3.1 Setting and preliminaries

Let $\mathcal{U}$ be an update family. Assuming they exist, we further fix two non-collinear rational stable directions $u_1$ and $u_2$ . We set $u_3=u_1+\pi$ , $u_4=u_2+\pi$ and $\mathcal{T}=\{u_1,u_2,u_3,u_4\}$ . We will simply call parallelogram a set of the form

\begin{align*}R(a,b;\ c,d)&=\left\{x\in{{\mathbb{R}}}^2\,|\,\langle x,u_3\rangle\in[a,c],\langle x,u_4\rangle\in[b,d]\right\}\\[3pt] &=\bar{{\mathbb{H}}}_{u_1}(\!-a) \cap \bar{{\mathbb{H}}}_{u_2}(\!-b) \cap \bar{{\mathbb{H}}}_{u_3}(c) \cap \bar{{\mathbb{H}}}_{u_4}(d)\end{align*}

for real numbers $a \leqslant b,c \leqslant d$ and denote by its topological interior. For parallelograms we will systematically extend definitions by translation and interchange of $u_1$ and $u_2$ (resp. $u_3$ and $u_4$ ).

Finally,

\begin{equation*}C_6\gg C_5\gg C_2^{\prime}\gg C_1\gg r=\max\{\|s-s'\| \,|\, s,s' \in U\cup\{0\},U \in \mathcal{U}\}\end{equation*}

are constants not depending on q, but only on $\mathcal{U}$ and $\mathcal{T}$ , each one sufficiently large with respect to functions of the next.Footnote 3 In addition, $K=K(q)$ will be a function such that $K \rightarrow +\infty$ when $q \rightarrow 0$ , so that $K\gg C_6$ . The value of K will depend on the class of models studied, hence will be specified in Section 4. Furthermore, we systematically assume that q is small enough, as we are interested in $q\rightarrow0$ .

If $\eta,\eta'$ are two configurations, we say that $\eta\leqslant\eta'$ when $\eta_s\leqslant\eta_s^{\prime}$ for all s. For instance, if a parallelogram D is spanned for $\eta$ , then it is also spanned for any $\eta'\leqslant\eta$ . This order should not be confused with the (inverted) one induced by inclusion when viewing $\eta$ as its set of infections.

Notice that the event that a given parallelogram D is spanned depends only on $\eta_D$ and does not occur when $\eta_D$ contains no infections. We further state two immediate consequences of Definition 3.1 for future reference.

Proof. By maximality of a strongly connected component, $[\eta\cap X]\subset X$ is at distance at least $C_2^{\prime}>2r$ from other strongly connected components X of $[\eta]$ . Thus,

\begin{equation*}[\eta]=\bigsqcup_Y [\eta\cap Y],\end{equation*}

where the union is on all strongly connected components of $[\eta]$ .

Another standard fact is the following Aizenman-Lebowitz lemma originating from [Reference Aizenman and Lebowitz1], whose proof can be found in the appendices of the arXiv version of this paper [Reference Hartarsky and Marêché18, Lemma A.9].

We next import and adapt the notion of crossing from [Reference Bollobás, Duminil-Copin, Morris and Smith4, Definition 8.16].

In words, a parallelogram is $u_1$ -crossed if, given an infected boundary condition on its side opposite to direction $u_1$ , the infections inside it are sufficient to infect a ‘path’ reaching its side in direction $u_1$ . Thus, a $u_1$ -crossing corresponds to the propagation of infection across the parallelogram in the direction $u_1$ . The event $C_R^{u_1}$ further demands that this crossing is achieved without the help of large spanned parallelograms (in view of Lemma 3.4). We should note that, while parallelograms smaller than critical are increasingly unlikely as their size grows, parallelograms larger than critical ones roughly become more likely with their size (hence the name of critical ones), so $C_R^{u_1}$ forces us to use only small unlikely parallelograms all along the crossing. Relying on this fact, it will be possible to establish very strong bounds on the probability of $C_R^{u_1}$ . This leads to the notion of a $(\ell,h)$ -crossing, which will one of the ‘good’ properties we will require the dynamics to satisfy.

Finally, we say that a site $s \in {{\mathbb{Z}}}^2$ is locally infectable in a configuration $\eta$ if $s \in [\eta \cap$ $(s+R(\!-2K,-2K;\ 2K,2K))]$ . We also denote by $\eta^s$ the configuration equal to $\eta$ everywhere except at s, that is $\eta^s_{s}=1-\eta_s$ and $\eta^s_{s'}=\eta_{s'}$ for any $s' \neq s$ . We then have the following property, originating from [Reference Hartarsky, Martinelli and Toninelli20, Section 2].

Proof. By definition, $0 \in [\eta^s\cap R]\backslash [\eta \cap R]$ . Therefore, $s \in R$ and $s+U\not\subset R$ . In particular, $d(s,0)> K$ . Let X be the strongly connected component of 0 in $[\eta^s\cap R]$ . We claim that $s\in X$ . Indeed, by Observation 3.3 we have $0\in X=[\eta^s\cap R \cap X]$ , so if $s\not\in X$ , we have $0\in [\eta\cap R]$ which is not the case.

Let $D_0$ be the smallest parallelogram containing X. Since $D_0$ contains 0 and s, its diameter is at least K. Moreover, $X=[\eta^s\cap R \cap X]\subset[\eta_R^s \cap D_0]$ , so X is a strongly connected set in $[\eta_R^s \cap D_0]$ . Therefore, $D_0$ is spanned in $\eta_R^s$ and has diameter at least K. In particular, by Lemma 3.4 there exists a critical parallelogram D spanned in $\eta^s_{R}$ . By Observation 3.2, D intersects R, and by Definition 3.1 ${{\textrm{diam}}}(D)\leqslant K$ . Thus, we have $D\subset R(\!-3K,-3K;\ 3K,3K)$ , which ends the proof of the lemma.

3.2 The combinatorial bottleneck

With the notation above we are ready to prove a very general deterministic bottleneck (Lemma 3.10 below), constituting the core of our work, which relatively straightforwardly translates into the following bound on ${{\mathbb{E}}}_{\mu}[\tau_0]$ .

The idea behind it is that for the centre of a parallelogram of size roughly $3^n$ to become infected, either an $(\ell,h)$ -crossing should occur or we should witness $n+1$ spanned critical parallelograms simultaneously. Assuming upper bounds on the probabilities of these two events, we deduce a lower bound on ${{\mathbb{E}}}_\mu[\tau_0]$ .

Proposition 3.7 Let $T,L,H,K,\ell,h$ be positive real numbers sufficiently large with respect to $C_2^{\prime}$ . Denote

\begin{equation*}\rho=\max_D\mu(D\text{ is spanned}),\end{equation*}

where the max is over all critical parallelograms. Also set

\begin{align*}{p_{\leftarrow}}&{}=\max_{a,b\in{{\mathbb{R}}}}\mu\left(C_{R(a,b;\ a+\ell,b+H)}^{u_1}\right)\\[3pt] {p_{\downarrow}}&{}=\max_{a,b\in{{\mathbb{R}}}}\mu\left(C_{R(a,b;\ a+L,b+h)}^{u_2}\right)\!.\end{align*}

Assume that for some integer $n\geqslant0$ we have the following inequalities on geometry:

(4) \begin{align}L&{}\geqslant 3^{n}(11K+\ell)&H&{}\geqslant 3^n(11K+h),\end{align}

and probability:

(5) \begin{align}1/8&{}\geqslant \mu(0\text{ is locally infectable})\end{align}

(6) \begin{align}1&{}\geqslant T(LH)^2\max({p_{\downarrow}},{p_{\leftarrow}})\end{align}

(7) \begin{align}1&{}\geqslant TLH(LHK^3\rho)^{n+1}.\end{align}

Then the $\mathcal{U}$ -KCM on ${{\mathbb{Z}}}^2$ satisfies ${{\mathbb{E}}}_\mu[\tau_0]\geqslant T$ .

The proof of Proposition 3.7 will occupy the rest of the present section. We start by fixing $T,L,H,K,\ell,h$ as in the statement and introducing the following definitions.

Recall that for any $R \subset {{\mathbb{Z}}}^2$ , we identify configurations $\eta\in\{0,1\}^R$ with their zero set $\{x\,{\in}\, R,\eta_x\,{=}\,0\}$ . Unless otherwise specified, configurations $\eta\in\{0,1\}^R$ are extended to $\{0,1\}^{{{\mathbb{Z}}}^2}$ by keeping the same zero set.

