Theorem 1.3 will follow directly from Theorem 5.1 below, which we will prove in this section. The proof is quite long, so we start with an overview. For clarity’s sake, we discuss the proof overview under the setting of Theorem 1.3, and only state Theorem 5.1 in Sect. 5.2.
Proof overview
We start with a high-level overview of the proof. Below we refer to Fig. 7. Since p is small enough, initially \(\eta ^1\) will grow without finding any seed of \(\eta ^2(0)\), as in Fig. 7a. When \(\eta ^1\) activates a seed of \(\eta ^2\), then we will apply Proposition 4.1 to establish that \(\eta ^1\) will go around \(\eta ^2\), encapsulating it inside a small ball (according to the norm \(|\cdot |\)). This is illustrated by the encapsulation of \(C_1\) in Fig. 7b. The yellow ball in the picture marks the region inside which the cluster of \(\eta ^2\) will be trapped; in Proposition 4.1 this corresponds to the ball \(\mathcal {B}\left( \frac{10R}{11-\lambda }\right) \). To ensure the encapsulation of a cluster of \(\eta ^2\), we will need to observe the passage times inside a larger ball; for example, only to ensure the measurability requirement of the event in Proposition 4.1 we need to observe the passage times in \(\mathcal {B}\left( R\left( \frac{11-\lambda }{10}\right) ^2\right) \). This larger ball is represented by the red circles in Fig. 7. As long as different red circles do not intersect one another, the encapsulation of different clusters of \(\eta ^2\) will happen independently. However, when \(\eta ^1\) encounters a large cluster of \(\eta ^2\) seeds, as it happens with the cluster \(C_3\) in Fig. 7d, the encapsulation procedure will require a larger region to succeed. We will carry this out by developing a multi-scale analysis of the encapsulation procedure, where the size of the region will depend, among other things, on the size of the clusters of \(\eta ^2(0)\). After the encapsulation takes place, as in Fig. 7e, we are left with a larger yellow ball and a larger red circle. Also, whenever two clusters of \(\eta ^2(0)\) are close enough such that their corresponding red circles intersect, as it happens with \(C_2\) in Fig. 7c, then the encapsulation cannot be guaranteed to succeed. In this case, we see these clusters as if they were a larger cluster, and perform the encapsulation procedure over a slightly larger region, as in Fig. 7c, d.
There is one caveat in the above description. Suppose \(\eta ^1\) encounters a very large cluster of \(\eta ^2\), for example \(C_3\) in Fig. 7d. It is likely that during the encapsulation of \(C_3\), inside the red circle of this encapsulation, we will find smaller clusters of \(\eta ^2\). This happens in Fig. 7d with \(C_4\). This does not pose a big problem, since as long as the red circle of the encapsulation of the small clusters do not intersect one another and do not intersect the yellow ball produced by the encapsulation of \(C_3\), the encapsulation of \(C_3\) will succeed. This is illustrated in Fig. 7e, where the encapsulation of \(C_4\) happened inside the encapsulation of \(C_3\). There is yet a subtlety. During the encapsulation of \(C_4\), the advance of \(\eta ^1\) is slowed down, as it needs to make a detour around the growing cluster of \(C_4\). This slowing down could cause the encapsulation of \(C_3\) to fail. Similarly, as \(\eta ^2\) spreads from \(C_3\), \(\eta ^2\) may find vertices that have already been occupied by \(\eta ^2\) due to the spread of \(\eta ^2\) from other non-encapsulated seeds. This would happen, for example, if the yellow ball that grows from \(C_3\) were to intersect the yellow ball that grows from \(C_4\). If this happens before the encapsulation of \(C_4\) ends, then the spread of \(C_3\) gets a small advantage. The area occupied by the spread of \(\eta ^2\) from \(C_4\) can in this case be regarded as being absorbed by the spread of \(\eta ^2\) from \(C_3\), causing \(C_3\) to spread faster than if \(C_4\) were not present. We will need to show that \(\eta ^1\) is not slowed down too much by possible detours around smaller clusters, and \(\eta ^2\) is not sped up too much by the absorption of smaller clusters.
To do this, we will define a sequence of scales \(R_1,R_2,\ldots \), with \(R_k\) increasing with k. The value of \(R_k\) represents the radius of the region inside which encapsulation takes place at scale k. (Later when making this argument rigorous, for each scale k we will need to introduce several radii, but to simplify the discussion here we can think for the moment that \(R_k\) gives the radius of the red circles in Fig. 7, and that the radius of the yellow circles at scale k is just a constant times \(R_k\).) The larger the cluster of seeds of \(\eta ^2\), the larger k must be. We will treat the scales in order, starting from scale 1. This procedure is illustrated in Fig. 8 for the encapsulation of the configuration in Fig. 7a. Once all clusters of scale \(k-1\) or below have been treated, we look at all remaining (untreated) clusters that are not too big to be encapsulated at scale k. If two clusters of scale k are too close to each other, so that their corresponding red circles intersect, we will not carry out the encapsulation and will treat these clusters as if they were one cluster from a larger scale, as illustrated in Fig. 8a. After disregarding these, all remaining clusters of scale k are disjoint and can be treated independently using the more refined Proposition 4.2. The \(\Pi _\iota \) will be the clusters of scale smaller than k that happen to fall inside the red circle of the cluster of scale k. Although small and going very fast to zero with k, the probability that the encapsulation procedure fails is still positive. So it will happen that some encapsulation will fail, as illustrated by the vertex at the top of Fig. 8a. If this happens for some cluster of scale k, which is an event measurable with respect to the passage times inside a red circle of scale k containing the \(\eta ^2\) seeds of that cluster, we then take the whole area inside this red circle and consider it as a larger cluster of \(\eta ^2(0)\) seeds, leaving it to be treated at a larger scale, as in Fig. 8b. Then we turn to the next scale, as in Fig. 8c, d.
