Polluted Bootstrap Percolation with Threshold Two in All Dimensions

TL;DR

This paper studies a bootstrap percolation model on high-dimensional cubic lattices, showing that the occupied site density approaches 1 as initial occupation and closure probabilities go to zero, resolving part of a conjecture.

Contribution

It proves that in all dimensions d≥3, the final occupied density tends to 1 as p,q→0, regardless of their scaling, partially resolving Morris's conjecture.

Findings

Final density approaches 1 as p,q→0

Contrasts with the 2D case where the critical parameter differs

Provides new insights into high-dimensional bootstrap percolation

Abstract

In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with threshold r=2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the critical parameter is q/{p^2}.

Equations

$$\begin{cases}\begin{array}[]{ll}\text{{\boldmath closed}}&\text{with probability }q;\\ \text{{\boldmath open} and {\boldmath initially occupied}}&\text{with probability }p;\\ \text{{\boldmath open} but not initially occupied}&\text{with probability }1-p-q.\end{array}\end{cases}$$

$$r$$

$$\mathbb{P}\bigl{(}\text{the origin is eventually occupied}\bigr{)}\to\begin{cases}1,&\text{if }q<cp^{2};\\ 0,&\text{if }q>Cp^{2}.\end{cases}$$

$$\mathbb{P}\bigl{(}\text{\rm the origin is eventually occupied}\bigr{)}\to 1\qquad\text{as }(p,q)\to(0,0).$$

$$r$$

$$q_c:=\inf \{q:{\varphi }^{+}(q)=0\}$$

$$\Lambda_{k}:=\mathbb{Z}^{2}\times\{k\}=\bigl{\{}x\in\mathbb{Z}^{3}:x_{3}=k\bigr{\}}.$$

$$(1,1,0).$$

$$\Bigl{\{}(-1,0,0),(0,-1,0),(0,0,-1),(-2,1,0),(1,-2,0),(-1,-1,1)\Bigr{\}}.$$

$$D:=\bigl{\{}x\notin A:x-(1,1,0)\in A\bigr{\}}.$$

$$H:=\{x:x_1+x_2+x_3\le 0\},$$

$$\displaystyle\mathbb{P}\bigl{(}(k,k,0)\in A\bigr{)}$$

$$\le \sum _{n\ge k}{7}^n{q}^{(2k+n)/3}$$

$$=(7q{)}^k\cdot \sum _{n\ge 0}(7q^{1/3}{)}^n$$

$$\le 8\cdot (7q{)}^k,$$

$$\sigma(\widehat{S}):=\bigl{\{}x,\;x+(0,0,1),\;x+(-1,-1,1):x\in\widehat{S}\bigr{\}}\cap\widehat{B};$$

$$\text{cell}(x)=(x_1,4x_2,16x_3)+\{0\}\times [0,4)\times [0,16).$$

$$S=\bigcup _{x\in \hat{S}}\text{cell}(x)$$

$$\mathbb{P}\Bigl{(}\widehat{D}\cap\bigl{\{}x+(t,t,0):t\in[1,k]\bigr{\}}\neq\emptyset\Bigr{)}\geq 1-e^{-ck},\qquad k>0,$$

$$\hat{S}:=\hat{D}\cap \hat{B}.$$

$$\mathbb{P}\bigl{(}E_{3}\mid E_{1}\cap E_{2}\bigr{)}\geq 1-2L(1-\widehat{p})^{L}\geq 1-2L\exp(-\widehat{p}L).$$

$$B⊳B_1⊳B_2⊳B_3,$$

$$B⊳B_1^{\prime }⊳B_2^{\prime }⊳B_3^{\prime },$$

$$B(a):=B+L{a}_{1}u+L{a}_2{u}^{\prime }.$$

$$\mathbb{P}(0\text{ is eventually occupied})$$

$$\le \mathbb{P}(0\text{ is eventually occupied starting from the adjusted configuration})$$

$$\le \mathbb{P}(\text{Z∪∂Z contains an initially occupied site})$$

$$\displaystyle\leq\sum_{k=1}^{\infty}\mathbb{P}\bigl{(}|\mathcal{Z}|=k\bigr{)}\bigl{(}1-(1-p)^{k+2dk}\bigr{)}.$$

$$\gamma (q):=\underset{p\to 0+}{\mathrm{limsup}}{p}^{-1}\varphi (p,q).$$

Full text

1. Introduction

Bootstrap percolation is a fundamental cellular automaton model for nucleation and growth from sparse random initial seeds. In this article we address how the model is affected by the presence of pollution in the form of sparse random permanent obstacles.

Let $\mathbb{Z}^{d}$ be the set of $d$-vectors of integers, which we call sites, and let $p,q\in[0,1]$ be parameters. In the initial (time zero) configuration, each site is chosen to have exactly one of three possible states:

[TABLE]

Initial states are chosen independently for different sites. Closed sites represent pollution or obstacles, while occupied sites represent a growing agent.

The configuration evolves in discrete time steps $t=0,1,2,\ldots$ as follows. As usual we make $\mathbb{Z}^{d}$ into a graph by declaring sites $u,v\in\mathbb{Z}^{d}$ to be neighbors if $\|u-v\|_{1}=1$. The threshold $r$ is an integer parameter. An open site $x$ that is unoccupied at time $t$ becomes occupied at time $t+1$ if and only if

[TABLE]

at time $t$. Closed sites remain closed forever and cannot become occupied. Open sites remain open. Once a site is occupied, it remains occupied. In the main cases of interest, $d\geq r\geq 2$.

Bootstrap percolation without pollution (the case $q=0$ in our formulation) has a long and rich history with many surprises. For $d\geq r\geq 1$, there is no phase transition in $p$, in the sense that every site of $\mathbb{Z}^{d}$ is eventually occupied almost surely for every $p>0$, as proved in [vE] ($d=2$) and [Sch] ($d\geq 3$). The metastability properties of the model on finite regions are understood in great depth (see e.g. [AL, Hol, BBDM, GHM]), while a broad range of variant growth rules have also been explored (e.g. [GG, DvE, BDMS]). For further background see the discussion later in the introduction, and the excellent recent survey [Mor].

The polluted bootstrap model (i.e. the case $q>0$) was introduced by Gravner and McDonald [GM2] in 1997. The principal quantity of interest is the final density of occupied sites, i.e. the probability that the origin is eventually occupied, in the regime where $p$ and $q$ are both small. In dimension $d=2$ with threshold $r=2$, Gravner and McDonald proved that the final density is strongly dependent on the relative scaling of $p$ and $q$. Specifically, there exist constants $c,C>0$ such that, as $p\to 0$ and $q\to 0$ simultaneously,

[TABLE]

In this article we give the first rigorous treatment of the polluted bootstrap percolation model in dimensions $d\geq 3$. We take the threshold $r$ to be $2$. (Threshold $r=3$ is addressed in a companion paper [GHS] by the current authors together with Sivakoff, as discussed below). Our main result is that, in contrast with dimension $d=2$, occupation prevails regardless of the $p$ versus $q$ scaling.