Definition 3.9 (Good paths and configurations) For any parallelogram $R \subset {{\mathbb{R}}}^2$ , configuration $\eta \in \{0,1\}^{R \cap {{\mathbb{Z}}}^2}$ and integer $n\geqslant0$ , we say that $\eta$ is n-good when the maximum number of critical parallelograms that are disjointlyFootnote 4 spanned in $\eta$ is at most n and R has no crossing for $\eta$ .

A legal path in R is a sequence $(\eta^{(j)})_{0 \leqslant j \leqslant m}$ of configurations in $\{0,1\}^{R \cap {{\mathbb{Z}}}^2}$ such that for every $j \in \{0,\dots,m-1\}$ , there exists $s\in R\cap{{\mathbb{Z}}}^2$ such that $\eta^{(j+1)}=(\eta^{(j)})^s$ and $(s+U)\cap R\subset \eta^{(j)}$ for some $U \in \mathcal{U}$ . For any integer $n\geqslant 0$ , the path is n-good if for every $j \in\{0,\dots,m\}$ , $\eta^{(j)}$ is n-good. For any $A,B \subset \{0,1\}^{R \cap {{\mathbb{Z}}}^2}$ , we say $(\eta^{(j)})_{0 \leqslant j \leqslant m}$ is a path from A to B when $\eta^{(0)} \in A$ and $\eta^{(m)} \in B$ (if A or $B=\{\eta\}$ , we will write $\eta$ to simplify).

We denote by G(R) the set of configurations in $\{0,1\}^{R \cap {{\mathbb{Z}}}^2}$ that contain no spanned critical parallelogram and such that R contains no crossing, that is the 0-good configurations. For any $n \in \mathbb{N}$ , we define

\begin{equation*}V(n,R) = \left\{\eta \in \{0,1\}^{R \cap {{\mathbb{Z}}}^2} \,|\text{ there is an {n}-good legal path from {G}{R} to $\eta$}\right\}.\end{equation*}

Finally, we define our domain sizes for the induction to come:

(8) \begin{align}L_n &{}= \frac{3^n-1}{2}\left(9K+\ell\right)+3^nK&H_n &{}= \frac{3^n-1}{2}\left(9K+h\right)+3^nK,\end{align}

so that $L_n-L_{n-1}=2L_{n-1}+9K+\ell$ and $H_{n}-H_{n-1}=2H_{n-1}+9K+h$ .

We first deduce Proposition 3.7 from Lemma 3.10.

Proof of Proposition 3.7, assuming Lemma 3.10. Clearly, it suffices to prove that ${{\mathbb{P}}}_{\mu}(\tau_0> 2T)\geqslant 1/2$ . Let $\tau'=\inf\{t\geqslant 0,0$ is locally infectable in $\eta(\textrm{t})\}$ . Clearly, $\tau'\leqslant \tau_0$ . We denote $R=R(\!-L/2,-H/2;\ L/2,H/2)$ and define the following events.

1. E ₁: $\tau'>0$ , that is 0 is not locally infectable in $\eta(0)$ .

2. E ₂: There is no critical parallelogram spanned in $(\eta(0))_R$ .

3. E ₃: For all $0\leqslant t\leqslant 2T$ , no $n+1$ critical parallelograms are disjointly spanned in $(\eta(t))_R$ .

4. E ₄: For all $0\leqslant t\leqslant 2T$ , R has no crossing for $(\eta(t))_R$ .

We claim that $\bigcap_{i=1}^4E_i\subset \{\tau'>2T\}$ . Indeed, let us assume $\bigcap_{i=1}^4E_i$ occurs. By Definition 3.9, $E_2$ and $E_4$ imply $(\eta(0))_R\in G(R)$ . Moreover, $E_3$ and $E_4$ give that $(\eta(t))_R$ is n-good for all $t\leqslant 2T$ . Thus, for all $t\leqslant 2T$ , the sequence of configurations $(\eta(t'))_R$ for $t'\in[0,t]$ yields a n-good legal path from $(\eta(0))_R\in G(R)$ to $(\eta(t))_R$ , therefore $(\eta(t))_R\in V(n,R)$ . Applying Lemma 3.10, this yields that for all $t\in[0,2T]$ there is no critical parallelogram spanned in $(\eta(t))_R$ intersecting

\begin{align*}&R(\!-L/2+L_n,-H/2+H_n;\ L/2-L_n,H/2-H_n)\\[3pt] &\quad \supset R\left(\!-\frac{9K+\ell}{2},-\frac{9K+h}{2};\ \frac{9K+\ell}{2},\frac{9K+h}{2}\right)\supset R(\!-\!3K,-3K;\ 3K,3K),\end{align*}

recalling (4) and (8). Finally, notice that 0 being locally infectable depends only on the configuration in $R(\!-2K,-2K;\ 2K,2K)$ . Thus, if $E_1$ occurs, we can apply Lemma 3.6 with $\eta^s\,{:\!=}\,(\eta(\tau'))_R$ and $\eta\,{:\!=}\,\lim_{t\rightarrow\tau',t<\tau'}(\eta(t))_R$ , to deduce that there is a critical parallelogram spanned in $(\eta(\tau'))_R$ and contained in $R(\!-3K,-3K;\ 3K,3K)$ . Hence, $\tau'>2T$ , as claimed.

We next brutally bound the probability of $E_1,\dots,E_4$ . By (5), $1-{{\mathbb{P}}}_{\mu}(E_1)\leqslant 1/8$ . By the union bound on all (discrete) critical parallelograms intersecting R (recall Observation 3.2) and (7) we have

\begin{equation*}1-{{\mathbb{P}}}_{\mu}(E_2)\leqslant O(LHK^2)\rho\leqslant \frac{LHK^3\rho}{8}\leqslant\frac{1}{8(TLH)^{1/(n+1)}}\leqslant \frac 1 8.\end{equation*}

In order to treat $E_3$ and $E_4$ , recall from Section 1.1 that the $\mathcal{U}$ -KCM may be constructed by associating to each site $x\in{{\mathbb{Z}}}^2$ a standard Possion process and attempting to update $\eta_x$ at the times given by its Poisson process. We will refer to these times as clock rings. Let N denote the number of clock rings in R between 0 and 2T. Since N is a Poisson random variable with parameter $2T|R|=\Theta(TLH)>1$ , we have ${{\mathbb{P}}}_{\mu}(N\geqslant C_1TLH)\leqslant 1/16$ (e.g. by the Bienaymé–Chebyshev inequality).

Let $\eta^{(j)}$ denote the restriction of the configuration to $R \cap{{\mathbb{Z}}}^2$ after the j-th clock ring in R. We next claim that $\eta^{(j)}$ is at equilibrium, the formal proof being postponed for the moment.

For any $\eta \in \{0,1\}^{R \cap {{\mathbb{Z}}}^2}$ we write $\mathcal{D}_{n+1}(\eta) = \{$ there are $n+1$ critical parallelograms disjointly spanned in $\eta\}$ , so that $E_3$ does not occur iff $\bigcup_{j=0}^N\mathcal{D}_{n+1}(\eta^{(j)})$ does not. Then the union bound and Claim 3.11 give

(9) \begin{align}1-{{\mathbb{P}}}_{\mu}(E_3)\leqslant{}& {{\mathbb{P}}}_{\mu}(N\geqslant C_1TLH)+\sum_{j=0}^{C_1TLH}{{\mathbb{P}}}_{\mu}\left(\mathcal{D}_{n+1}\left(\eta^{(j)}\right)\right)\nonumber\\{}\leqslant{}& \frac{1}{16}+2C_1TLH\mu(\mathcal{D}_{n+1}(\eta_R)).\end{align}

In order to bound $\mu(\mathcal{D}_{n+1}(\eta_R))$ , we use the union bound on all $(O(LHK^2))^{n+1}$ possible choices of $n+1$ critical parallelograms intersecting R (by Observation 3.2, a parallelogram spanned in $\eta_R$ intersects R) together with the BK inequality [Reference van den Berg and Kesten33] to get that (9) is at most

\begin{equation*}\frac{1}{16}+2C_1TLH(O(LHK^2)\rho)^{n+1}\leqslant \frac{1}{8},\end{equation*}

using (7) in the last inequality.