In order to handle the slow down of \(\eta ^1\) due to detours imposed by smaller scales, and the sped up of \(\eta ^2\) due to absorption of smaller scales, we will introduce a decreasing sequence of positive numbers \(\epsilon _1,\epsilon _2,\ldots \), as follows. In the encapsulation of a cluster C of scale k, we will show not only that \(\eta ^1\) is able to encapsulate C, but also that \(\eta ^1\) does that sufficiently fast. We do this by coupling the spread of \(\eta ^1\) inside the red circle of C with a slower first passage percolation process of rate \(\prod _{i=1}^ke^{-\epsilon _i}\) that evolves independently of \(\eta ^2\). In other words, this slower first passage percolation process does not need to do a detour around C, but pay the price by having slower passage times. We show that the spread of \(\eta ^1\) around C is faster than that of this slower first passage percolation process. Similarly, we show that, even after absorbing smaller scales, \(\eta ^2\) still spreads slow enough inside the red circle of C, so that we can couple it with a faster first passage percolation process of rate \(\lambda \prod _{i=1}^ke^{\epsilon _i}\), which evolves independently of everything else. We show using this coupling that the spread of \(\eta ^2\) is slower than that of the faster first passage percolation process. Thus at scale k, \(\eta ^1\) is spreading at rate at least \(\prod _{i=1}^ke^{-\epsilon _i}\) while \(\eta ^2\) is spreading at rate at most \(\lambda \prod _{i=1}^ke^{\epsilon _i}\), regardless of what happened at smaller scales. By adequately setting \(\epsilon _k\), we can ensure that \(\prod _{i=1}^ke^{-\epsilon _k}> \lambda \prod _{i=1}^ke^{\epsilon _k}\) for all k, allowing us to apply Proposition 4.2 at all scales.
The final ingredient is to develop a systematic way to argue that \(\eta ^1\) produces an infinite cluster. For this we introduce two types of regions, which we call contagious and infected. We start at scale 1, where all vertices of \(\eta ^2(0)\) are contagious. Using the configuration in Fig. 7a as an example, all white balls there are contagious. The contagious vertices that do not belong to large clusters or are not close to other contagious vertices, are treated at scale 1. The other contagious vertices remain contagious for scale 2. Then, for each cluster treated at scale 1, either the encapsulation procedure is successful or not. If it is successful, then the yellow balls produced by the encapsulation of these clusters are declared infected, and the vertices in these clusters are removed from the set of contagious vertices. In Fig. 8b, the yellow area represents the infected vertices after clusters of scale 1 have been treated. Recall that when an encapsulation is successful, all vertices reached by \(\eta ^2\) from that cluster must be contained inside the yellow area. On the other hand, if the encapsulation is not successful, then all vertices inside the red circle become contagious and go to scale 2, together with the other preselected vertices. An example of this situation is given by the cluster at the top-right corner of Fig. 8b. We carry out this procedure iteratively until there are no more contagious vertices or the origin has been disconnected from infinity by infected vertices. The proof is concluded by showing that \(\eta ^2\) is confined to the set of infected vertices, and that with positive probability the infected vertices will not disconnect the origin from infinity.
Roadmap of the proof We now proceed to the details of the proof. We split the proof in few sections. In Sect. 5.2, we state Theorem 5.1, the more general version of Theorem 1.3. Then in Sect. 5.3 we set up the multi-scale analysis, specifying the sizes of the scales and some parameters. This will define boxes of multiple scales, and we will classify boxes as being either good or bad. Roughly speaking, a box will be good if the encapsulation procedure inside the box is successful. The concrete definition of good boxes is done in Sect. 5.4. In Sect. 5.5 we estimate the probability that a box is good, independent of what happens outside the box. We then introduce contagious and infected sets in Sect. 5.6, and show that \(\eta ^2\) is confined to the set of infected vertices. At this point, it remains to show that the set of infected vertices does not disconnect the origin from infinity. For this, we need to control the set of contagious vertices, which can actually grow as we move to larger scales (for example, this happens when some encapsulation procedure fails). The event that a vertex is contagious at some scale k depends on what happens at previous scale. We estimate the probability of such event by establishing a recursion over scales, which we carry out in Sect. 5.7. With this we have a way to control whether a vertex is infected. In order to show that the origin is not disconnected from infinity by infected vertices, we apply the first moment method. We sum, over all contours around the origin, the probability that this contour contains only infected vertices. Since infected vertices can arise at any scale, we need to look at multi-scale paths and contours of infected vertices, which we do in Sect. 5.8. We then put all ingredients together and complete the proof of Theorem 1.3 in Sect. 5.9.
General version of Theorem 1.3
In this section we will consider a generalization of FPPHE, where the passage times of \(\eta ^2\) can be given by any distribution, while the passage times of \(\eta ^1\) are exponential random variables of rate 1.
Let \(\upsilon \) be a probability distribution on \((0,\infty )\), with no atoms, and such it has a finite exponential moment. It holds by [1, Theorem 2.16] that a first passage percolation with passage times given by i.i.d. random variables with distribution \(\upsilon \) has a limit shape \(\mathcal {B}_\upsilon \), as in (4). Recall that \(\mathcal {B}\left( r\right) =r \mathcal {B}\) denotes the ball of radius r according to the norm induced by the shape theorem of first passage percolation with passage times that are exponential random variables of rate 1.
For any edge (x, y) of the lattice, let \(\zeta ^1_{x,y}\) be an independent exponential random variable of rate 1, and let \(\zeta ^2_{x,y}\) be an independent random variable distributed according to \(\upsilon \). For \(i\in \{1,2\}\), \(\zeta ^i_{x,y}\) is regarded as the passage time of \(\eta ^i\) through (x, y); that is, when \(\eta ^i\) occupies x, then after time \(\zeta ^i_{x,y}\) we have that \(\eta ^i\) will occupy y provided that y has not been occupied by the other type.
Recall that, for any t, we define \({\bar{\eta }}^1(t)\) as the set of vertices of \(\mathbb {Z}^d\) that are not contained in the infinite component of \(\mathbb {Z}^d{\setminus } \eta ^1(t)\), which comprises \(\eta ^1(t)\) and all vertices of \(\mathbb {Z}^d{\setminus } \eta ^1(t)\) that are separated from infinity by \(\eta ^1(t)\). Theorem 1.3 follows immediately from the theorem below by taking \(\upsilon \) to be the exponential distribution of rate \(\lambda \).