Theorem 1**.**

Consider polluted bootstrap percolation on $\mathbb{Z}^{d}$ with $d\geq 3$, threshold $r=2$, density $p>0$ of initially occupied sites, and density $q>0$ of closed sites. We have

[TABLE]

Moreover, the probability that the origin lies in an infinite connected set of eventually occupied sites also tends to $1$. The same statements hold for modified bootstrap percolation.

In the above statement, a set of sites is called connected if it induces a connected subgraph of $\mathbb{Z}^{d}$. The modified bootstrap percolation model is a well-known variant of the standard model, in which the condition (1) for a site to become occupied is replaced with:

[TABLE]

where $e_{i}$ is the $i$th coordinate vector. (As before, closed sites cannot become occupied, and occupied sites remain occupied forever).

Theorem 1 resolves Conjecture 4.6 of Morris [Mor] in the key case $r=2$. To be precise, this conjecture may be expressed as: for all $d>r\geq 1$, there exists an infinite connected eventually occupied set with probability at least $1/2$ for $(p,q)$ sufficiently close to $(0,0)$. The author states that the conjecture seems to be very difficult.

Defining $\phi(p,q)=\phi_{d,r}(p,q)$ to be the probability that the origin is eventually occupied, it follows from the obvious monotonicities of the model that $\phi$ is (weakly) increasing in $p$ and decreasing in $q$. Therefore, the convergence in Theorem 1 is equivalent to $\lim_{q\to 0}\lim_{p\to 0}\,\phi(p,q)=1$. This formulation will be reflected in our proof. We will show that for $q$ sufficiently small there is an infinite structure of open sites on which occupation can spread, no matter how small $p$, and that the density of this structure tends to $1$ as $q\to 0$. Our methods are very different from those in previous works on bootstrap percolation, and involve the technology of oriented surfaces introduced recently in [DDG*+*].

Our result reveals an interesting phase transition. Let $r=2$ and $d\geq 3$ and consider the decreasing function $\phi^{+}(q):=\phi(0^{+},q)=\lim_{p\to 0^{+}}\phi(p,q)$. Theorem 1 implies that $\phi^{+}(q)>0$ for $q$ sufficiently close to [math]. On the other hand, standard arguments imply that $\phi^{+}(q)=0$ if $q$ exceeds one minus the critical probability $p_{c}^{\textrm{site}}(\mathbb{Z}^{d})$ of site percolation. Therefore the critical probability

[TABLE]

is nontrivial. In fact, we show the following slightly stronger fact involving a strict inequality.

Corollary 2**.**

Consider the setting of Theorem 1. The critical value $q_{c}$ defined above satisfies $0<q_{c}\leq 1-p_{c}^{\text{\rm site}}(\mathbb{Z}^{d})$. For $d=3$, the latter inequality is strict. The function $\phi^{+}$ vanishes on $(q_{c},1]$, is strictly positive on $[0,q_{c})$, and converges to $1$ as $q\to 0$.

Our methods do not produce a good lower bound on $q_{c}$, and give no information on the behavior of $\phi^{+}$ near $q_{c}$.

As mentioned earlier, the companion paper [GHS] treats polluted bootstrap percolation with threshold $r=3$. The strongest result of [GHS] is for the modified bootstrap percolation model with $d=r=3$. Similarly to the case $d=r=2$ of [GM2], but in contrast with the $d>r=2$ case of Theorem 1, the final density here depends on the $p$ versus $q$ scaling, but now with a cube law (modulo logarithmic factors). Specifically, as $p,q\to 0$, the final occupied density converges to $1$ if $q<c\,(p/\log p^{-1})^{3}$, and to [math] if $q>Cp^{3}$. Interestingly, the first of these bounds relies crucially on Theorem 1 of the current article (together with a straightforward renormalization argument). The second bound (which is far from straightforward) again uses oriented surfaces, but in a completely different way: to block growth rather than to facilitate it.

We record some simple observations about other choices of the threshold $r$. For $d=r$, notwithstanding the detailed results of [GM2, GHS], an easy argument rules out the conclusion $\lim_{(p,q)\to(0,0)}\phi(p,q)=1$ of Theorem 1. Indeed, if $p,q\to 0$ with $p=o(q^{2^{d}})$ then with high probability there exist $M<0<N$ such that no site in the box $\{0,1\}^{d-1}\times[M,N]$ is initially occupied but every site on the two ends $\{0,1\}^{d-1}\times\{M,N\}$ is closed. On this event, the origin cannot become occupied. (For the modified model, the same argument works even for the line $\{0\}^{d-1}\times[M,N]$, giving the same conclusion under the weaker assumption $p=o(q)$. Similar comparisons involving $\{0,1\}^{d-d^{\prime}}\times\mathbb{Z}^{d^{\prime}}$ or $\{0\}^{d-d^{\prime}}\times\mathbb{Z}^{d^{\prime}}$ for $d>d^{\prime}$ are available, which, when combined with the results of [GM2] for $d^{\prime}=2$ or [GHS] for $d^{\prime}=3$, yield further improvements.) On the other hand, the case of threshold $r=1$ is easily understood via standard site percolation: the final occupied set is simply the union of all open clusters that contain initially occupied sites. (This observation is relevant to Corollary 2.) Finally, thresholds $r>d$ are less interesting to us, since, even with no closed sites, there are finite sets such as $\{0,1\}^{d}$ that remain unoccupied forever if unoccupied initially, so $\lim_{(p,q)\to(0,0)}\phi(p,q)=0$.

Background

Bootstrap percolation is an established model for nucleation and metastability, and one of very few cellular automaton models with a well-developed mathematical theory. It has been applied in physics, biology, and social science to various growth phenomena, including crack formation, crystal growth, and spread of information or infection. See [GZH] for a recent example. Bootstrap percolation has been used in the rigorous analysis of other models such as sandpile and Ising models; see e.g. [Mor]. The evolving set method in Markov mixing theory can be viewed as bootstrap percolation with a randomly varying threshold [MP].

Bootstrap percolation was first considered on trees [CLR], but the lattice $\mathbb{Z}^{d}$ with its physics connotations has received the most attention. There has been recent interest in mean-field and power-law graphs, motivated in part by applications to social networks; see e.g. [JŁTV, AFP, KL].