Similarly, using (6), we have

\begin{equation*}1-{{\mathbb{P}}}_{\mu}(E_4)\leqslant \frac{1}{16}+2C_1TLH\cdot O(L+H)\max({p_{\downarrow}},{p_{\leftarrow}}) \leqslant \frac{1}{8}.\end{equation*}

Putting the bounds $1-{{\mathbb{P}}}_\mu(E_i)\leqslant 1/8$ for all i together and recalling that $\bigcap_{i=1}^4E_i\subset \{\tau'>2T\}$ , we conclude that

\begin{align*}{{\mathbb{P}}}_{\mu}(\tau_0> 2T)\geqslant {{\mathbb{P}}}_\mu(\tau'>2T)\geqslant {{\mathbb{P}}}_\mu\left(\bigcap_{i=1}^4E_i\right)\geqslant 1- \sum_{i=1}^4(1-{{\mathbb{P}}}_{\mu}(E_i))\geqslant \frac{1}{2}.\\[-40pt]\end{align*}

Proof of Claim 3.11. We fix j, and denote by t the time of the j-th clock ring in R. For all $s\in[0,t]$ , we will construct a (random) set of sites $X_s$ such that the configuration in R at time t can be reconstructed from the configuration in $X_s$ at time s and the updates since time s. The construction is as follows. For all $s\in[0,t]$ , $X_s$ contains all sites in R. Moreover, for any $x_0\in R$ which had a clock ring before time t, for any s before this clock ring we add to $X_s$ the sites $x_1$ such that $x_1-x_0\in{{\mathbb{U}}}$ , where ${{\mathbb{U}}}=\bigcup_{U\in\mathcal{U}}U$ . Now, for any of these $x_1$ that had a clock ring before the clock ring at $x_0$ , for any s before this clock ring at $x_1$ we add the sites $x_2$ such that $x_2-x_1\in{{\mathbb{U}}}$ , etc. It is classical to see that the $X_s$ are a.s. finite. For example, one can see there are exactly j clock rings in R before time t, so clock rings in R may add at most $j|{{\mathbb{U}}}|$ sites to the $X_s$ . Now, for any of these sites that is not in R, the number of sites it brings can be bounded from above by a continuous time branching process with reproduction law $\delta_{|{{\mathbb{U}}}|+1}$ , so stays finite.

We now consider all the clock rings such that there exists $s\in[0,t]$ such that the clock ring occurs in $X_s$ at time at most s, and order them chronologically. If s is the time of the j $^{\prime}$ -th such clock ring, $\bar\eta^{(j')}$ will denote the configuration in $X_s$ at time s. Let us denote by $\mathcal{F}$ the sigma-algebra generated by all clock rings in ${{\mathbb{Z}}}^2$ . In particular, for any $s\in[0,t]$ , $X_s$ is $\mathcal{F}$ -measurable. By induction on j $^{\prime}$ , one can prove that conditionally to $\mathcal{F}$ , $\bar\eta^{(j')}$ has law $\mu$ . Now, the j $^{\prime}$ corresponding to j depends only on $\mathcal{F}$ , hence $\eta^{(j)}$ has law $\mu$ , which proves the claim.

Proof of Lemma 3.10. We will prove the lemma by induction on n. For any n let us call $\mathcal{H}_n$ the statement of the lemma for n. $\mathcal{H}_0$ holds by definition. Let $n \geqslant 1$ and assume that $\mathcal{H}_{n-1}$ holds. Let $R=R(a,b;\ c,d)$ be such that $c-a \geqslant 2L_n$ and $d-b \geqslant 2H_n$ . We define a smaller parallelogram R $^{\prime}$ in the middle of R:

\begin{equation*}R'=R(a+L_n-L_{n-1},b+H_n-H_{n-1};\ c-(L_n-L_{n-1}),d-(H_n-H_{n-1}))\end{equation*}

(see Figure 2). We will prove Lemma 3.10 by showing $\mathcal{H}_{n}$ , using the following result, whose proof we postpone for the moment.

Let $\eta \in V(n,R)$ . The idea is that since $\eta$ is n-good, there are at most n spanned critical parallelograms in $\eta$ , and Lemma 3.12 implies one of them has to be outside R $^{\prime}$ . Thus, there can only be $n-1$ such parallelograms in R $^{\prime}$ . This will allow us to use $\mathcal{H}_{n-1}$ in R $^{\prime}$ , which will prevent the existence of a spanned critical parallelogram intersecting $R(a+L_n,b+H_n;\ c-L_n,d-H_n)$ .

We now give the rigorous argument. Since $\eta \in V(n,R)$ , there exists an n-good legal path from G(R) to $\eta$ , denoted by $(\eta^{(j)})_{0 \leqslant j \leqslant m}$ . For any $j\in\{0,\dots,m\}$ , we have $\eta^{(j)}\in V(n,R)$ . Let us prove that there are at most $n-1$ critical parallelograms disjointly spanned in $\eta^{(j)}_{R'}$ . If $\eta^{(j)}\in G(R)$ , it is clear. If $\eta^{(j)}\not\in G(R)$ , Lemma 3.12 guarantees the existence of a critical parallelogram D spanned in $\eta^{(j)}$ that does not intersect R $^{\prime}$ . Now, let k be the maximal number of critical parallelograms disjointly spanned in $\eta^{(j)}_{R'}$ , and $D_1,\dots,D_k$ be such parallelograms. Since D does not intersect R $^{\prime}$ , we have that $D,D_1,\dots,D_k$ are critical parallelograms disjointly spanned in $\eta^{(j)}$ . Since $\eta^{(j)}$ is n-good, it contains at most n disjointly spanned critical parallelograms, so $k\leqslant n-1$ . Furthermore, if R $^{\prime}$ had a crossing for $\eta^{(j)}_{R'}$ , R would have a crossing for $\eta^{(j)}$ , which is not the case, as $\eta^{(j)}$ is n-good. Hence, R $^{\prime}$ has no crossing for $\eta^{(j)}_{R'}$ . We deduce that $\eta^{(j)}_{R'}$ is $(n-1)$ -good in the parallelogram R $^{\prime}$ . Thus, if we consider the path $(\eta^{(j)}_{R'})_{0 \leqslant j \leqslant m}$ and keep only the steps $\eta^{(j)}_{R'}$ that differ from $\eta^{(j-1)}_{R'}$ , we obtain a $(n-1)$ -good legal path, going from G(R $^{\prime}$ ) to $\eta_{R'}$ . This implies $\eta_{R'}\in V(n-1,R')$ . Therefore, we can apply $\mathcal{H}_{n-1}$ to $\eta_{R'}$ , which yields that there is no spanned critical parallelogram in $\eta_{R'}$ intersecting $R(a+L_n,b+H_n;\ c-L_n,d-H_n)$ . This implies $\mathcal{H}_n$ and concludes the proof of Lemma 3.10.

Consequently, it remains to prove Lemma 3.12.

Proof of Lemma 3.12. Another way to state Lemma 3.12 is to say that if we start a n-good legal path from G(R), we cannot reach a configuration in which all spanned critical parallelograms intersect R $^{\prime}$ . As legal paths are reversible, we will prove that if we start an n-good legal path from a configuration in which all spanned critical parallelograms intersect R $^{\prime}$ , it is impossible to reach G(R), because there will always be a spanned critical parallelogram ‘near’ R $^{\prime}$ .

Figure 2.

The setting of the proof of Lemma 3.12. For the figure we assume that $u_3=0$ and $u_4=\pi/2$ . B $^{\prime}$ is the frame with thickened boundary, $R_\ell$ and $R_h$ are the overlapping regions in dark grey. The regions $R_{\ell,\downarrow}^{\prime}$ , $R_{h,\downarrow}^{\prime}$ , $R_{\ell,\uparrow}^{\prime}$ and $R_{h,\uparrow}^{\prime}$ are in lighter grey and the frame formed by their union is B. The horizontally (resp. vertically) hatched regions are $R_{h,\downarrow}$ and $R_{h,\uparrow}$ (resp. $R_{\ell,\downarrow}$ and $R_{\ell,\uparrow}$ ). The contour inside B $^{\prime}$ is $\gamma$ and its diagonally hatched interior is . All the regions drawn are closed subsets of ${{\mathbb{R}}}^2$ with the exception of $R_\ell$ , $R_h$ and , which are open. The thicker version, $\bar\gamma$ , of $\gamma$ and the set $F\subset{{\mathbb{Z}}}^2$ defined in (10) are not drawn.

In order to do that, we start by introducing the following geometric regions, represented in Figure 2.