Theorem 5.1
For any \(\lambda <1\), there exists a value \(p_0\in (0,1)\) such that the following holds. For all \(p\in (0,p_0)\) and all \(\upsilon \) satisfying \(\mathcal {B}_\upsilon \subseteq \mathcal {B}\left( \lambda \right) \), there are positive constants \(c_1=c_1(p,d,\upsilon )\) and \(c_2=c_2(p,d,\upsilon )\) so that
$$\begin{aligned} \mathbb {P}\left( {\bar{\eta }}^1(t) \supseteq B(0,c_1t) \text { for all } t\ge 0\right) >c_2. \end{aligned}$$
Multi-scale setup
Let \(\epsilon \in (0,1/2)\) be fixed and small enough so that all inequalities below hold:
$$\begin{aligned} \lambda e^{2x}<\lambda (1+3x) < 1-2x \quad \text {for all } x\in (0,\epsilon ]. \end{aligned}$$
(10)
We can define positive constants \(C_\mathrm {FPP}<C_\mathrm {FPP}'\), depending only on d, such that for all \(r>0\) we have
$$\begin{aligned}{}[-C_\mathrm {FPP}r,C_\mathrm {FPP}r]^d \subset \mathcal {B}\left( r\right) \subset [-C_\mathrm {FPP}' r,C_\mathrm {FPP}' r]^d. \end{aligned}$$
(11)
Set \(C_\mathrm {FPP}\) to be the largest constant and \(C_\mathrm {FPP}'\) to be the smallest constant satisfying (11). Since \(\mathcal {B}\left( r\right) \) is convex and has all the symmetries of the lattice \(\mathbb {Z}^d\), we not only obtain that \(\mathcal {B}\left( r\right) \) is contained in the \(\ell _\infty \)-ball of radius \(C'_\mathrm {FPP}r\) but it contains the \(\ell _1\)-ball of radius \(C'_\mathrm {FPP}r\). Using that the latter contains the \(\ell _\infty \)-ball of radius \(\frac{C'_\mathrm {FPP}r}{d}\), we obtain that
$$\begin{aligned} \frac{C'_\mathrm {FPP}}{C_\mathrm {FPP}} \le d. \end{aligned}$$
(12)
Given \(\upsilon \), we can define \(\Delta _\upsilon \ge 1\) as the smallest number such that
$$\begin{aligned} \mathcal {B}_\upsilon \Delta _\upsilon \supseteq \mathcal {B}\left( \lambda \right) . \end{aligned}$$
(13)
Equivalently, we have \(\Delta _\upsilon = \sup _{x \in \mathcal {B}\left( \lambda \right) } |x|_\upsilon \). If \(\upsilon \) is an exponential distribution of rate \(\lambda \), we have \(\Delta _\upsilon =1\).
Let \(L_1\) be a large number, and fix \(\alpha >1\) so that it satisfies the conditions in Proposition 4.2. We let k be an index for the scales. For \(k\ge 1\), once \(L_k\) has been defined, we set
$$\begin{aligned} R_k = \inf \left\{ r \ge L_k :\mathcal {B}\left( r\right) \supset \left[ -10d^2 \tfrac{C_\mathrm {FPP}'}{C_\mathrm {FPP}} L_k,10d^2 \tfrac{C_\mathrm {FPP}'}{C_\mathrm {FPP}} L_k\right] ^d\right\} . \end{aligned}$$
(14)
Also, for \(k\ge 1\), define
$$\begin{aligned} R_k^\mathrm {enc}= 2\alpha R_k \exp \left( \frac{1+c_1}{2\epsilon }\right) \quad \text {and}\quad R_k^{\mathrm {outer}}= \frac{72k^2\Delta _\upsilon }{\epsilon }R_k^\mathrm {enc}, \end{aligned}$$
(15)
where \(c_1\) is the constant in Proposition 4.2. Since \(1-\lambda > 2\epsilon \), we have that \(\mathcal {B}\left( R_k^\mathrm {enc}\right) \) contains all the passage times according to which the event in Proposition 4.2 with \(r=R_k\) is measurable. For \(k\ge 2\), let
$$\begin{aligned} L_k = \inf \left\{ \ell \ge 12 C_\mathrm {FPP}'R_{k-1}^{\mathrm {outer}}: [-\ell /2,\ell /2]^d\supseteq \mathcal {B}\left( 100k^d R_{k-1}^{\mathrm {outer}}\right) \right\} . \end{aligned}$$
(16)
We then obtain the following bounds for \(L_k\):
$$\begin{aligned} 200 C_\mathrm {FPP}k^d R_{k-1}^{\mathrm {outer}}\le L_k \le 200 C_\mathrm {FPP}' k^d R_{k-1}^{\mathrm {outer}} \end{aligned}$$
(17)
and
$$\begin{aligned} \frac{C^2_\mathrm {FPP}R_k}{10d^2 C'_\mathrm {FPP}} \le L_k \le \frac{C_\mathrm {FPP}R_k}{10d^2}. \end{aligned}$$
(18)
The first bound follows from (16) and (11), and the fact that in (16) \(L_k\) is obtained via an infimum, so any cube containing \(\mathcal {B}\left( 100 k^d R_{k-1}^{\mathrm {outer}}\right) \) must have side length at least \(L_k\). The second bound follows from similar considerations, but applying (14) and (11).
The intuition is that \(L_k\) is the size of scale k, and \(R_k\) is the radius of the clusters of \(\eta ^2(0)\) to be treated at scale k. The value of \(R_k^\mathrm {enc}\) gives the radius inside which the encapsulation takes place; in the overview in Sect. 5.1 and in Figs. 7 and 8, \(R_k^\mathrm {enc}\) will be larger than the radius of each yellow ball so that each \(\eta ^2\) cluster treated at scale k will be contained inside a ball of radius \(R_k^\mathrm {enc}\). Regarding \(R_k^{\mathrm {outer}}\), it represents a larger radius, which will be needed for the development of some couplings between scales; in the overview in Sect. 5.1 and in Figs. 7 and 8, \(R_k^{\mathrm {outer}}\) gives the radius of the red circles.
With the definitions above we obtain
$$\begin{aligned} R_k^{\mathrm {outer}}= & {} \frac{144\alpha \exp \left( \frac{1+c_1}{2\epsilon }\right) }{\epsilon }\Delta _\upsilon k^2R_k \le \frac{1440 C_\mathrm {FPP}'\,d^2\,\alpha \exp \left( \frac{1+c_1}{2\epsilon }\right) }{\epsilon C_\mathrm {FPP}^2}\Delta _\upsilon k^2 L_k \nonumber \\\le & {} ck^{d+2} R_{k-1}^{\mathrm {outer}}, \end{aligned}$$
(19)
for some constant \(c=c(d,\epsilon ,\alpha ,\upsilon )>\frac{288000 d^2 \alpha }{\epsilon }\exp \left( \frac{1+c_1}{2\epsilon }\right) \Delta _\upsilon \). Iterating the above bound, we obtain
$$\begin{aligned} R_k^{\mathrm {outer}}\le & {} c^{k-1} \left( k!\right) ^{d+2} R_{1}^{\mathrm {outer}}\nonumber \\\le & {} \left( \frac{1440 C_\mathrm {FPP}'\,d^2\,\alpha \exp \left( \frac{1+c_1}{2\epsilon }\right) \Delta _\upsilon }{\epsilon C_\mathrm {FPP}^2}\right) c^{k-1} \left( k!\right) ^{d+2} L_{1}. \end{aligned}$$
(20)
Using similar reasons we can see that
$$\begin{aligned} R_k \ge \frac{10d^2 L_k}{C_\mathrm {FPP}} \ge 2000 d^2 k^d R_{k-1}^{\mathrm {outer}}= \frac{288000 d^2 \alpha k^{d+2}\exp \left( \frac{1+c_1}{2\epsilon }\right) \Delta _\upsilon }{\epsilon } R_{k-1}, \nonumber \\ \end{aligned}$$
(21)
which allows us to conclude that
$$\begin{aligned} R_k \ge {\tilde{c}}^{k-1} (k!)^{d+2} R_1 \ge \left( 100\frac{k^2}{\epsilon }\right) ^{3d+6} \quad \text { for all } k\ge 1, \end{aligned}$$
where \({\tilde{c}}\) is a positive constant depending on \(\alpha ,\epsilon ,d\) and \(\upsilon \), and the last step follows for all \(k\ge 1\) by setting \(L_1\) large enough.