Polluted bootstrap percolation was introduced in [GM2] on the two dimensional lattice. Potential areas of application include the effects of impurities on crystal growth, of immunization on epidemics, or of interventions on spread of rumors. Since [GM2], rigorous progress on growth processes in random environments has been limited, and the case of polluted bootstrap percolation in three and higher dimensions has been entirely open until now. Here are some examples of work on related models. Investigation of asymptotic shapes in models related to polluted bootstrap percolation with $r=1$ was initiated in [GM1]; a recent paper [JŁTV] studies such processes on a complete graph with excluded edges; and [DEK*+*] addresses a Glauber dynamics (which can be viewed as a non-monotone version of bootstrap percolation) with “frozen” vertices. Polluted bootstrap percolation and closely related models have been used in empirical studies of complex networks with “damaged” vertices [BDGM1, BDGM2].

A key element in our proof will be the simple but powerful method of random oriented surfaces recently introduced in [DDG*+*]. This method has been further used and developed in a variety of contexts [DDS, GH1, GH2, GH3, HM, BT], but ours is the first application to cellular automata so far as we are aware. A distinct application to polluted bootstrap percolation will appear in [GHS].

Another useful tool will be the results of [LSS] concerning domination of finitely dependent processes. A random configuration $X=(X_{v})_{v\in\mathbb{Z}^{d}}$ taking values in $\{0,1\}^{\mathbb{Z}^{d}}$ is called $m$-dependent if $(X_{v})_{v\in A}$ and $(X_{v})_{v\in B}$ are independent of each other whenever the sets $A$ and $B$ are at distance greater than $m$. The relevant result of [LSS] is that for any $p<1$ there exists $p^{\prime}=p^{\prime}(p,d,m)<1$ such that, if $X$ is $m$-dependent and satisfies $\mathbb{E}X_{v}\geq p^{\prime}$ for all $v$, then $X$ stochastically dominates an i.i.d. process with parameter $p$.

Outline of proof and organization

The modified bootstrap percolation model is “weaker” than the standard model, in the sense that it is more difficult for a site to become occupied, so that for a given initial configuration, the occupied set for the modified model is a subset of that for the standard model at each time $t$. Therefore, it suffices to prove the conclusions of Theorem 1 for the modified model. Moreover, we may without loss of generality assume that $d=3$. Indeed, for $d\geq 4$ we may restrict to the $3$-dimensional subspace $\mathbb{Z}^{3}\times\{0\}^{d-3}$. Any site that becomes occupied in the $d=3$ model restricted to the subspace also becomes occupied in the full model on $\mathbb{Z}^{d}$ (where in both cases $r=2$). Therefore, for the remainder of the paper we consider the modified bootstrap percolation model with $r=2$ on $\mathbb{Z}^{3}$ except where explicitly stated otherwise.

In the absence of closed sites, the two-dimensional bootstrap rule fills $\mathbb{Z}^{2}$ from any positive density $p$ of occupied sites. This suggests the following approach. For $q$ sufficiently small we may attempt to construct an infinite two-dimensional surface that avoids closed sites and behaves like $\mathbb{Z}^{2}$, in the sense that it also admits growth by the $r=2$ model for any $p>0$. In Section 2 we indeed construct an oriented surface, called a curtain, with some of the required properties. In particular, starting from an infinite fully occupied half space of $\mathbb{Z}^{3}$, a curtain will become fully occupied almost surely for any $p>0$. The construction of the curtain itself does not involve $p$, and does not depend on the locations of initially occupied sites.

A curtain alone is not sufficient to prove Theorem 1, because a finite occupied nucleus does not lead to indefinite growth on a curtain. To address this, we will use a renormalization argument involving curtains with different orientations that intersect each other. This part of the argument will involve $p$, in the determination of a length scale. In Section 3 we construct the unit of our renormalization, which is a curtain restricted to a finite box, with carefully constrained geometry, and scaled to facilitate the required intersections. This modified curtain is called a sail. The size of the box is chosen to be a power of $p^{-1}$, which allows the sail to contain sufficient initially occupied sites for growth similar to that on a curtain. In Section 4 we use comparison methods to show that if two sails intersect appropriately then occupation is transmitted from one to the other. Finally, Section 5 completes the renormalization argument, which involves comparison of an infinite network of sails with supercritical oriented percolation, together with “sprinkling” for the initial nucleation.

We conclude the paper with a list of open problems.

Notation and conventions

As stated earlier, we work with the polluted modified bootstrap percolation model with threshold $r=2$ on $\mathbb{Z}^{3}$ unless stated otherwise. The cubic lattice, also denoted $\mathbb{Z}^{3}$, is the graph with vertex set $\mathbb{Z}^{3}$ and with an edge between sites $u$ and $v$ whenever $\|u-v\|_{1}=1$. When discussing sets of sites, connectivity and components always refer to this graph.

When describing subsets of $\mathbb{Z}^{3}$, intervals will be understood to denote their intersections with $\mathbb{Z}$, so $[a,b)$ denotes $[a,b)\cap\mathbb{Z}=\{a,a+1,\ldots,b-1\}$, etc. Let $\mathbb{N}$ be the set of nonnegative integers. We will frequently wish to consider $2$-dimensional layers of $\mathbb{Z}^{3}$, which by convention will be taken perpendicular to the $3$rd coordinate. Thus, for $k\in\mathbb{Z}$ we define the $k$th layer to be

[TABLE]

Let $\langle\cdot,\cdot\rangle$ denote the standard inner product on $\mathbb{Z}^{3}$, and let $e_{1},e_{2},e_{3}$ be the standard coordinate vectors.

We will consider paths of various types, not always with nearest-neighbor steps. In general, a path is a finite or infinite sequence of sites $(\ldots,)x_{0},x_{1},\ldots,x_{n}(,\ldots)$. Its steps are the vectors $(\ldots,)x_{1}-x_{0},x_{2}-x_{1},\ldots,x_{n}-x_{n-1}(,\ldots)$. It is a nearest-neighbor path if all steps are of the form $\pm e_{i}$. It is self-avoiding if all its sites are distinct.

2. Curtains

In this section we introduce the oriented surfaces underlying our construction in their pure form. Later they will be modified by scaling and restricting to finite boxes.

Definition**.**

A curtain is a set $D\subset\mathbb{Z}^{3}$ satisfying the following.

1. (C1)

For any $k\in\mathbb{Z}$, the intersection $D\cap\Lambda_{k}$ with layer $k$ is an infinite path comprising steps $e_{1}$ and $-e_{2}$, with no three consecutive steps in the same direction; i.e. no $e_{1},e_{1},e_{1}$ or $-e_{2},-e_{2},-e_{2}$. 2. (C2)

For all $x\in D$, either $x+(0,0,-1)\in D$ or $x+(1,1,-1)\in D$.