$R_\ell$ will contain the aforementioned spanned critical parallelogram ‘near’ R $^{\prime}$ . In the parallelograms $R_{\ell,\downarrow}$ and $R_{\ell,\uparrow}$ , we will use $\mathcal{H}_{n-1}$ , and $R^{\prime}_{\ell,\downarrow}$ , $R^{\prime}_{\ell,\uparrow}$ are the respective ‘central’ parallelograms inside that will not be intersected by spanned critical parallelograms. We also define similar regions with index h instead of $\ell$ .

We further define the two frames (see Figure 2)

As the union of $R^{\prime}_{\ell,\downarrow}$ , $R^{\prime}_{\ell,\uparrow}$ , $R^{\prime}_{h,\downarrow}$ , $R^{\prime}_{h,\uparrow}$ , the frame B will not be intersected by spanned critical parallelograms. B will be a ‘buffer’ between the inner and outer parts of R, and B $^{\prime}$ its central part. We will be able to find a contour $\gamma$ contained in B $^{\prime}$ such that the dynamics in the interior of the contour is ‘isolated’ from the dynamics outside in a specific way. This is the goal of the following claim.

Proof. We proceed by renormalisation. Let H denote the regular hexagon centred at the origin with diameter $C_1^3$ and having two horizontal sides. Consider the tiling of the plane with translates of H and denote by ${{\mathbb{T}}}$ the triangular lattice formed by their centres. Let $T=\{t\in{{\mathbb{T}}}|H+t\subset B'\}$ be the sites of ${{\mathbb{T}}}$ corresponding to B $^{\prime}$ . We say that a site $t\in T$ is open if no site in $(t+H)\cap{{\mathbb{Z}}}^2$ is locally infectable in $\eta$ .

If there exists a contour of open sites in T surrounding R $^{\prime}$ (where a contour in ${{\mathbb{T}}}$ is a self-avoiding and closed path in the graph $({{\mathbb{T}}},\{(t,t')\in{{\mathbb{T}}}^2|t+H$ and $t'+H$ share a side $\}$ ), we may choose $\gamma$ approximating this contour, which clearly satisfies the conditions of the claim. Assume for a contradiction that such a contour does not exist. In this case, there is a path of closed sites in T from the inner to the outer boundary of T. In particular, this path yields a strongly connected (recall Definition 3.1) set X of sites of ${{\mathbb{Z}}}^2$ that are locally infectable in $\eta$ , with diameter at least $K-4C_1^3$ , contained in either the ‘left part’ of the frame B $^{\prime}$ , defined as

\begin{equation*}R^{\prime\prime}_{\ell,\downarrow} = R(a+L_{n-1}+3K,b+H_{n-1}+3K;\ a+L_{n-1}+4K,d-H_{n-1}-3K),\end{equation*}

(see Figure 2) or in the top, right or bottom part of B $^{\prime}$ , defined similarly. Without loss of generality, assume that X is contained in $R^{\prime\prime}_{\ell,\downarrow}$ . Since the sites of X are locally infectable in $\eta$ , they are infectable in $\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}$ , where $R^{\prime\prime\prime}_{\ell,\downarrow}$ is ‘ $R^{\prime\prime}_{\ell,\downarrow}$ enlarged by 2K on each side,’ that is

\begin{equation*}R^{\prime\prime\prime}_{\ell,\downarrow} = R(a+L_{n-1}+K,b+H_{n-1}+K;\ a+L_{n-1}+6K,d-H_{n-1}-K).\end{equation*}

X is then a strongly connected set contained in $[\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}]$ .

We denote by X $^{\prime}$ the strongly connected component of $[\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}]$ containing X, and we consider the smallest parallelogram D containing X $^{\prime}$ . We claim that D is spanned in $\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}$ . Indeed, Observation 3.3 yields $X'=[\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}} \cap X'] \subset [\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}} \cap D]$ , so X $^{\prime}$ is a strongly connected set in $[\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}} \cap D]$ such that the smallest parallelogram containing X $^{\prime}$ is D, which means that D is spanned in $\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}$ . Furthermore, D contains X, which has diameter at least $K-4C_1^3$ . Therefore, by Lemma 3.4 there exists a critical parallelogram D $^{\prime}$ spanned in $\eta_{R^{\prime\prime\prime}_{\ell,\downarrow}}$ . D $^{\prime}$ is then spanned in $\eta$ . Moreover, by Observation 3.2, D $^{\prime}$ intersects $R^{\prime\prime\prime}_{\ell,\downarrow}$ , and, since D $^{\prime}$ is critical, it has diameter at most K. Hence, D $^{\prime}$ is contained in $R^{\prime}_{\ell,\downarrow}$ . We deduce the existence of a critical parallelogram spanned in $\eta$ not intersecting R $^{\prime}$ , hence a contradiction.

Now that most of the needed geometric regions are defined, we may start the proof of Lemma 3.12 itself. We fix $\eta \in \{0,1\}^{R \cap {{\mathbb{Z}}}^2} \backslash G(R)$ such that every critical parallelogram spanned in $\eta$ intersects R $^{\prime}$ . We will prove that there is no n-good legal path from $\eta$ to G(R). Since legal paths can be reversed, this implies $\eta \not\in V(n,R)$ , which proves Lemma 3.12.

We fix a contour $\gamma$ as provided by Claim 3.13 for the configuration $\eta$ (see Figure 2), as well as its thickened version $\bar\gamma$ and its interior . In order to isolate the dynamics in from the dynamics outside, we need to define

(10) \begin{equation}F=\{s\in B'\cap{{\mathbb{Z}}}^2|s\text{ is not locally infectable in }\eta\}.\end{equation}

We then have $\bar\gamma\subset F$ . F will ‘shield from outside interference.’

Let $(\eta^{(j)})_{0 \leqslant j \leqslant m}$ be an n-good legal path with $\eta^{(0)}=\eta$ . We will use an induction on $j \in \{0,\dots,m\}$ to prove that $\eta^{(m)} \not\in G(R)$ . More precisely, we will prove by induction on j that the following properties hold for $j \in \{0,\dots,m\}$ .

1. $\mathcal{P}_j^1$ For every $\zeta \in \{\ell,h\}$ , there exists a critical parallelogram contained in $R_\zeta$ spanned in $\eta^{(j)}$ .

2. $\mathcal{P}_j^2$ The sites of F are not locally infectable in $\eta^{(j)}$ .

3. $\mathcal{P}_j^3$ For every $(\zeta,\xi) \in \{\ell,h\}\times\{\downarrow,\uparrow\}$ , $\eta^{(j)}_{R_{\zeta,\xi}} \in V(n-1,R_{\zeta,\xi})$ .

4. $\mathcal{P}_j^4$ .

$\mathcal{P}_j^4$ is what we mean by ‘the dynamics inside is isolated from the outside’: from time 0 to time j, the sites infectable by the configuration in have not changed. $\mathcal{P}_j^2$ is necessary for F to play its role as a shield throughout the path. $\mathcal{P}_j^3$ , along with $\mathcal{H}_{n-1}$ , will ensure that there are no spanned critical parallelograms at the centre of the $R_{\zeta,\xi}$ , which will help to preserve the shield. Finally, if $\mathcal{P}^1_m$ is satisfied, then there exists a critical parallelogram spanned in $\eta^{(m)}$ , so $\eta^{(m)} \not \in G(R)$ , which proves Lemma 3.12, so it suffices to establish the induction.

We begin with a quick claim.

Proof. Since $\eta$ belongs to an n-good legal path, R has no crossing for $\eta$ , but by the definition of $\eta$ , $\eta\not\in G(R)$ , hence there exists a critical parallelogram spanned in $\eta$ . Moreover, by assumption, every critical parallelogram spanned in $\eta$ intersects R $^{\prime}$ , hence the claim.

With this, we can start work on the induction.

Base: j = 0. $\mathcal{P}_0^4$ is trivial, and the definition (10) of F implies $\mathcal{P}_0^2$ . For $\mathcal{P}_0^1$ , Claim 3.14 gives the existence of a critical parallelogram spanned in $\eta$ intersecting R $^{\prime}$ , hence contained in $R_\ell\cap R_h$ , which yields $\mathcal{P}_0^1$ .