At each scale \(k\ge 1\), tessellate \(\mathbb {Z}^d\) into cubes of side-length \(L_k\), producing a collection of disjoint cubes
$$\begin{aligned} \{Q_k^\mathrm {core}(i)\}_{i\in \mathbb {Z}^d}, \text { where } Q_k^\mathrm {core}(i) = L_ki + [-L_k/2,L_k/2)^d. \end{aligned}$$
(22)
Whenever we refer to a cube in \(\mathbb {Z}^d\), we will only consider cubes of the form \(\prod _{i=1}^d[a_i,b_i]\) for reals \(a_i<b_i\), \(i\in \{1,2,\ldots ,d\}\). We will need cubes at each scale to overlap. We then define the following collection of cubes
$$\begin{aligned} \{Q_k(i)\}_{i\in \mathbb {Z}^d}, \text { where } Q_k(i) = L_ki + [-10dL_k,10dL_k]^d. \end{aligned}$$
We refer to each such cube \(Q_k(i)\) of scale k as a k-box, and note that \(Q_k(i)\supset Q_k^\mathrm {core}(i)\). One important property is that
$$\begin{aligned}&\text {if a subset } A\subset \mathbb {Z}^d\text { is completely contained inside a cube of side length } 18dL_k,\\&\quad \text {then }\exists i\in \mathbb {Z}^d\text { such that }A\subset Q_k(i). \end{aligned}$$
As described in the proof overview (see Sect. 5.1), when going from scale k to scale \(k+1\), we will need to consider a slowed down version of \(\eta ^1\) and a sped up version of \(\eta ^2\). For this reason we set \(\epsilon _1=0\) and define for \(k\ge 2\)
$$\begin{aligned} \epsilon _k = \frac{\epsilon }{k^2}. \end{aligned}$$
Set \(\lambda ^1_1 = 1\) and \(\lambda _1^2=\lambda \), and let \(\zeta ^1_1=\zeta ^1\) and \(\zeta ^2_1=\zeta ^2\) be the passage times used by \(\eta ^1\) and \(\eta ^2\), respectively. For \(k\ge 2\), define
$$\begin{aligned} \lambda ^1_k = \exp \left( -\sum \nolimits _{i=2}^k \epsilon _i\right) \quad \text {and}\quad \lambda ^2_k = \lambda \,\exp \left( \sum \nolimits _{i=2}^k \epsilon _i\right) . \end{aligned}$$
We have that \(\lambda ^1_k > \lambda ^1_{k+1}\) and \(\lambda ^2_k < \lambda ^2_{k+1}\) for all \(k\ge 1\). Also, note that
$$\begin{aligned} \sum _{k=2}^{\infty } \epsilon _k < \epsilon \int _1^\infty x^{-2}\,dx = \epsilon , \end{aligned}$$
which gives
$$\begin{aligned} \lambda ^1_k> e^{-\epsilon }> 1-\epsilon> \lambda e^\epsilon > \lambda ^2_k \quad \text {for all } k\ge 1, \end{aligned}$$
where the third inequality follows from the bound on \(\epsilon \) via (10).
For each \(k\ge 2\), consider two collections of passage times \(\zeta _k^1\) and \(\zeta _k^2\) on the edges of \(\mathbb {Z}^d\), which are given by \(\frac{\zeta ^1}{\lambda ^1_k}\) and \(\frac{\zeta ^2\lambda }{\lambda ^2_k}\), respectively. These will be the passage times we will use in the analysis at scale k. Note that, for any given k, the passage times of \(\zeta _k^1\) are independent exponential random variables of parameter \(\lambda _k^1\), while for the passage times of \(\zeta _k^2\) we obtain that its limit shape is contained in \(\mathcal {B}\left( \lambda _k^2\right) \).
Moreover, up to a time scaling, having passage times \(\zeta _k^1,\zeta _k^2\) is equivalent to having type 1 spreading at rate 1, while type 2 spreads according to a random variable whose limit shape is contained in \(\mathcal {B}\left( \frac{\lambda _k^2}{\lambda _k^1}\right) \). Therefore, let \(\lambda _k^{\mathrm {eff}}=\frac{\lambda _k^2}{\lambda _k^1}\) be the effective rate of type 2 in comparison with that of type 1 at scale k. From now on, we will refer to the \(\lambda ^2_k\) as the rate of spread of type 2 at scale k, even if type 2 may not have exponential passage times.
We obtain that
$$\begin{aligned} \lambda< \lambda _k^{\mathrm {eff}}\le \frac{\lambda e^\epsilon }{e^{-\epsilon }} = \lambda e^{2\epsilon } < 1-2\epsilon . \end{aligned}$$
(23)
Thus the effective rate of spread of type 2 is smaller than 1 at all scales. We can also define the effective passage time of type 2 at scale k as
$$\begin{aligned} \zeta _k^{\mathrm {eff}}= \frac{\zeta ^2 \lambda \lambda _k^1}{\lambda _k^2}=\frac{\zeta ^2 \lambda }{\lambda _k^{\mathrm {eff}}}; \end{aligned}$$
(24)
in this way, at scale k, when employing Proposition 4.2, we will take the passage times \(\zeta ^1_k,\zeta ^2_k\) and scale time by a factor of \(\lambda _k^1\), so that type 1 spreads according to the passage times \(\zeta ^1\) and type 2 spreads according to the passage times \(\zeta _k^{\mathrm {eff}}\). Finally, for \(k\ge 1\), define
$$\begin{aligned} T_k^1 = R_k^\mathrm {enc}\left( \frac{11-\lambda }{10}\right) ^2\frac{1}{\lambda _k^1} \ge R_k^\mathrm {enc}\left( \frac{11-\lambda _k^{\mathrm {eff}}}{10}\right) ^2\frac{1}{\lambda _k^1}. \end{aligned}$$
(25)
Note that, using the passage times \(\zeta ^1_k,\zeta ^2_k\), we have that \(T_k^1\) represents the time required to run each encapsulation procedure at scale k (before time is scaled by a factor of \(\lambda _k^1\) as mentioned above).