Figure 2 in the next section shows the intersection of a curtain with a box. The main goal of this section is to construct an infinite open curtain when $q$ is sufficiently small. This will be done adapting the duality technique introduced in [DDG*+*] for construction of Lipshitz surfaces. The curtain will form the outer boundary of a set reachable by certain paths from a fixed half space. Before giving the construction, we illustrate the relevance of curtains to bootstrap percolation with the following lemma. (Formally, the lemma will not be used in the proof of Theorem 1. Instead we will use a more specialized variant, Lemma 8.)

Lemma 3**.**

Let $D$ be a curtain. Suppose that for every $x\in D$, the three sites $x$ and $x+(0,0,1)$ and $x+(-1,-1,1)$ are all open. Moreover, suppose that for every $k\in\mathbb{N}$, the set $(D\cap\Lambda_{k})+e_{3}$ contains some initially occupied site. If $D\cap\Lambda_{0}$ is initially entirely occupied, then $D\cap\bigcup_{k\in\mathbb{N}}\Lambda_{k}$ becomes entirely occupied in the modified bootstrap model on $\mathbb{Z}^{3}$.

Proof.

By induction on the layer, it suffices to prove that $D\cap\Lambda_{1}$ becomes entirely occupied. This verification is given in two steps below, and is illustrated in Figure 1. Let $Y:=(D\cap\Lambda_{0})+e_{3}$ be the set above the intersection with the bottom layer.

First we claim that every site in $Y$ eventually becomes occupied. Indeed, $Y$ is connected, open, and contains an occupied site, and every $y\in Y$ has an occupied neighbor $y-e_{3}\notin Y$. The claim therefore follows from the bootstrap rule.

We now claim that every site in $Y-(1,1,0)$ also eventually becomes occupied. Indeed, consider such a site $z=y-(1,1,0)$ where $y\in Y$. Since $Y$ is a path with the properties given in (C1), there exist sites $z+ae_{1}$ and $z+be_{2}$ in $Y$, where $a,b\in[0,3]$. Moreover, the intervening sites $z+ie_{1}$ and $z+je_{2}$ for $i\in(0,a)$ and $j\in(0,b)$ are open by (C2), and each has a neighbor in $Y$ distinct from $z+ae_{1}$ and $z+be_{2}$. Since all sites in $Y$ become occupied, so do all these sites, whence so does $z$.

The proof is now concluded by observing that $D\cap\Lambda_{1}\subseteq Y\cup(Y-(1,1,0))$. ∎

Now we proceed with the construction of a curtain. A permissible path is a finite sequence of sites $x_{0},\ldots,x_{n}\in\mathbb{Z}^{3}$ such that every step $x_{i+1}-x_{i}$ satisfies the following. Either it is a taxed step, which is to say that $x_{i+1}$ is closed, and $x_{i+1}-x_{i}$ equals

[TABLE]

Otherwise, the step is free, that is, $x_{i+1}-x_{i}$ lies in

[TABLE]

(with no restriction on the states of sites).

Fix any (deterministic) set $H\subset\mathbb{Z}^{3}$ and let $A$ be the (random) set reachable by permissible paths from $H$. Then define the following outer boundary:

[TABLE]

Lemma 4**.**

For any choice of $H$, the set $D$ is either empty or an open curtain.

The lemma is of course only useful when $D$ is nonempty. This will be proved to hold under suitable circumstances in Proposition 5 below.

Proof of Lemma 4.

We must prove that if $D$ is nonempty then it is open and has properties (C1) and (C2). Consider any $x\in D$. By translation invariance of the definition, we assume without loss of generality that $x$ is the origin $0=(0,0,0)$.

Clearly, [math] is open, since otherwise the taxed step from $(-1,-1,0)\in A$ would make $0\in A$.

Turning to property (C1), we have $(-1,-1,0)\in A$ but $0\notin A$, so using the definition of free steps, $(-1,-2,0)\in A$ but $(1,0,0)\notin A$. We claim that either $(1,0,0)\in D$ or $(0,-1,0)\in D$, but not both. Indeed, if $(0,-1,0)\in A$ then $(1,0,0)\in D$, while if $(0,-1,0)\notin A$ then $(0,-1,0)\in D$. A similar argument shows that either $(-1,0,0)\in D$ or $(0,1,0)\in D$ but not both. This shows that $D\cap\Lambda_{0}$ is a union of disjoint paths with steps $e_{1}$ and $-e_{2}$. To check the restriction on three consecutive steps, note that $(0,-3,0)\in A$ but $(2,-1,0)\notin A$, which implies $(0,-3,0),(3,0,0)\notin D$. To show that there is only one path, note that $(-1,-1,0)$ is a sum of two free steps, so the diagonal $\{(k,k,0):k\in\mathbb{Z}\}$ is partitioned into an interval belonging to $A$ and an interval belonging to $A^{C}$. If $D$ is nonempty then both intervals are nonempty, and so the diagonal contains exactly one site in $D$.

To prove property (C2), note that $(-1,-1,-1)\in A$ but $(1,1,-1)\notin A$. Consequently, if $(0,0,-1)\notin A$, then $(0,0,-1)\in D$. On the other hand, if $(0,0,-1)\in A$, then $(1,1,-1)\in D$. ∎

We now choose $H$ to be the half-space

[TABLE]

and let $A$ and $D$ by defined as above. Note that, by property (C1), a curtain intersects the line $\{(t,t,0):t\in\mathbb{Z}\}$ in exactly one site.

Proposition 5**.**

There exist positive constants $q_{0}$ and $c$ such that the following holds. For $q<q_{0}$, the set $D$ constructed above is, almost surely, an open curtain. Furthermore, the probability that $(1,1,0)\in D$ tends to $1$ as $q\to 0$, while for any $q<q_{0}$, the probability that $D$ intersects the ray $\{(t,t,0):t>k\}$ is less than $e^{-ck}$ for all $k>0$.

Proof.

Let $h=(1,1,1)$. Observe that the scalar product $\langle x,h\rangle$: equals $2$ when $x$ is the taxed step; equals $-1$ when $x$ is a free step; and is nonpositive when $x\in H$.

Fix $k\geq 1$ and suppose $(k,k,0)\in A$. Then there exists a permissible from $H$ to $(k,k,0)$. By erasing loops, we may assume that the path is self-avoiding. Let $n_{F}$ and $n_{T}$ be the number of free and taxed steps of the path, respectively, and let $n=n_{F}+n_{T}$ be the total length. As $\langle(k,k,0),h\rangle=2k$, the above observations about scalar products imply $-n_{F}+2n_{T}\geq 2k$. It follows that $n\geq n_{T}\geq k$ and $n_{T}\geq(2k+n)/3$. Therefore,

[TABLE]

provided $q<q_{0}:=8^{-3}$.