We now show $\mathcal{P}_0^3$ . Let $(\zeta,\xi) \in \{\ell,h\}\times\{\downarrow,\uparrow\}$ . We claim that there is no critical parallelogram spanned in $\eta_{R_{\zeta,\xi}}$ . Indeed, by Observation 3.2 such a parallelogram would have to intersect $R_{\zeta,\xi}$ so could not intersect R $^{\prime}$ , which would contradict the definition of $\eta$ . Moreover, R has no crossing for $\eta$ since $\eta$ is n-good, so $R_{\zeta,\xi}$ has no crossing for $\eta_{R_{\zeta,\xi}}$ . Thus, $\eta_{R_{\zeta,\xi}} \in G(R_{\zeta,\xi})$ , hence $\eta_{R_{\zeta,\xi}} \in V(n-1,R_{\zeta,\xi})$ , so $\mathcal{P}_0^3$ holds.

Induction step. Let $j \in \{0,\dots,m-1\}$ , and suppose that $\mathcal{P}^1_{j}$ , $\mathcal{P}^2_{j}$ , $\mathcal{P}^3_{j}$ and $\mathcal{P}_j^4$ hold. Since $(\eta^{(k)})_{0 \leqslant k \leqslant m}$ is a legal path, we have $\eta^{(j+1)}=(\eta^{(j)})^s$ and $(s+U)\cap R\subset \eta^{(j)}$ for some $s\in R \cap {{\mathbb{Z}}}^2$ and $U\in\mathcal{U}$ .

We first prove $\mathcal{P}^4_{j+1}$ , that is , using that ‘F shields from the influence of the exterior.’

Proof. If , then , so by $\mathcal{P}_j^4$ . Moreover, if , then . Furthermore, the sites of $\bar \gamma$ are in F, so by $\mathcal{P}_{j}^2$ they are not locally infectable in $\eta^{(j)}$ and, in particular $s'\not\in \eta^{(j)}$ for all $s'\in\bar\gamma$ . Since $(s+U)\cap R\subset \eta^{(j)}$ , this implies and so by $\mathcal{P}_j^4$ .

We cannot prove $\mathcal{P}^1_{j+1}$ yet, which would be that $R_\ell$ and $R_h$ contain a critical parallelogram spanned in $\eta^{(j+1)}$ . Instead, we establish a weaker result, that there exists at least one spanned critical parallelogram of ‘to the left’ of $R_{\ell,\uparrow}$ (see Figure 2), as well as one ‘below’ $R_{h,\uparrow}$ (these two parallelograms may be the same). The idea of the proof is that since at the beginning of the path all spanned critical parallelograms intersect R $^{\prime}$ , if at step $j+1$ they have moved too much to the right, then they must have left a trail of infectable sites behind them, which constitutes a crossing.

Proof. We will only treat $H={{\mathbb{H}}}_{u_3}(c-2L_{n-1}-7K)$ as the argument for the other half-plane is the same. Assume for a contradiction that there is no critical parallelogram contained in H that is spanned in . We will construct a crossing for $\eta^{(j+1)}$ , which contradicts the fact that $\eta^{(j+1)}$ is n-good.

By Claim 3.14, there exists a critical parallelogram D spanned in $\eta^{(0)}$ intersecting R $^{\prime}$ . D is then contained in (see Figure 2), hence spanned in . Let X be a strongly connected set of such that the smallest parallelogram containing X is D. Then, since D is ‘sufficiently to the left’ to intersect R $^{\prime}$ , X intersects $\bar{{\mathbb{H}}}_{u_3}(c-2L_{n-1}-9K-\ell)$ (see Figure 2). By Claim 3.15 we have , and , so we can consider X $^{\prime}$ the strongly connected component of containing X. Then X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_3}(c-2L_{n-1}-9K-\ell)$ .

We will now prove that X $^{\prime}$ also intersects $\bar{{\mathbb{H}}}_{u_1}(\!-(c-2L_{n-1}-8K))$ . The smallest parallelogram D $^{\prime}$ containing X $^{\prime}$ contains X, thus it contains D, so it has diameter at least $K/C_1$ , since D is critical (recall Definition 3.1). By Observation 3.3, , so D $^{\prime}$ is spanned in . Hence, by Lemma 3.4, there exists a critical parallelogram D $^{\prime\prime}$ spanned for . By the assumption made at the beginning of the proof of the claim, D $^{\prime\prime}$ cannot be contained in H, so D $^{\prime\prime}$ intersects $\bar{{\mathbb{H}}}_{u_1}(\!-(c-2L_{n-1}-7K))$ . Furthermore, D $^{\prime\prime}$ is critical, so its diameter is at most K. Thus, D $^{\prime\prime}$ is contained in $\bar{{\mathbb{H}}}_{u_1}(\!-(c-2L_{n-1}-8K))$ . In addition, since D $^{\prime\prime}$ is spanned for , by Observation 3.2 D $^{\prime\prime}$ intersects X $^{\prime}$ , so X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_1}(\!-(c-2L_{n-1}-8K))$ .

We now construct the rectangle in which the crossing will take place. We denote $a_0=\max\{a'\,|\,X' \subset \bar{{\mathbb{H}}}_{u_1}(\!-a')\}$ (the ‘left end’ of X $^{\prime}$ ). We claim $a_0 \geqslant a$ . Indeed, , and since we have and $u_1$ is a stable direction, we have , so $X' \subset {{\mathbb{H}}}_{u_1}(\!-a)$ . Moreover, $a_0 \leqslant c-2L_{n-1}-9K-\ell$ , since we proved that X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_3}(c-2L_{n-1}-9K-\ell)$ . Furthermore, we saw that by Observation 3.3, , so if we denote $R_{X'}=R(a_0,b;a_0+\ell,d)$ , then . Together with the fact that X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_1}(\!-(c-2L_{n-1}-8K)) \subset {{\mathbb{H}}}_{u_1}(\!-(a_0+\ell))$ , this yields that $R_{X'}$ is $u_1$ -crossed for (recall Definition 3.5).

Moreover, since critical parallelograms have diameter at most K and $a_0 \leqslant c\,{-}\,2L_{n-1}\,{-}\,9K\,{-}\,\ell$ , any critical parallelogram intersecting $R_{X'}$ is contained in H. This will imply that there is no critical parallelogram spanned for , as by Observation 3.2 such a parallelogram would intersect $R_{X'}$ , thus would be contained in H, which is impossible by the assumption made at the beginning of the proof of the claim. Since $R_{X'}$ is $u_1$ -crossed for and there is no critical parallelogram spanned for , the event $C_{R_{X'}}^{u_1}$ occurs for $\eta^{(j+1)}$ (recall Definition 3.5). But this is a contradiction with the fact that R has no crossing for $\eta^{(j+1)}$ , as it is a configuration in a n-good legal path. This contradiction concludes the proof of the claim.

Claim 3.16 will allow us to prove half of $\mathcal{P}_{j+1}^3$ , more precisely $\eta^{(j+1)}_{R_{\ell,\uparrow}} \in V(n-1,R_{\ell,\uparrow})$ and $\eta^{(j+1)}_{R_{h,\uparrow}} \in V(n-1,R_{h,\uparrow})$ . The idea of the proof is that since Claim 3.16 yields that at least one spanned critical parallelogram is to the left of $R_{\ell,\uparrow}$ , and as the configurations are n-good, there are at most n spanned critical parallelograms in total, this implies there are at most $n-1$ spanned critical parallelograms inside $R_{\ell,\uparrow}$ (and similarly for $R_{h,\uparrow}$ ).

Proof. We will only prove $\eta^{(j+1)}_{R_{\ell,\uparrow}} \in V(n-1,R_{\ell,\uparrow})$ , as the other proof is similar. It will suffice to prove that $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ is $(n-1)$ -good. Indeed, by $\mathcal{P}^3_j$ we have $\eta^{(j)}_{R_{\ell,\uparrow}} \in V(n-1,R_{\ell,\uparrow})$ , hence in the case $\eta^{(j+1)}_{R_{\ell,\uparrow}}\neq\eta^{(j)}_{R_{\ell,\uparrow}}$ we can say that there exists an $(n-1)$ -good legal path from $G(R_{\ell,\uparrow})$ to $\eta^{(j)}_{R_{\ell,\uparrow}}$ . If we add $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ to this path, we then obtain an $(n-1)$ -good legal path from $G(R_{\ell,\uparrow})$ to $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ , hence $\eta^{(j+1)}_{R_{\ell,\uparrow}} \in V(n-1,R_{\ell,\uparrow})$ .