Definition of good boxes
For each \(Q_k(i)\), we will apply Proposition 4.2 to handle the situation where \(Q_k(i)\) entirely contains a cluster of \(\eta ^2(0)\). At scale k we will only handle the clusters that have not already been handled at a scale smaller than k. By the relation between \(L_k\) and \(R_k\) in (18), the cluster of \(\eta ^2\) inside \(Q_k(i)\) will not start growing before \(\eta ^1\) reaches the boundary of \(L_ki+\mathcal {B}\left( R_k\right) \). By the time \(\eta ^1\) reaches the boundary of \(L_ki+\mathcal {B}\left( R_k\right) \), \(\eta ^1\) must have crossed the boundary of \(L_ki+\mathcal {B}\left( \alpha R_k\right) \). (For the moment we assume that \(L_ki+\mathcal {B}\left( \alpha R_k\right) \) does not contain the origin, otherwise we will later consider that the origin has already been disconnected from infinity by \(\eta ^2\).) At this point we apply Proposition 4.2 with \(r=R_k\) and \(\lambda =\lambda _k^{\mathrm {eff}}\), obtaining values R and T such that
$$\begin{aligned} R\le \alpha R_k \exp \left( \frac{c_1}{1-\lambda _k^{\mathrm {eff}}}\right) \le \alpha R_k \exp \left( \frac{c_1}{2\epsilon }\right) \le R_k^\mathrm {enc}, \end{aligned}$$
(26)
where the second inequality follows from (15) and the last inequality follows from (23), and
$$\begin{aligned} T\le R \left( \frac{11-\lambda _k^{\mathrm {eff}}}{10}\right) ^2 \le R_k^\mathrm {enc}\left( \frac{11-\lambda _k^{\mathrm {eff}}}{10}\right) ^2 \le \lambda _k^1 T_k^1, \end{aligned}$$
(27)
where the last two inequalities follow from (26) and (25), respectively. Note that in our application of Proposition 4.2 above time has been scaled by \(\lambda _k^1\), since we apply it with type 1 (resp., type 2) spreading at rate 1 (resp., \(\lambda _k^{\mathrm {eff}}\)) instead of the actual rate \(\lambda _k^1\) (resp., \(\lambda _k^2\)). This is the reason why the term \(\frac{1}{\lambda _k^1}\) appears in the definition of \(T_k^1\) in (25). With this we get \(\lambda _{k}^1 T_k^1\) in the right-hand side of (27), and the actual time that the encapsulation procedure takes is \(\frac{1}{\lambda _k^1}T\le T_k^1\). At the moment we have not yet defined the sets \(\{\Pi _\iota \}_\iota ,\{\Pi '_\iota \}_\iota \); they will only be defined precisely in Sect. 5.6.
Now let \(E_k(i,x)\) with \(x\in L_ki+\mathcal {B}\left( \alpha R_k\right) {\setminus } \mathcal {B}\left( \alpha R_k/2\right) \), be the event in the application of Proposition 4.2 with the origin at \(L_ki\), \(r=R_k\), passage times given by \(\zeta ^1,\zeta ^{\mathrm {eff}}_k\), and \(\eta ^1\) starting from x. Here x represents the first vertex of \(\partial ^\mathrm {o}\left( L_ki+ \mathcal {B}\left( \alpha R_k\right) \right) \) occupied by \(\eta ^1\), from where the encapsulation of the cluster of \(\eta ^2\) inside \(L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) \) will start. Letting \(B_k(i) = \left( L_ki+\mathcal {B}\left( \alpha R_k\right) {\setminus } \mathcal {B}\left( \alpha R_k/2\right) \right) \cup \partial ^\mathrm {o}\left( L_ki+\mathcal {B}\left( \alpha R_k\right) \right) \), define
$$\begin{aligned} G_k^\mathrm {enc}(i) \text { to be the event that } E_k(i,x)\text { holds for all } x\in B_k(i). \end{aligned}$$
The event \(G_k^\mathrm {enc}(i)\) implies that \(\eta ^1\) encapsulates \(\eta ^2\) inside \(L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) \) during a time interval of length \(T_k^1\), unless \(\eta ^2\) “invades” \(L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) \) from outside, that is, unless another cluster of \(\eta ^2\) starts growing and reaches the boundary of \(L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) \) before \(\eta ^1\) manages to encapsulate \(\eta ^2\) inside \(L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) \). (When we apply the above argument later in the proof, we will only try to encapsulate a cluster of \(\eta ^2\) at scale k if the ball \(L_ki+\mathcal {B}\left( R_k^{\mathrm {outer}}\right) \supset L_ki+ \mathcal {B}\left( R_k^\mathrm {enc}\right) \) does not intersect other balls being treated at the same scale. If there is another ball being treated at the same scale k and intersecting \(L_ki+\mathcal {B}\left( R_k^{\mathrm {outer}}\right) \), then these balls will be only treated at a larger scale, not allowing different clusters of \(\eta ^2\) of the same scale to interfere in each other’s encapsulation.)
For each \(i\in \mathbb {Z}^d\), define
$$\begin{aligned} Q_k^\alpha (i) = L_ki + \mathcal {B}\left( \alpha R_k\right) , \quad Q_k^\mathrm {enc}(i) = L_ki + \mathcal {B}\left( R_k^\mathrm {enc}\right) ,\\ Q_k^{\mathrm {outer}}(i) = L_ki + \mathcal {B}\left( R_k^{\mathrm {outer}}\right) \quad \text {and}\quad Q_k^{{\mathrm {outer}}/3}(i) = L_ki + \mathcal {B}\left( R_k^{\mathrm {outer}}/3\right) . \end{aligned}$$
We will also define two other events \(G_k^{1}(i)\) and \(G_k^{2}(i)\), which will be measurable with respect to \(\zeta ^1_k,\zeta ^2_k\) inside \(Q_k^{\mathrm {outer}}(i)\). For any \(X\subset \mathbb {Z}^d\), let \(\zeta ^1_{k}|_X\) be the passage times that are equal to \(\zeta ^1_k\) inside X and are equal to infinity everywhere else; define \(\zeta ^2_{k}|_X\) analogously. Define the event \(G_k^{1}(i)\) as
$$\begin{aligned} \begin{aligned} \Big \{&D\big (\partial ^\mathrm {o}Q_k^\mathrm {enc}(i),\partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i); \zeta _{k+1}^1|_{Q_k^{\mathrm {outer}}(i)}\big ) \\&\quad \ge T_k^1 + \sup \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)} D\big (\partial ^\mathrm {o}Q_k^\alpha (i),x; \zeta _{k}^1|_{Q_k^{\mathrm {outer}}(i)}\big )\Big \}. \end{aligned} \end{aligned}$$
The main intuition behind this event is that, during the encapsulation of a \((k+1)\)-box, \(\eta ^1\) will need to perform some small local detours when encapsulating clusters of scale k or smaller. We can capture this by using the slower passage times \(\zeta _{k+1}^1\). If \(G_k^{1}\) holds for the k-boxes that are traversed during the encapsulation of a \((k+1)\)-box, then using the slower passage times \(\zeta _{k+1}^1\) but ignoring the actual detours around k-boxes will only slow down \(\eta ^1\).