Note that $0\in A$, so that if $(k,k,0)\notin A$ then $\{(t,t,0):t\in[1,k]\}$ intersects $D$ while $\{(t,t,0):t>k\}$ does not. Thus, if $q<q_{0}$ then Lemma 4 implies that $D$ is almost surely nonempty, and thus is an open curtain by Lemma 4. Equation 3 also gives the claimed exponential bound, since $8\cdot(7q)^{k}\leq(56q_{0})^{k}$. Taking $k=1$ in 3, we get $\mathbb{P}((1,1,0)\notin D)\leq 56q$, giving the second claim. ∎

The results from this section are already strongly suggestive of the conclusions of Theorem 1, although by no means sufficient to prove them. Indeed, consider the initial configuration consisting of the fully occupied half-space $\{x:\langle x,e_{3}\rangle\leq 0\}$, and elsewhere product measure with densities $p$ and $q$ as usual. It follows easily from Lemmas 3, 4 and 5 that the probability that any fixed site $x\in\mathbb{Z}^{3}$ is eventually occupied converges to $1$ as $(p,q)\to(0,0)$. Indeed, with high probability $x$ lies in a curtain that has the properties in Lemma 3: the presence of the appropriately placed open sites can be guaranteed via [LSS], while the presence of an occupied site in each layer of the curtain holds almost surely. The remaining difficulty in the proof of Theorem 1 is the need to replace the occupied half-space with a finite nucleus.

3. Sails

A box in $\mathbb{Z}^{3}$ is a Cartesian product of any three integer intervals. Its dimensions are the cardinalities of the three intervals (in order). An oriented box is a box with a distinguished corner.

Fix an integer length scale $L$. This scale will be later chosen to be a suitable function of $p$. A brick is an oriented box of dimensions $4L$, $16L$, and $32L$, in any order. Bricks will be the units of our renormalization. We will formulate the required properties of bricks by translating and scaling a smaller box. The proto-brick $\widehat{B}$ is the oriented box $[0,4L)\times[0,4L)\times[0,2L)$ with the distinguished corner at the origin.

We now formulate the key definition in our renormalization argument. The idea is that the proto-brick contains a suitably placed portion of a curtain, with properties analogous to those in Lemma 3, but restricted to the proto-brick. See Figure 2 for an illustration.

Definition**.**

The proto-brick $\widehat{B}$ is good if there exists a set $\widehat{S}\subseteq\widehat{B}$ with the following properties:

1. (G1)

all sites in the following set are open:

[TABLE] 2. (G2)

$\widehat{S}$ satisfies (C2) in the definition of a curtain except at the bottom layer: for all $x\in\widehat{S}\setminus\Lambda_{0}$, either $x+(0,0,-1)\in\widehat{S}$ or $x+(1,1,-1)\in\widehat{S}$; 3. (G3)

$\widehat{S}\subseteq\{x:3L<x_{1}+x_{2}+x_{3}<4L\}$; 4. (G4)

for each layer $k\in[0,2L)$, the intersection $\widehat{S}\cap\Lambda_{k}$ is an oriented path that starts on $\{x:x_{1}=0\}$, ends on $\{x:x_{2}=0\}$ and makes steps $-e_{2}$ or $e_{1}$ with no consecutive three steps of the same type; and 5. (G5)

for each layer except the top, there is an occupied site immediately above its intersection with $\widehat{S}$, i.e. $(\widehat{S}\cap\Lambda_{k})+e_{3}$ contains an occupied site for each $k\in[0,2L-1)$.

Next we scale up this definition to a brick, starting with one in a standard location and orientation. Let $B$ be the brick $[0,4L)\times[0,16L)\times[0,32L)$ with the distinguished corner at the origin. For $x\in\widehat{B}$, define the following subset of $B$:

[TABLE]

See Figure 3. For a given configuration on $B$, we define an auxiliary configuration on $\widehat{B}$ by declaring a site $x\in\widehat{B}$ open if all sites in ${\text{\tt cell}}(x)$ are open; otherwise, we declare $x$ closed. We also call $x$ initially occupied if all sites in ${\text{\tt cell}}(x)$ are initially occupied. We call $B$ good if, in the auxiliary configuration, $\widehat{B}$ is good. See Figure 3.

If $B$ is good and $\widehat{S}$ is any set satisfying the above conditions, then we call

[TABLE]

a sail for $B$. Thus $B$ is good if and only it has a sail.

Define the tail and head of $B$ to be its lower and upper halves, $[0,4L)\times[0,16L)\times[0,16L)$ and $[0,4L)\times[0,16L)\times[16L,32L)$ respectively. The tip of $B$ is the box $[0,4L)\times[0,4L)\times[16L,32L)$, which is a quarter of the head. See Figure 3. The base of $B$ is the bottom layer of cells $\bigcup_{x\in\widehat{B}\cap\Lambda_{0}}{\text{\tt cell}}(x).$ If $B$ is good and $S$ is a sail for $B$, then the head, tail, base, and tip of $S$ are the intersections of $S$ with the corresponding subsets of $B$.

If the brick $B$ is good, and $S$ is a sail for $B$, then we say that $S$ is activated by time $t$ if every site in the the head of $S$ is occupied at time $t$.

Now we transfer all the above definitions to an arbitrary brick $B^{\prime}$ by isometry. More precisely, let $\eta$ be an isometry of $\mathbb{Z}^{3}$ that maps $B$ to $B^{\prime}$, respecting the distinguished corners. The head of $B^{\prime}$ is the image under $\eta$ of the head of $B$. The brick $B^{\prime}$ is good if applying $\eta^{-1}$ to the configuration makes $B$ good, in which case a sail for $B^{\prime}$ is an image under $\eta$ of a sail for $B$ in that configuration, and so on.

We next show that with high probability a brick is good, and moreover the sail can be chosen to contains a specific site.

Proposition 6**.**

Assume $L=\lceil p^{-128}\rceil$. Then the probability that $B$ is good and has a sail $S$ that contains the site $x_{0}:=(L+1,4L+4,16L)$ converges to $1$ as $(p,q)\to(0,0)$.

We remark that $L$ does not need to be such a large power of $p^{-1}$; with suitable modifications to the definitions and proofs (perhaps at the expense of increased complexity), order $p^{-1}\log p^{-1}$ would suffice.

Proof of Proposition 6.

Fix $\epsilon>0$. For most of the proof we will consider the relevant event on the proto-brick $\widehat{B}$. Therefore let $\widehat{p}=p^{64}$ and $\widehat{q}=1-(1-q)^{64}$, which are the probabilities that a site is, respectively, initially occupied and open in the auxiliary configuration. Let $L=\lceil p^{-128}\rceil$.