We now prove that $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ is $(n-1)$ -good. Firstly, $R_{\ell,\uparrow}$ has no crossing for $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ because R has no crossing for $\eta^{(j+1)}$ . It remains only to show that the maximal number k of critical parallelograms that are disjointly spanned in $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ is at most $n-1$ . Let $D_1,\dots,D_k$ be critical parallelograms that are disjointly spanned in $\eta^{(j+1)}_{R_{\ell,\uparrow}}$ . By Claim 3.16, there exists a critical parallelogram $D\subset {{\mathbb{H}}}_{u_3}(c-2L_{n-1}-7K)$ that is spanned in and, therefore, also in $\eta^{(j+1)}_D$ . D is then disjoint from $R_{\ell,\uparrow}$ (see Figure 2), so we deduce that $D_1,\dots,D_k,D$ are disjointly spanned in $\eta^{(j+1)}$ , so $\eta^{(j+1)}$ contains $k+1$ disjointly spanned critical parallelograms. Since $\eta^{(j+1)}$ is n-good, we get $k \leqslant n-1$ , which ends the proof.

We are now ready to prove $\mathcal{P}_{j+1}^1$ , that is, $R_\ell$ and $R_h$ contain a critical parallelogram spanned in $\eta^{(j+1)}$ . To do that, we will prove that a spanned critical parallelogram in ${{\mathbb{H}}}_{u_4}(d-2H_{n-1}-7K)$ provided by Claim 3.16 is in fact in $R_\ell$ (and similarly for $R_h$ ). The idea of the proof is that Claim 3.16 prevents the parallelogram from being ‘too far up,’ being ‘too far left or down’ would induce a crossing, so will be impossible, and being ‘too far right’ will be prevented because Claim 3.17 will allow us to use $\mathcal{H}_{n-1}$ in $R_{\ell,\uparrow}$ .

Proof. We only treat $R_\ell$ , as $R_h$ is similar. Assume for a contradiction that there is no critical parallelogram contained in $R_\ell$ spanned in $\eta^{(j+1)}$ .

By Claim 3.16, there exists a critical parallelogram D contained in ${{\mathbb{H}}}_{u_4}(d-2H_{n-1}-7K)$ that is spanned in . By assumption $D\not\subset R_\ell$ . There are three possibilities (see Figure 2):

1. (a) $D\cap R_{\ell,\downarrow}\neq\varnothing$ , that is D is ‘too far left;’

2. (b) $D\cap R_{h,\downarrow}\neq \varnothing$ , that is D is ‘too far down;’

3. (c) $D \cap \bar{{\mathbb{H}}}_{u_1}(\!-(c-L_{n-1}-7K))\neq\varnothing$ , that is D is ‘too far right.’

We first assume case (c) occurs. Since D is spanned in , by Observation 3.2, D intersects . In addition, since D is critical it has diameter at most K. This yields that the intersection of D and $\bar{{\mathbb{H}}}_{u_1}(\!-(c-L_{n-1}-7K))$ is in $R^{\prime}_{\ell,\uparrow}$ (see Figure 2). Since D intersects $R^{\prime}_{\ell,\uparrow}$ and has diameter at most K, it is contained in $R_{\ell,\uparrow}$ , hence spanned in $\eta_{R_{\ell,\uparrow}}^{(j+1)}$ . However, by Claim 3.17 we have $\eta^{(j+1)}_{R_{\ell,\uparrow}}\in V(n-1,R_{\ell,\uparrow})$ , so $\mathcal{H}_{n-1}$ implies there is no critical parallelogram intersecting $R^{\prime}_{\ell,\uparrow}$ spanned in $\eta_{R_{\ell,\uparrow}}^{(j+1)}$ , so we get a contradiction.

Cases (a) and (b) being analogous, we only treat case (a). Assume case (a) occurs. The argument will resemble the one in the proof of Claim 3.16. Since D is spanned in , there exists a strongly connected set such that D is the smallest parallelogram containing X. Let X $^{\prime}$ be the strongly connected component of containing X, and D $^{\prime}$ be the smallest parallelogram containing X $^{\prime}$ . Then D $^{\prime}$ contains X, hence it contains D and, since D is critical, ${{\textrm{diam}}}(D')\geqslant{{\textrm{diam}}}(D)\geqslant K/C_1$ . Furthermore, X $^{\prime}$ is a strongly connected component of by Claim 3.15, so by Observation 3.3. This implies , hence D $^{\prime}$ is spanned in .

Since ${{\textrm{diam}}}(D')\geqslant K/C_1$ , Lemma 3.4 implies that there exists a critical parallelogram D $^{\prime\prime}$ that is spanned in . Then D $^{\prime\prime}$ is spanned in $\eta^{(0)}=\eta$ and therefore intersects R $^{\prime}$ by the definition of $\eta$ . Since D $^{\prime\prime}$ is critical, its diameter is at most K, so, since D $^{\prime\prime}$ intersects R $^{\prime}$ , it is contained in $\bar{{\mathbb{H}}}_{u_1}(\!-a-2L_{n-1}-8K-\ell)$ (see Figure 2). Moreover, since D $^{\prime\prime}$ is spanned in , by Observation 3.2, X $^{\prime}$ intersects D $^{\prime\prime}$ hence X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_1}(\!-a-2L_{n-1}-8K-\ell)$ . In addition, since D intersects $R_{\ell,\downarrow}$ by assumption a) and is the smallest parallelogram containing X, then X intersects $\bar{{\mathbb{H}}}_{u_3}(a+2L_{n-1}+7K)$ , thus X $^{\prime}$ intersects this half-plane as well.

Denote $a_0=\max\{a'\,|\,X' \subset \bar{{\mathbb{H}}}_{u_1}(\!-a')\}$ (the ‘left end’ of X $^{\prime}$ ). Then $a_0 \geqslant a$ , since as $u_1$ is a stable direction, and $a_0 \leqslant a+2L_{n-1}+7K$ , as X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_3}(a+2L_{n-1}+7K)$ . As in Claim 3.16 (recalling that X $^{\prime}$ intersects $\bar{{\mathbb{H}}}_{u_1}(\!-a-2L_{n-1}-8K-\ell)$ ), this entails that $R_{X'}=R(a_0,b;\ a_0+\ell,d)$ is $u_1$ -crossed for $\eta^{(0)}$ (recall Definition 3.5). However, R has no crossing for $\eta^{(0)}$ as $\eta^{(0)}$ is n-good, so $C_{R_{X'}}^{u_1}$ does not occur for $\eta^{(0)}$ . Consequently, there exists a critical parallelogram spanned for $\eta^{(0)}_{R_{X'}}$ , hence for $\eta^{(0)}$ . By Observation 3.2 this parallelogram intersects $R_{X'}$ , so since critical parallelograms have diameter at most K and $a_0 \leqslant a+2L_{n-1}+7K$ it does not intersect R $^{\prime}$ , which contradicts the fact that all critical parallelograms spanned in $\eta^{(0)}=\eta$ intersect R $^{\prime}$ .

The previous claim will allow us to complete the proof of $\mathcal{P}_{j+1}^3$ . We already proved in Claim 3.17 that $\eta^{(j+1)}_{R_{\ell,\uparrow}} \in V(n-1,R_{\ell,\uparrow})$ and $\eta^{(j+1)}_{R_{h,\uparrow}} \in V(n-1,R_{h,\uparrow})$ , so we need $\eta^{(j+1)}_{R_{\ell,\downarrow}} \in V(n-1,R_{\ell,\downarrow})$ and $\eta^{(j+1)}_{R_{h,\downarrow}} \in V(n-1,R_{h,\downarrow})$ . The argument is identical to that of Claim 3.17: by Claim 3.18 we have a spanned critical parallelogram in $R_\ell$ , which entails that there are at most $n-1$ critical spanned parallelograms in $R_{\ell,\downarrow}$ (and similarly for $R_{h,\downarrow}$ ), hence the result.

Proof. The proof is actually the same as in Claim 3.17, replacing ${{\mathbb{H}}}_{u_3}(c-2L_{n-1}-7K)$ by $R_\ell$ or $R_h$ and Claim 3.16 by Claim 3.18.

It remains only to prove $\mathcal{P}_{j+1}^2$ , that is that the sites of F are not locally infectable in $\eta^{(j+1)}$ .