We also need to handle the case where the growth of \(\eta ^2\) is sped up by absorption of smaller scales. For \(i\in \mathbb {Z}^d\), define
$$\begin{aligned} G_k^{2}(i) =\bigcap _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)}\Big \{&D\big (x,\partial ^\mathrm {o}Q_k^\mathrm {enc}(i); \zeta _{k}^2|_{Q_k^{\mathrm {outer}}(i)}\big ) \\&\quad \ge \sup \nolimits _{y\in Q_k^\mathrm {enc}(i)} D\big (x,y; \zeta _{k+1}^2|_{Q_k^{\mathrm {outer}}(i)}\big )\Big \}. \end{aligned}$$
Note that the event \(G_k^2(i)\) implies the following. Let \(x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)\) be the first vertex of \(Q_k^{{\mathrm {outer}}/3}(i)\) reached by \(\eta ^2\) from outside \(Q_k^{{\mathrm {outer}}/3}(i)\). While \(\eta ^2\) travels from x to \(Q_k^\mathrm {enc}(i)\), the encapsulation of \(Q_k^\mathrm {enc}(i)\) may start taking place. Then, \(\eta ^2\) can only get a sped up inside \(Q_k^\mathrm {enc}(i)\) if \(\eta ^2\) enters \(Q_k^\mathrm {enc}(i)\) before the encapsulation of \(Q_k^\mathrm {enc}(i)\) is completed. However, under \(G_k^2(i)\) and the passage times \(\zeta _k^2\), the time that \(\eta ^2\) takes to go from x to \(Q_k^\mathrm {enc}(i)\) is larger than the time, under \(\zeta _{k+1}^2\), that \(\eta ^2\) takes to go from x to all vertices in \(Q_k^\mathrm {enc}(i)\). Therefore, under \(G_k^2(i)\), we can use the faster passage times \(\zeta _{k+1}^2\) to absorb the possible sped up that \(\eta ^2\) may get by the cluster growing inside \(Q_k^\mathrm {enc}(i)\).
For \(i\in \mathbb {Z}^d\) and \(k\ge 1\), we define
$$\begin{aligned} G_k(i)= G_k^\mathrm {enc}(i) \cap G_k^{1}(i) \cap G_k^{2}(i), \end{aligned}$$
and say that
$$\begin{aligned} Q_k(i) \text { is }{} \textit{good} \text { if } G_k(i) \text { holds}. \end{aligned}$$
Hence, intuitively, \(Q_k(i)\) being good means that \(\eta ^1\) successfully encapsulates the growing cluster of \(\eta ^2\) inside \(Q_k(i)\), and this happens in such a way that the detour of \(\eta ^1\) during this encapsulation is faster than letting \(\eta ^1\) use passage times \(\zeta _{k+1}^1\), and also the possible sped up that \(\eta ^2\) may get from clusters of \(\eta ^2\) coming from outside \(Q_k(i)\) is slower than letting \(\eta ^2\) use passage times \(\zeta _{k+1}^2\).
We now explain why in the definition of \(G_k^1(i)\) and \(G_k^2(i)\) we calculate passage times from \(\partial Q_k^{{\mathrm {outer}}/3}(i)\) instead of from \(\partial Q_k^{\mathrm {outer}}(i)\). The reason is that we had to define \(G_k^1(i)\) and \(G_k^2(i)\) in such a way that they are measurable with respect to the passage times inside \(Q_k^{\mathrm {outer}}(i)\). We do this by forcing to use only passage times inside \(Q_k^{\mathrm {outer}}(i)\). By using the distance between \(\partial Q_k^{\mathrm {outer}}(i)\) and \(\partial Q_k^{{\mathrm {outer}}/3}(i)\), we can ensure that this constraint does not change much the probability that the corresponding events occur.
Probability of good boxes
In this section we show that the events \(G_k^\mathrm {enc}(i)\), \(G_k^1(i)\) and \(G_k^2(i)\), defined in Sect. 5.4, are likely to occur.
Lemma 5.2
There exist positive constants \(L_0=L_0(d,\epsilon )\) and \(c=c(d,\upsilon )\) such that if \(L_1\ge L_0\), then for any \(k\ge 1\) and any \(i\in \mathbb {Z}^d\) we have
$$\begin{aligned} \mathbb {P}\left( G_k(i)\right) \ge 1 - \exp \left( -c\left( \epsilon \lambda R_k^\mathrm {enc}\right) ^\frac{d+1}{2d+4}\right) . \end{aligned}$$
Moreover, the event \(G_k(i)\) is measurable with respect to the passage times inside \(Q_k^{\mathrm {outer}}(i)\).
Before proving the lemma above, we state and prove two lemmas regarding the probability of the events \(G_k^1(i)\) and \(G_k^2(i)\).