Call a site $x$ swell if $x$ and $x+(0,0,1)$ and $x+(-1,-1,1)$ are all open in the auxilliary configuration. By the results of [LSS], the configuration of swell sites dominates a product measure on $\mathbb{Z}^{3}$ with parameter $1-\widehat{q}{\,}^{\prime}$, where $\widehat{q}{\,}^{\prime}=\widehat{q}{\,}^{\prime}(\widehat{q})\to 0$ as $\widehat{q}\to 0$.

Next, we apply Proposition 5 and translation invariance to construct a swell curtain close to the half-space $(L,L,L)+H$, rather than $H$. To be precise, translate the configuration of swell sites by $-(L,L,L)$, construct the set $D$ according to the last section, but using swell sites in place of open sites, and translate it back by $(L,L,L)$ to obtain a set $\widehat{D}$ of swell sites that lies in $((L,L,L)+H)^{C}=\{x:x_{1}+x_{2}+x_{3}>3L\}$.

Let $E_{1}$ be the event that $\widehat{D}$ is a curtain and contains the site $\widehat{x}_{0}=(1,1,0)+(L,L,L)$. By the construction of $\widehat{D}$ in the previous section, $E_{1}$ is an increasing event with respect to the configuration of swell sites. (This follows because, in the notation of that section, the set $A$ of sites reachable from $H$ via permissible paths is decreasing). Therefore, by Proposition 5 and [LSS], there exists $q_{1}>0$ such that if $\widehat{q}<q_{1}$ then $\mathbb{P}(E_{1})>1-\epsilon$.

Moreover, by Proposition 5, [LSS], and translation invariance, for any deterministic $x=(x_{1},x_{2},x_{3})$ with $x_{1}+x_{2}+x_{3}=3L$ we have

[TABLE]

where $c>0$ is an absolute constant. (This event is again increasing in the configuration of swell sites, by the construction of $\widehat{D}$.) Now let

[TABLE]

Let $E_{2}$ be the event that every $x\in\widehat{S}$ satisfies $x_{1}+x_{2}+x_{3}<4L$. Then 4 and a union bound imply that $\mathbb{P}(E_{2})\geq 1-16L^{2}\exp(-cL/2).$ Since $L\to\infty$ as $p\to 0$ (i.e. as $\widehat{p}\to 0$), for $\widehat{p}$ is sufficiently small we have $\mathbb{P}(E_{2})\geq 1-\epsilon$.

We have shown that $\widehat{p}$ and $\widehat{q}$ are both sufficiently small then $\mathbb{P}(E_{1}\cup E_{2})\geq 1-2\epsilon$. On $E_{1}\cup E_{2}$, the set $\widehat{S}$ satisfies properties (G1)–(G4) in the definition of a good proto-brick. So far we have not considered initially occupied sites (although the parameter $p$ has appeared in the definition of the length scale $L$). One way to sample the auxiliary configuration is as follows. First declare each site closed independently with probability $\widehat{q}$. Then, conditional on the resulting configuration, declare each open site to be initially occupied independently with probability $\widehat{p}/(1-\widehat{q})$ ($\geq\widehat{p}$). Let $E_{3}$ be the event that $\widehat{S}$ satisfies property (G5). On $E_{1}\cap E_{2}$, each intersection with a layer $\widehat{S}\cap\Lambda_{k}$ for $k\in[0,2L-1)$ contains at least $L$ sites (by (G3) and (G4)). Moreover, all sites in $(\widehat{S}\cap\Lambda_{k})+e_{3}$ are open (by (G1)). Hence,

[TABLE]

Since $L\sim\widehat{p}\,^{-2}$, this is at least $1-\epsilon$ for $\widehat{p}$ sufficiently small.

We have shown that for $\widehat{p}$ and $\widehat{q}$ sufficiently small, with probability at least $1-3\epsilon$ the set $\widehat{S}$ satisfies (G1)–(G5) and contains $\widehat{x}_{0}$. Finally, recalling the definition of the auxilliary configuration, we deduce that for $p$ and $q$ sufficiently small, the brick $B$ is likewise good and has a sail containing $x_{0}\in{\text{\tt cell}}(\widehat{x}_{0})$ with probability at least $1-3\epsilon$. ∎

4. Activation

Recall that a sail of a good brick is said to be activated if its head is fully occupied (at some time). To enable our renormalization argument, we now show that for appropriately placed good bricks, activation of one sail leads to activation of another.

Let $B$ be the brick in standard position as before, and let $B^{\prime}$ be a brick with dimensions $(32L,4L,16L)$ such that the centroid of its tail coincides with the centroid of the tip of $B$. (The idea is that the tip of $B$ cuts the tail of $B^{\prime}$ in two. There are eight possible choices of $B^{\prime}$: two possible boxes that share a tail, each with four possible orientations. See Figure 4 in the next section for examples.) Then we write $B\rhd B^{\prime}$. Similarly for any isometry $\eta$ of $\mathbb{Z}^{3}$ we write $\eta(B)\rhd\eta(B^{\prime})$.

Proposition 7**.**

Let $B$ and $B^{\prime}$ be as described above. Suppose that they are both good and let $S$ and $S^{\prime}$ be any respective sails. In the modified bootstrap percolation model, if $S$ is activated by some time, then $S^{\prime}$ is activated by some later time.

We separate the proof into the following four lemmas, starting with the underlying growth mechanism.

Lemma 8**.**

Suppose that the proto-brick $\widehat{B}$ is good, and let $\widehat{S}$ be any set satisfying the conditions in the definition of good. Assume also that the intersection $\widehat{S}\cap\Lambda_{0}$ with the bottom layer is entirely occupied initially, and that $\mathbb{Z}^{3}\setminus\sigma(\widehat{S})$ is entirely closed. Then $\widehat{S}$ is entirely occupied at some time.

Proof.

The argument is essentially the same as for Lemma 3, except that one must verify that the relevant sites lie in the proto-brick. We prove by induction on $k=0,\ldots,2L-1$ that the layer $\widehat{S}\cap\Lambda_{k}$ is eventually occupied. For $k=0$ this holds by assumption.