Proof. Assume for a contradiction that there exists $s' \in F\subset B'$ that is locally infectable in $\eta^{(j+1)}$ . By $\mathcal{P}_j^2$ , s $^{\prime}$ is not locally infectable in $\eta^{(j)}$ . Therefore, by Lemma 3.6, there exist $\zeta\in\{\ell,h\}$ , $\xi\in\{\downarrow,\uparrow\}$ and a critical parallelogram $D\subset R^{\prime}_{\zeta,\xi}$ spanned in $\eta^{(j+1)}_{R_{\zeta,\xi}}$ . However, by Claim 3.19 $\eta^{(j+1)}_{R_{\zeta,\xi}}\,{\in}\, V(n\,{-}\,1,R_{\zeta,\xi})$ , so $\mathcal{H}_{n-1}$ yields the desired contradiction.

Claims 3.15, 3.18, 3.19 and 3.20 together establish the induction step, which completes the proof of Lemma 3.12.

4.1 Proof of Theorem 1.7

In this section we assume that $\mathcal{U}$ has an infinite number of stable directions. We then choose two rational directions $u_1<u_2<u_1+\pi$ sufficiently close to each other, such that all directions in $[2u_1-u_2,2u_2-u_1]$ are stable and $u_1$ , $u_2$ satisfy a technical condition which the reader is advised to ignore, namely that $u_1$ and $u_2$ are constructed like the eponymous directions in the proof of [Reference Hartarsky, Marêché and Toninelli19, Lemma 4.1].

4.1.1 Proof of Theorem 1.7(a)

For this section we further assume that $\mathcal{U}$ is unbalanced. We fix the values of the parameters of Proposition 3.7 as follows.

\begin{align*}K&{}=q^{-\alpha-1/4}&\ell&{}=q^{-4\alpha}&L&{}=\exp\left(\frac{(\log(1/q))^2}{C_6q^{\alpha}}\right)\\T&{}=\exp\left(\frac{(\log (1/q))^4}{C_6^2q^{2\alpha}}\right)&h&{}=q^{-4\alpha}&H&{}=\exp\left(\frac{(\log(1/q))^2}{C_6q^{\alpha}}\right)\!.\end{align*}

Proof of Theorem 1.7(a). The upper bound was proved in [Reference Martinelli, Morris and Toninelli27, Theorem 2(a)], so we focus on the lower one. We will apply Proposition 3.7 with the above choice of parameters, so that we obtain the desired conclusion: ${{\mathbb{E}}}_\mu[\tau_0]\geqslant T$ . Hence, it suffices to verify the hypotheses of the proposition. Indeed, setting $n=(\log(1/q))^2/(2C_6q^\alpha)$ , we have

\begin{equation*}L=\exp\left(\frac{(\log(1/q))^2}{C_6q^{\alpha}}\right)\geqslant 3^n\cdot2q^{-4\alpha}\geqslant 3^n(11K+\ell)\end{equation*}

and similarly for H. By Lemma 4.1 we do have $\mu(0\text{ is locally infectable})\leqslant 1/8$ . Moreover, recalling that $\rho\leqslant \exp(\!-(\log(1/q))^2/(C_5q^\alpha))$ by Lemma 4.2, we obtain

\begin{align*}TLH(LHK^3\rho)^{n+1}\leqslant{}& \exp\left(\frac{2(\log(1/q))^4}{C_6^2q^{2\alpha}}+n\left(\frac{3(\log(1/q))^2}{C_6q^\alpha}-\frac{(\log(1/q))^2}{C_5q^\alpha}\right)\right)\\\leqslant{}&\exp\left(\frac{(\log(1/q))^4}{C_6q^{2\alpha}}\left(\frac{2}{C_6}+\frac{3}{2C_6}-\frac{1}{2C_5}\right)\right)\leqslant 1.\end{align*}

Finally,

\begin{equation*}T(LH)^2\leqslant \exp\left(\frac{2(\log(1/q))^4}{C_6^2q^{2\alpha}}\right)\leqslant \exp\left(q^{-3\alpha}\right)\end{equation*}

and Lemma 4.4 below yields $\max({p_{\leftarrow}},{p_{\downarrow}})\leqslant \exp(\!-q^{-3\alpha})$ . Therefore, we obtain $T(LH)^2\max({p_{\leftarrow}},{p_{\downarrow}})\leqslant 1$ and all the hypotheses of Proposition 3.7 are verified once we establish Lemma 4.4.

Lemma 4.4 With the notation and assumptions above we have

\begin{equation*}\max({p_{\leftarrow}},{p_{\downarrow}})\leqslant \exp\left(\!-q^{-3\alpha}\right)\!.\end{equation*}

This bound is proved in the appendices of the arXiv version of this paper [Reference Hartarsky and Marêché18, Lemma B.4], but let us provide a rough sketch for the reader’s convenience.

Sketch of the proof of Lemma 4.4. The proof is based on the concept of hierarchies introduced by Holroyd in [Reference Holroyd23] and extensively studied since then.

Consider the rectangle $R=R(\!-\ell,0;\ 0,H)$ and assume that $C_R^{u_1}$ occurs with $\eta'=\eta$ in Definition 3.5. Fix some $u_1$ -crossing $X\subset[{{\mathbb{H}}}_{u_1}\cup(R\cap\eta)]$ of R. One can retrace how X becomes infected by the bootstrap percolation process by a method known as a spanning algorithm as follows. Starting from single infections, we lump them into groups of infections by progressively merging two groups if their closures are close to each other. Since X has diameter at least $\ell$ , this process will eventually produce a set of infections of diameter roughly $\ell$ . We associate to each set of infections a droplet (appropriately shaped polygon) containing its closure.

We can then view the spanning algorithm run backwards as a progressive shattering of the initial droplet of size $\ell$ and record its history. However, we do not wish to register the entire history of all splittings. When a droplet splits into two large droplets, we study the subsequent splittings of both droplets, but when a large droplet splits into a large one plus a small one, we ignore the small one. Still, if there are many successive such small splittings, we occasionally write down one of the resulting large droplets. We stop studying the splitting when the droplets are of size roughly K. We then call these droplets of size K seeds. In total, this gives us a tree of large droplets called hierarchy recording how the droplet of size $\ell$ is produced.

Recall that the event $C_R^{u_1}$ requires the absence of spanned critical parallelograms, which prevents the existence of clusters of infectable sites of size at least K, hence of droplets of size at least K. This seemingly forbids the hierarchy to exist. However, in the above we overlooked the presence of the infected boundary ${{\mathbb{H}}}_{u_1}$ . To take it into account, we actually consider hierarchies of cut droplets which are ‘very flat triangles sticking out of the boundary.’

To bound the probability of $C_R^{u_1}$ , we bound the probability that such a hierarchy occurs. There are two possibilities: either there are many splittings into two large droplets, or much of the splitting is done into a large and a small droplet. In the first case, there will be many seeds. We establish a preliminary upper bound on the probability of occurrence of a seed of order $\exp(\!-\!q^{-\alpha})$ . Since each seed should be contained in the original droplet of size $\ell$ , there are few possible choices for them. We deduce an appropriate bound on the probability of such a hierarchy.

Assuming that most of the size $\ell$ of the initial droplet is gained by splittings into a large and a small droplet, let us consider one such step. This means that in the spanning algorithm, when we add to the infections of a large (cut) droplet the few infections contained in a small droplet, we get to infect a slightly larger droplet. This essentially implies the existence of an infectable set going from the boundary of the large droplet to that of the larger one. This in turn yields the occurrence of another cut droplet of size at least K. Hence, the preliminary bound on the probability of occurrence of a seed allows us to bound the probability of such a splitting, concluding the proof.

4.1.2 Proof of Theorem 1.7(b)

For this section we further assume that $\mathcal{U}$ is balanced. We fix the values of the parameters of Proposition 3.7 as follows.

\begin{align*}K&{}=q^{-\alpha}&\ell&{}=q^{-4\alpha}&L&{}=\exp\left(\frac{1}{C_6q^{\alpha}}\right)\\T&{}=\exp\left(\frac{1}{C_6^2q^{2\alpha}}\right)&h&{}=q^{-4\alpha}&H&{}=\exp\left(\frac{1}{C_6q^{\alpha}}\right)\!.\end{align*}

Then Theorem 1.7(b) follows directly from Proposition 3.7 and the upper bound from [Reference Hartarsky17, Theorem 1(b)]. Setting $n=1/(2C_6q^\alpha)$ , the hypotheses of the proposition follow like in Section 4.1.1 from the choice of parameters, Lemmas 4.1 and 4.3, and Lemma 4.4, which still applies.

