{W^{O(1)}}{\mu}\left(\left.\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right).\end{multlined}{\mu}\left(\left.\overline{\mathcal{S}\mathcal{T}}\left({\Lambda}_{1}^{(n)}\right)\cap\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right)\\\
\geqslant
q^{W^{O(1)}}{\mu}\left(\left.\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right).$
(24)

To deal with the last term we prove the following.

######  Lemma 5.5 (Contraction rate).

Setting $m=\lfloor n/(2k)\rfloor\geqslant 1$, we have

${\mu}\left(\left.\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}_{1}^{(n)}\cup{\Lambda}_{2}^{(n)}\right)\right)\\\
\geqslant\begin{cases}{\mu}\left(\overline{\mathcal{S}\mathcal{G}}\left({\Lambda}_{2}^{(n)}\right)\right)&2^{m}\leqslant
1/\left(\log^{C}(1/q)q^{\alpha}\right),\\\
q^{O(C)}\frac{{\mu}(\overline{\mathcal{S}\mathcal{G}}({\Lambda}_{2}^{(n)}))}{{\mu}\left(\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{(n-2k)}\right)\right)}&2^{m}\geqslant\log^{C}(1/q)/q^{\alpha},\\\
\exp\left(-2^{m}q^{1-o(1)}\right)&\text{otherwise}.\end{cases}$ (25)

###### Proof.

The first case follows from the Harris inequality Eq. 7.

For the other two cases we start by noting that
${\Lambda}^{(n)}_{1}\cup{\Lambda}^{(n)}_{2}={\Lambda}^{(n)}-{\lambda}_{r}u_{r}$
may be viewed as a $2k$-fold CBSEP-extension of ${\Lambda}^{(n-2k)}$.
Recalling the offset in Definition 4.7, set

$\displaystyle\mathcal{S}\mathcal{G}^{\bullet}_{0}$
$\displaystyle{}=\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{(n)}-{\lambda}_{r}u_{r}\right),$
$\displaystyle\mathcal{S}\mathcal{G}^{\bullet}_{i}$
$\displaystyle{}=\bigcap_{j=1}^{i}\mathcal{S}\mathcal{G}^{\mathbf{1}}_{l^{(n-j)}/2}\left({\Lambda}^{(n-j+1)}-{\lambda}_{r}u_{r}\right)\qquad
i{}\in[1,2k-1],$ $\displaystyle\mathcal{S}\mathcal{G}^{\bullet}_{2k}$
$\displaystyle{}=\mathcal{S}\mathcal{G}^{\bullet}_{2k-1}\cap\mathcal{S}\mathcal{G}^{\mathbf{1}}_{l^{(n-2k)}/2-{\lambda}_{r}}\left({\Lambda}^{(n-2k+1)}-{\lambda}_{r}u_{r}\right),$

so that $\mathcal{S}\mathcal{G}^{\bullet}_{i}$ corresponds to fixing the
position of the core, which is a translate of ${\Lambda}^{(n-i)}$, inside
${\Lambda}^{(n)}-{\lambda}_{r}u_{r}$, but leaving its internal offsets
unconstraint (see Fig. 5(b)). Thus, Lemma 4.10 applied $2k$ times gives

${\mu}\left(\left.\mathcal{S}\mathcal{G}^{\bullet}_{2k}\right|\mathcal{S}\mathcal{G}^{\mathbf{1}}\left({\Lambda}^{(n)}_{1}\cup{\Lambda}^{(n)}_{2}\right)\right)=\prod_{i=1}^{2k}{\mu}(\mathcal{S}\mathcal{G}^{\bullet}_{i}|\mathcal{S}\mathcal{G}^{\bullet}_{i-1})\geqslant
q^{O(C)}.$

Expanding the definition of $\mathcal{S}\mathcal{G}^{\bullet}_{2k}$ via
Definition 4.7, we see that this event is the intersection of
$\mathcal{S}\mathcal{G}^{\mathbf{1}}({\Lambda}^{(n-2k)})$ with some increasing