Lemma 5.3
There exist positive constants \(L_0=L_0(d,\epsilon )\) and \(c=c(d)\) such that if \(L_1\ge L_0\), then for any \(k\ge 1\) and any \(i\in \mathbb {Z}^d\) we have
$$\begin{aligned} \mathbb {P}\left( G_k^1(i)\right) \ge 1 - \exp \left( -c \left( R_k^{\mathrm {outer}}\right) ^\frac{d+1}{2d+4}\right) . \end{aligned}$$
Proof
Set \(\delta =\frac{\epsilon }{120 k^{2}}\). Define
$$\begin{aligned} \tau _1 = \left( \frac{24k^2}{\epsilon }-1\right) \frac{1}{\lambda _{k+1}^1}R_k^\mathrm {enc}= \frac{1}{\lambda _{k+1}^1}\left( \frac{1}{3}R_k^{\mathrm {outer}}- R_k^\mathrm {enc}\right) \end{aligned}$$
and
$$\begin{aligned} \tau _2 = \frac{24k^2}{\epsilon \lambda _k^1}R_k^\mathrm {enc}= \frac{1}{3\lambda _k^1}R_k^{\mathrm {outer}}. \end{aligned}$$
We will show that there exists a constant \(c=c(d)>0\) such that
$$\begin{aligned}&\mathbb {P}\left( D\big (\partial ^\mathrm {o}Q_k^\mathrm {enc}(i), \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i);\zeta _{k+1}^1|_{Q_k^{\mathrm {outer}}(i)}\big ) \le (1-\delta )\tau _1\right) \nonumber \\&\quad \le (R_k^{\mathrm {outer}})^{3d} \exp \left( -c (R_k^{\mathrm {outer}})^{\frac{d+1}{2d+4}}\right) , \end{aligned}$$
(28)
and
$$\begin{aligned} \mathbb {P}\left( \sup \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)} D\big ( \partial ^\mathrm {o}Q_k^\alpha (i), x;\zeta _{k}^1|_{Q_k^{\mathrm {outer}}(i)}\big ) \ge (1+\delta )\tau _2\right)&\nonumber \\ \le \frac{c}{1-2\epsilon } (R_k^{\mathrm {outer}})^{2d} \exp \left( -c (R_k^{\mathrm {outer}})^{\frac{d+1}{2d+4}}\right) .&\end{aligned}$$
(29)
Using (28) and (29), it remains to show that
$$\begin{aligned} (1-\delta )\tau _1 \ge T_k^1 + (1+\delta )\tau _2. \end{aligned}$$
Note that
$$\begin{aligned} (1-\delta )\tau _1&= (1-\delta )\left( \frac{24k^2}{\epsilon }-1\right) \frac{\exp \left( \epsilon (k+1)^{-2}\right) }{\lambda _k^1}R_k^\mathrm {enc}\\&= T_k^1 + \left( (1-\delta )\left( \frac{24k^2}{\epsilon }-1\right) \exp \left( \epsilon (k+1)^{-2}\right) -\left( \frac{11-\lambda }{10}\right) ^2\right) \frac{R_k^\mathrm {enc}}{\lambda _k^1}. \end{aligned}$$
Thus we need to show that the last term in the right-hand side above is at least \((1+\delta )\tau _2\), which is equivalent to showing that
$$\begin{aligned} \begin{aligned}&\frac{24k^2}{\epsilon }\Bigl ( (1-\delta )\exp \left( \epsilon (k+1)^{-2}\right) -1-\delta \Bigr ) \\ {}&\quad \ge \left( \frac{11-\lambda }{10}\right) ^2 + (1-\delta )\exp \left( \epsilon (k+1)^{-2}\right) . \end{aligned} \end{aligned}$$
Rearranging the terms, the inequality above translates to
$$\begin{aligned}&\frac{24k^2}{\epsilon }\Bigl ( (1-\delta )\exp \left( \epsilon (k+1)^{-2}\right) -1-\delta \Bigr ) \\&\quad \ge \left( \frac{11-\lambda }{10}\right) ^2 + (1-\delta )\exp \left( \epsilon (k+1)^{-2}\right) . \end{aligned}$$
Using that \(\exp \left( \epsilon (k+1)^{-2}\right) \ge 1+\epsilon (k+1)^{-2}\) and then applying the value of \(\delta \), we obtain that the left-hand side above is at least
$$\begin{aligned} \frac{24k^2}{\epsilon }\left( \epsilon (k+1)^{-2}-2\delta -\delta \epsilon (k+1)^{-2}\right)&= 24\left( \frac{k^2}{(k+1)^{2}}-\frac{1}{60}-\frac{\epsilon }{120(k+1)^{2}}\right) \\&\ge 24\left( \frac{1}{4}-\frac{1}{60}-\frac{1}{480}\right) >5. \end{aligned}$$
Hence, it now suffices to show that
$$\begin{aligned} 5 \ge \left( \frac{11-\lambda }{10}\right) ^2 + (1-\delta )\exp \left( \epsilon (k+1)^{-2}\right) , \end{aligned}$$
which is true since the right-hand side above is at most \(\left( \frac{11-\lambda }{10}\right) ^2 + \exp \left( \epsilon /4\right) \le \left( \frac{11}{10}\right) ^2 + \exp \left( 1/4\right) \le 3\).
Now we turn to establish (28) and (29). We start with (28). First note that
$$\begin{aligned} R_k^{\mathrm {outer}}/3-R_k^\mathrm {enc}= \left( \frac{24k^2}{\epsilon }-1\right) R_k^\mathrm {enc}= \tau _1\lambda _{k+1}^1. \end{aligned}$$
Recall the notation \(S_t^\delta \) from Proposition 3.1, which is the (unlikely) event that at time t first passage percolation of rate 1 does not contain \(\mathcal {B}\left( (1-\delta )t\right) \) or is not contained in \(\mathcal {B}\left( (1+\delta )t\right) \). Then using time scaling to go from passage times of rate \(\lambda _{k+1}^1\) to passage times of rate 1, and using the union bound on x, we obtain
$$\begin{aligned}&\mathbb {P}\left( D\big (\partial ^\mathrm {o}Q_k^\mathrm {enc}(i), \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i);\zeta _{k+1}^1|_{Q_k^{\mathrm {outer}}(i)}\big ) \le (1-\delta )\tau _1\right) \\&=\mathbb {P}\left( D\big (\partial ^\mathrm {o}Q_k^\mathrm {enc}(i), \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i);\zeta _{1}^1|_{Q_k^{\mathrm {outer}}(i)}\big ) \le (1-\delta )\tau _1\lambda _{k+1}^1\right) \\&\le \sum \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)}\mathbb {Q}\left( S_{(1-\delta )\tau _1\lambda _{k+1}^1}^\delta \right) \\&\le c_1 \left( (1-\delta )\tau _1\lambda _{k+1}^1\right) ^{3d} \exp \left( -c_2 \left( (1-\delta )\tau _1\lambda _{k+1}^1\right) ^\frac{d+1}{2d+4}\right) \\&\le c_1 (R_k^{\mathrm {outer}})^{3d} \exp \left( -c_3 (R_k^{\mathrm {outer}})^\frac{d+1}{2d+4}\right) , \end{aligned}$$
where in the first inequality we used that \((1+\delta )(1-\delta )\tau _1\lambda _{k+1}^1\le \tau _1\lambda _{k+1}^1 = R_k^{\mathrm {outer}}/3-R_k^\mathrm {enc}\), and in the second inequality we applied Proposition 3.1.