Fix $k\geq 1$ and let $Y=(\widehat{S}\cap\Lambda_{k-1})+e_{3}$. Then $Y$ becomes occupied, since it is connected and open, it contains an occupied site, and it is adjacent to $\widehat{S}\cap\Lambda_{k-1}$ which becomes occupied by the inductive hypothesis. If $z\in\widehat{S}\cap\Lambda_{k}$ then either $z\in Y$ or $z=y-(1,1,0)$ where $y\in Y$. In the latter case there exist $z+ae_{1},z+be_{2}\in\pi+e_{3}\in Y$ with $a,b\in[0,3]$. The bootstrap rule then guarantees that $z+ie_{1}$ and $z+je_{2}$ become occupied for $i\in(0,a)$ and $j\in(0,b)$, and then $z$ becomes occupied. ∎

The following comparison lemma states that cutting off part of a configuration only increases the eventually occupied set, provided we make the cut surface occupied. This will enable us to make use of sails that intersect each other.

Lemma 9**.**

Consider a set of sites $A$, and a subset $F\subseteq A$. Let $B$ be a connected component of $A\setminus F$. Suppose that every site in $A^{C}$ is closed but that the initial configuration is otherwise arbitrary. Now alter the initial configuration by making $F$ initially occupied but $A\setminus(F\cup B)$ closed. The alteration (weakly) increases the set of eventually occupied sites in $B$.

Proof.

We proceed by induction on the time step. Suppose that at all times prior to $t$, the set of occupied sites of $B$ in the altered dynamics dominates the set in the original dynamics. Assume that a site $x\in B$ becomes occupied in the original dynamics at time $t$. Any neighbor of $x$ that was occupied in the original dynamics at time $t-1$ either lies in $B$, in which case it is also occupied in the altered dynamics by the induction hypothesis, or it lies in $F$, in which case it was initially occupied in the altered dynamics. Thus $x$ also becomes occupied in the altered dynamics. ∎

Lemma 10**.**

From any configuration on $B$, form the auxiliary configuration on $\widehat{B}$, and perform the modified bootstrap percolation dynamics from the auxiliary configuration with all sites outside $\widehat{B}$ closed. If $x$ becomes occupied in the auxiliary dynamics, then ${\text{\tt cell}}(x)$ becomes fully occupied in the original dynamics.

Proof.

This follows by straightforward induction on time step. ∎

Next we state a geometric fact about sails. Let $A\subseteq\mathbb{Z}^{3}$ and let $F,B_{1},B_{2}$ be disjoint subsets of $A$. We say that $F$ separates $B_{1}$ and $B_{2}$ in $A$ if $A\setminus F$ contains no nearest-neighbor path from $B_{1}$ to $B_{2}$.

Lemma 11**.**

Suppose that the brick $B$ is good. Then any sail $S$ for $B$ separates, in the tip of $B$, the two faces of the tip $\{0\}\times[0,4L)\times[16L,32L)$ and $\{4L-1\}\times[0,4L)\times[16L,32L)$.

Proof.

By property (G4) of a good proto-brick, the intersection of $S$ with a layer $\Lambda_{k}$ is an oriented path, thickened by conversion of sites to cells. Therefore its complement $\Lambda_{k}\setminus S$ clearly has two components, $U_{k}$ and $V_{k}$ say, which contain the intersections of the first and second faces respectively with $\Lambda_{k}$, by (G3).

It remains to check that no site of $U_{k-1}$ is adjacent to a site of $V_{k}$, and likewise for $V_{k-1}$ and $U_{k}$, for $k=1,\ldots,4L-1$. Since such adjacent sites would differ by $e_{3}$, this is easily verified from property (G2). (Also see Figure 1). ∎

Now we prove the main result of this section.

Proof of Proposition 7.

By definition of $S$ from $\widehat{S}$ and Lemma 11, the head of $S$ separates the base of $S^{\prime}$ from the head of $S^{\prime}$ in $\sigma(S^{\prime})$.

Consider the dynamics from the following new configuration. Make every site outside $\sigma(S^{\prime})$ closed. Make the base of $S^{\prime}$ occupied. Otherwise, retain the initial configuration in $\sigma(S^{\prime})$. By Lemmas 10 and 8, the entire sail $S^{\prime}$ becomes occupied. The proof is concluded by applying Lemma 9 to $\sigma(S^{\prime})$. ∎

5. Renormalization

In this section we prove the main result, Theorem 1, as well as Corollary 2. We start with a simple geometric ingredient. Recall that $B$ is the brick in standard position.

Lemma 12**.**

There exist bricks $B_{i}$, $B_{i}^{\prime}$, for $i=1,2,3$, with

[TABLE]

such that $B$, $B_{3}$, and $B_{3}^{\prime}$ are distinct and have the same orientation. Furthermore, there exist vectors $u,u^{\prime}\in\mathbb{Z}^{3}$ and a constant $C$, none of them depending on $L$, such that $B_{3}=B+Lu$ and $B_{3}^{\prime}=B+Lu^{\prime}$ (so in particular $Lu$ and $Lu^{\prime}$ are the distinguished corners of $B_{3}$ and $B_{3}^{\prime}$ respectively), and all seven bricks lie within distance $CL$ of the origin.

Proof.

See Figure 4. Recall that $B$ has dimensions $(4L,16L,32L)$. We choose $B_{1}$ and $B_{1}^{\prime}$ equal to each other, with dimensions $(32L,4L,16L)$, and satisfying $B\rhd B_{1}$. Then take $B_{2}$ and $B_{2}^{\prime}$ to be the same box as each other, with dimensions $(16L,32L,4L)$, but with different orientations and in particular different tips. Finally take $B_{3}$ and $B_{3}^{\prime}$ to be suitable translations of $B$, as determined by these tips. ∎

Proof of Theorem 1.

As discussed in the introduction, it suffices to prove the case of the modified bootstrap model on $\mathbb{Z}^{3}$.

We will compare with oriented percolation in $\mathbb{Z}^{2}$. Let $L=\lceil p^{-128}\rceil$ and let $u,u^{\prime}\in\mathbb{Z}^{3}$ be as in Lemma 12. Also fix $\epsilon>0$. For $a=(a_{1},a_{2})\in\mathbb{Z}^{2}$, define the associated brick

[TABLE]

Call the site $a$ excellent if the translations by $La_{1}u+La_{2}u^{\prime}$ of the seven bricks $B,B_{i},B_{i}^{\prime}$ of Lemma 12 are all good.

Suppose that $a$ and $b$ are excellent, so that in particular the bricks $B(a)$ and $B(b)$ are good. Suppose also that there is a path of excellent sites in $\mathbb{Z}^{2}$ from $a$ to $b$ consisting of steps $e_{1}$ and $e_{2}$. (We call a path with these steps oriented). Then by Lemmas 12 and 7, if some sail of $B(a)$ is activated then any sail of $B(b)$ is activated at some later time.