4.2 Proof of Theorem 1.8

4.2.1 Proof of Theorem 1.8(c)

In this section we assume that $\mathcal{U}$ is unbalanced, rooted and has a finite number of stable directions. Therefore, we can find rational directions $u_1,u_2,u_3$ such that $u_1+\pi=u_3$ , $u_2\in(u_1,u_3)$ and $\alpha(u_i)\geqslant\alpha+1$ for all $i\in\{1,2,3\}$ .

We fix the values of the parameters of Proposition 3.7 as follows.

\begin{align*}K&{}=q^{-\alpha-1/4}&\ell&{}=q^{-\alpha-5/8}&L&{}=q^{-\alpha-3/4}\\T&{}=\exp\left(\frac{(\log (1/q))^3}{C_6q^{\alpha}}\right)&h&{}=q^{-\alpha-5/8}&H&{}=q^{-\alpha-3/4}.\end{align*}

Proof of Theorem 1.7(c). The upper bound was proved in [Reference Hartarsky, Martinelli and Toninelli21, Theorem 1], so we focus on the lower one. As previously, it suffices to verify the hypotheses of Proposition 3.7 with the above choice of parameters. Indeed, setting $n=\log (1/q)/C_1$ , we have

\begin{equation*}L=q^{-\alpha-3/4}\geqslant 3^n\cdot2q^{-\alpha-5/8}\geqslant 3^n(11K+\ell)\end{equation*}

and similarly for H. By Lemma 4.1 we do have $\mu(0\text{ is locally infectable})\leqslant 1/8$ . Moreover, recalling that $\rho\leqslant \exp(\!-(\log(1/q))^2/(C_5q^\alpha))$ by Lemma 4.2, we obtain

\begin{align*}TLH(LHK^3\rho)^{n+1}\leqslant{}&\exp\left(\frac{2(\log(1/q))^3}{C_6q^{\alpha}}-\frac{n(\log(1/q))^2}{2C_5q^\alpha}\right)\\[3pt] \leqslant{}&\exp\left(\frac{(\log(1/q))^3}{q^{\alpha}}\left(\frac{2}{C_6}-\frac{1}{2C_1C_5}\right)\right)\leqslant 1.\end{align*}

Finally,

\begin{equation*}T(LH)^2\leqslant \exp\left(\frac{2(\log(1/q))^3}{C_6q^{\alpha}}\right)\leqslant \exp\left(q^{-\alpha-1/4}\right)\!.\end{equation*}

Thus, once we establish Lemma 4.5 below, all the hypotheses of Proposition 3.7 are verified.

Lemma 4.5 With the notation and assumptions above we have

\begin{equation*}\max({p_{\leftarrow}},{p_{\downarrow}})\leqslant \exp\left(\!-q^{-\alpha-1/4}\right)\!.\end{equation*}

This bound is proved in the appendices of the arXiv version of this paper [Reference Hartarsky and Marêché18, Lemma B.3]. Since the proof is quite different from the one of Lemma 4.4, we also provide a sketch.

Sketch of the proof of Lemma 4.5. Consider $R=R(\!-\ell,0;\ 0,H)$ . Our goal is essentially to prove that $\mu(C_R^{u_1})\leqslant e^{-\ell}$ . To do this, we use an improvement of the partition method of [Reference Bollobás, Duminil-Copin, Morris and Smith4, Section 8.3]. We cut the rectangle into strips of a large constant width and lump the strips together into groups crossed by a spanned parallelogram. We establish that it is deterministically necessary for all strips to either be intersected by a spanned parallelogram as above or to contain a set of $\alpha(u_1)\geqslant \alpha+1$ infections. Yet, H is much smaller than $q^{-\alpha-1}$ , so it is unlikely that a strip contains $\alpha+1$ infections. As for the strips intersected by spanned parallelograms, since $C_R^{u_1}$ assumes that no critical parallelogram is spanned, we only need to care about smaller spanned parallelograms, for which one may prove a preliminary probability bound exponentially small in their size. Combining these facts and taking into account that the size of R is only polynomial in $1/q$ , we can control the entropy and conclude that the probability of a crossing is indeed exponentially small in its width.

4.2.2 Proof of Theorem 1.8(e)

In this section we assume that $\mathcal{U}$ is balanced, rooted and with a finite number of stable directions. Therefore, we can find non-opposite rational directions $u_1,u_2$ such that $\alpha(u_1)\geqslant\alpha+1$ and $\alpha(u_2)\geqslant \alpha+1$ . We fix the values of the parameters of Proposition 3.7 as follows.

\begin{align*}K&{}=1/(C_5q^{\alpha})&\ell&{}=q^{-\alpha-1/2}&L&{}=q^{-\alpha-3/4}\\[3pt]T&{}=\exp\left(\frac{\log (1/q)}{C_6q^{\alpha}}\right)&h&{}=q^{-\alpha-1/2}&H&{}=q^{-\alpha-3/4}.\end{align*}

Then Theorem 1.8(e) follows directly from Proposition 3.7 and the upper bound in [Reference Hartarsky17, Theorem 1(e)]. Setting $n=\log (1/q)/C_1$ , the hypotheses of the proposition follow like in Section 4.2.1 from the choice of parameters, Lemmas 4.1 and 4.3, and Lemma 4.5, which still holds.

4.2.3 Proof of Theorem 1.8(f)

In this section we assume that $\mathcal{U}$ is semi-directed. Therefore, we can find non-opposite rational directions $u_1,u_2$ such that $\alpha(u_1)=\alpha$ and $\alpha(u_2)\geqslant \alpha+1$ . We fix the values of the parameters of Proposition 3.7 as follows.

\begin{align*}K&{}=1/(C_5q^{\alpha})&\ell&{}=q^{-\alpha-1/2}&L&{}=q^{-\alpha-3/4}\\[3pt]T&{}=\exp\left(\frac{\log\log(1/q)}{C_6^3q^{\alpha}}\right)&h&{}=\frac{\log\log (1/q)}{q^{\alpha}}&H&{}=\frac{(\log (1/q))^{1/4}}{q^{\alpha}}.\end{align*}

Proof of Theorem 1.7(f). The upper bound is proved in [Reference Hartarsky17, Theorem 1(f)], so we focus on the lower one. As previously, it suffices to verify the hypotheses of Proposition 3.7 with the above choice of parameters. Indeed, setting $n=\log\log(1/q)/C_1$ , we have

\begin{align*}L={}&q^{-\alpha-3/4}\geqslant 3^n\cdot 2q^{-\alpha-1/2}\geqslant 3^n(11K+\ell),\\[3pt]H={}&\frac{(\log(1/q))^{1/4}}{q^\alpha}\geqslant 3^n\frac{2\log\log(1/q)}{q^\alpha}\geqslant 3^n(11K+h).\end{align*}

By Lemma 4.1 we do have $\mu(0\text{ is locally infectable})\leqslant 1/8$ . Moreover, recalling that $\rho\leqslant \exp(\!-1/(C_5q^\alpha))$ by Lemma 4.3, we obtain

\begin{align*}TLH(LHK^3\rho)^{n+1}\leqslant{}&\exp\left(\frac{2\log\log(1/q)}{C_6^3q^{\alpha}}-\frac{n}{2C_5q^\alpha}\right)\\[3pt]\leqslant{}&\exp\left(\frac{\log\log(1/q)}{q^{\alpha}}\left(\frac{2}{C_6^3}-\frac{1}{2C_1C_5}\right)\right)\leqslant 1.\end{align*}

Finally,

\begin{equation*}T(LH)^2\leqslant \exp\left(\frac{2\log\log(1/q)}{C_6^3q^{\alpha}}\right)\leqslant \exp\left(\frac{\log\log(1/q)}{2C_6^2q^\alpha}\right)\leqslant \exp(q^{-\alpha-1/4}).\end{equation*}

Thus, once we establish Lemma 4.6 below, all the hypotheses of Proposition 3.7 are verified.

Lemma 4.6 With the notation and assumptions above we have

\begin{align*} {p_{\leftarrow}}&{}\leqslant\exp\left(\!-q^{-\alpha-1/4}\right)& {p_{\downarrow}}&{}\leqslant\exp\left(\!-\frac{\log\log(1/q)}{2C_6^2q^\alpha}\right)\!.\end{align*}

This bound is proved in the appendices of the arXiv version of this paper [Reference Hartarsky and Marêché18, Lemma B.3] like Lemma 4.5.