Let $E$ be the event that there exists an excellent bi-infinite oriented path $\pi$ in $\mathbb{Z}^{2}$ containing $0=(0,0)$, and that moreover $B=B(0)$ has a good sail containing $x_{0}:=(L+1,4L+4,16L)$ in its head. By Lemma 12, the random configuration of excellent sites is $m$-dependent for some fixed $m$ not depending on $L$. Therefore, by [LSS], Proposition 6, and the fact that oriented percolation on $\mathbb{Z}^{2}$ has a nontrivial phase transition (see e.g. [Gri]), if $p$ and $q$ are sufficiently small then $\mathbb{P}(E)\geq 1-\epsilon$.

It remains to show that some sail on the path is activated, for which a rather crude sprinkling argument will suffice. Assuming $2p+q<1$, we consider two coupled initial configurations. The level-$1$ configuration has parameters $p$ and $q$ as before. Conditional on the level-$1$ configuration, the level-$2$ configuration is obtained by adding some further occupied sites; specifically, we declare each open site that was not initially occupied at level $1$ to be initially occupied at level $2$ independently with probability $p/(1-p-q)$, and leave the configuration otherwise unchanged. The law of the level-$2$ configuration is simply a product measure with parameters $2p$ and $q$. Now condition on the level-$1$ configuration, and suppose that it is such that $E$ occurs at level $1$. Fix an excellent oriented path $\pi$ as in the definition of $E$, and let $\pi^{+}$ and $\pi^{-}$ be the forward and backward halves of $\pi$ that start at [math] and end at [math] respectively. Then for each site $a$ of $\pi^{-}$, all open sites in the brick $B(a)$ are initially occupied at level $2$ with probability at least $p^{|B|}$, independently for each such $a$. Therefore, conditionally almost surely, some site $a$ on $\pi^{-}$ has this property, which implies in particular that any sail of the associated brick $B(a)$ is activated at level $2$.

We conclude that if $2p$ and $q$ are sufficiently small then with probability at least $1-\epsilon$ there exists an infinite sequence of distinct activated sails, each intersecting the next, one of which contains $x_{0}$ in its head. By translation invariance we conclude that with probability at least $1-\epsilon$, the origin lies in an infinite connected eventually occupied set, as required. ∎

Proof of Corollary 2.

It follows from Theorem 1 that $\lim_{q\to 0^{+}}\phi^{+}(q)=1$, which implies that $q_{c}>0$. As $\phi$ is a decreasing function, it is positive on $[0,q_{c})$. Our remaining task is to prove the claimed upper bound on $q_{c}$, for which it suffices to consider the standard (as opposed to modified) bootstrap model with $r=2$ on $\mathbb{Z}^{d}$ for $d\geq 3$.

Call a site $3$-open if it is open and has at least $3$ open sites among its $2d$ neighbors. Let $p_{c}^{\prime}$ be the critical probability for existence of an infinite connected set of $3$-open sites in $\mathbb{Z}^{d}$. Then clearly $p_{c}^{\prime}\geq p_{c}^{\text{\rm site}}$. For $d=3$, the method of essential enhancements [AG, BBR] shows that $p_{c}^{\prime}>p_{c}^{\text{\rm site}}$. (The strict inequality is expected to hold for $d\geq 4$ also, but no complete proof is available – see [BBR]).

For any set $Z\subseteq\mathbb{Z}^{d}$, let the external boundary $\partial Z$ be the set of sites in $\mathbb{Z}^{d}\setminus Z$ that have a neighbor in $Z$. Note that $|\partial Z|\leq 2d|Z|$. Assume that no site in $\partial Z$ is $3$-open, and that no site in $Z\cup\partial Z$ is initially occupied. Then we claim that no site in $Z\cup\partial Z$ is ever occupied. Indeed, suppose on the contrary that $x\in Z\cup\partial Z$ is a first site in the set to become occupied, say at time $t\geq 1$. Then $x\in\partial Z$, and so $x$ has at most $2$ open neighbors, of which at least one is in $Z$, which by assumption is unoccupied at time $t-1$. So $x$ has at most one occupied neighbor at time $t-1$. Since $r=2$, this contradicts the assumption that $x$ becomes occupied at time $t$.

Now let $q>1-p_{c}^{\prime}$. Given the random configuration on $\mathbb{Z}^{d}$, create an adjusted configuration by converting all closed sites among the origin and its $2d$ neighbors to open (but not initially occupied) sites. Let $\mathcal{Z}$ be the maximal connected set of $3$-open sites containing the origin in the adjusted configuration. Clearly, $0<|\mathcal{Z}|<\infty$ almost surely. Then we have

[TABLE]

This tends to [math] as $p\to 0^{+}$, by dominated convergence. ∎

6. Open Problems

Recall that $\phi(p,q)$ is the probability that the origin is eventually occupied with densities $p$ and $q$ of initially occupied and closed sites respectively, and that we define $\phi^{+}(q)=\phi(0^{+},q)$ and $q_{c}=\inf\{q:\phi^{+}(q)=0\}$.

1. (i)

For which dimensions and thresholds $d>r\geq 3$ is it the case that $\phi(p,q)\to 1$ as $(p,q)\to(0,0)$? As conjectured in [Mor], the answer “all” seems plausible. (The current paper proves the cases $d>r=2$, while the conclusion fails for $d=r$.) 2. (ii)

Where the convergence in (i) above does not hold (presumably, only for $d=r$), suppose that $p,q\to 0$ in such a way that $\log q/\log p\to\alpha.$ For which $\alpha$ does $\phi$ converge to [math], or to $1$? The articles [GM2] and [GHS] address $d=r=2$ and $d=r=3$ respectively. 3. (iii)

Is $\phi^{+}$ continuous at $q_{c}$? 4. (iv)

Consider the critical value $q_{c}=q_{c}(d)$ as a function of dimension (with $r=2$, say). Does $q_{c}$ approach $1$ as $d\to\infty$ and, if so, at what rate? 5. (v)

Let $T$ be the first time the origin is occupied. What is the asymptotic behavior of $T$ as $p,q\to 0$? (For example, find “close” functions $f$ and $g$ of $p$ and $q$ for which $f\leq T\leq g$ with high probability). 6. (vi)

For $r=2$ and $d=3$, consider

[TABLE]

Is there a $q>q_{c}$ for which this is infinite? If so, this would distinguish the phase transition in the case $r=2$ from that of the case $r=1$ (where $q_{c}=1-p_{c}^{\text{\rm site}}(\mathbb{Z}^{d})$, and $\gamma(q)$ is finite for all $q>q_{c}$.)

Acknowledgements

We thank David Sivakoff for many valuable discussions. Janko Gravner was partially supported by the NSF grant DMS–1513340, Simons Foundation Award #281309, and the Republic of Slovenia’s Ministry of Science program P1–285. He also gratefully acknowledges the hospitality of the Theory Group at Microsoft Research, where most of this work was completed.