Now we turn to (29). We again use time scaling and the fact that \(\tau _2\lambda _k^1=R^{\mathrm {outer}}_k/3\) to write
$$\begin{aligned}&\mathbb {P}\left( \sup \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)} D\Big ( \partial ^\mathrm {o}Q_k^\alpha (i), x;\zeta _{k}^1|_{Q_k^{\mathrm {outer}}(i)}\Big ) \ge (1+\delta )\tau _2\right) \\&\le \mathbb {P}\left( \sup \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)}D\Big ( L_k i, x;\zeta _{k}^1|_{Q_k^{\mathrm {outer}}(i)}\Big ) \ge (1+\delta )\tau _2\right) \\&\le \sum \nolimits _{x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)}\mathbb {Q}\left( S_{(1+\delta )\tau _2\lambda _k^1}^{\delta /2}\right) \\&\le c_1 \left( R_k^{\mathrm {outer}}\right) ^{3d} \exp \left( -c_2 \left( R_k^{\mathrm {outer}}\right) ^\frac{d+1}{2d+4}\right) , \end{aligned}$$
where the second inequality follows since \((1-\delta /2)(1+\delta )\tau _2\lambda _k^1\ge R_k^{\mathrm {outer}}/3\) for all \(\delta \in [0,1]\). Moreover, \((1+\delta /2)(1+\delta )\tau _2\lambda _k^1< \frac{2 R_k^{\mathrm {outer}}}{3}\), implying that \(S_{(1+\delta )\tau _2\lambda _k^1}^{\delta /2}\) is measurable with respect to the passage times inside \(Q_k^{\mathrm {outer}}(i)\). Finally, the last step of the derivation above follows from Propositon 3.1. \(\square \)
The next lemma shows that \(G_k^2(i)\) occurs with high probability.
Lemma 5.4
There exist positive constants \(L_0=L_0(d,\epsilon )\) and \(c=c(d,\nu )\) such that if \(L_1\ge L_0\), then for any \(k\ge 1\) and any \(i\in \mathbb {Z}^d\) we have
$$\begin{aligned} \mathbb {P}\left( G_k^2(i)\right) \ge 1 - \exp \left( -c \left( R_k^{\mathrm {outer}}\right) ^\frac{d+1}{2d+4}\right) . \end{aligned}$$
Proof
Set \(\delta =\frac{\epsilon }{20 k^{2}}\) and fix an arbitrary \(x\in \partial ^\mathrm {i}Q_k^{{\mathrm {outer}}/3}(i)\). Define the smallest distance between x and \(Q_k^\mathrm {enc}(i)\) with respect to the norm \(\upsilon \) as
$$\begin{aligned} m = \min _{y \in \partial ^\mathrm {i}Q_k^\mathrm {enc}(i)} |x-y|_\upsilon . \end{aligned}$$
Since \(\mathcal {B}_\upsilon \subseteq \mathcal {B}\left( \lambda \right) \), we have that
$$\begin{aligned} m \ge \frac{1}{\lambda } \left( \frac{R_k^{\mathrm {outer}}}{3}- R_k^\mathrm {enc}\right) . \end{aligned}$$
Under the passage times \(\zeta _k^2\), the time it takes to reach \(Q_k^\mathrm {enc}(i)\) from x is roughly \(m \frac{\lambda }{\lambda _k^2}\). Therefore, we define
$$\begin{aligned} \tau _1 = \frac{m\lambda }{\lambda _k^2}, \end{aligned}$$
and will show later that there exists a constant \(c'>0\) such that, uniformly over x,
$$\begin{aligned}&\mathbb {P}\left( D\big ( x, \partial ^\mathrm {o}Q_k^\mathrm {enc}(i);\zeta _{k}^2|_{Q_k^{\mathrm {outer}}(i)}\big ) \le (1-\delta )\tau _1\right) \le \exp \left( -c' (R_k^{\mathrm {outer}})^{\frac{d+1}{2d+4}}\right) . \end{aligned}$$
(30)
Now, under the faster passage times \(\zeta _{k+1}^2\), the time it takes to reach \(Q_k^\mathrm {enc}(i)\) from x is roughly \(m \frac{\lambda }{\lambda _{k+1}^2}\). Let \(x'\in \partial ^\mathrm {i}Q_k^\mathrm {enc}(i)\) be the first vertex of \(Q_k^\mathrm {enc}(i)\) reached from x. Note that
$$\begin{aligned} x'+\mathcal {B}_\upsilon \Delta _\upsilon \frac{2 R_k^\mathrm {enc}}{\lambda } \supseteq x'+\mathcal {B}\left( \lambda \right) \frac{2 R_k^\mathrm {enc}}{\lambda } = x'+\mathcal {B}\left( 2 R_k^\mathrm {enc}\right) \supseteq Q_k^\mathrm {enc}(i). \end{aligned}$$
Under the passage times \(\zeta _{k+1}^2\), which is a scale of \(\zeta ^2\) by a factor of \(\frac{\lambda }{\lambda _{k+1}^2}\), the time until \(x'+\mathcal {B}_\upsilon \Delta _\upsilon \frac{2 R_k^\mathrm {enc}}{\lambda }\) is fully occuppied starting from y is roughly \(\Delta _\upsilon \frac{2 R_k^\mathrm {enc}}{\lambda _{k+1}^2}\). Therefore, we set
$$\begin{aligned} \tau _2 = m \frac{\lambda }{\lambda _{k+1}^2} + \Delta _\upsilon \frac{2 R_k^\mathrm {enc}}{\lambda _{k+1}^2}, \end{aligned}$$
and will show that there exists a constant \(c''>0\) such that
$$\begin{aligned}&\mathbb {P}\left( \sup \nolimits _{y\in \partial ^\mathrm {o}Q_k^\mathrm {enc}(i)} D\Big ( x, y;\zeta _{k+1}^2|_{Q_k^{\mathrm {outer}}(i)}\Big ) \ge (1+\delta )\tau _2\right) \nonumber \\&\quad \le \exp \left( -c'' (R_k^{\mathrm {outer}})^{\frac{d+1}{2d+4}}\right) . \end{aligned}$$
(31)
Assuming (30) and (31) for the moment, it remains to show that
$$\begin{aligned} (1-\delta )\tau _1 \ge (1+\delta )\tau _2. \end{aligned}$$
(32)
Replacing \(\lambda _{k+1}^2\) with \(\lambda _k^2 \exp (\epsilon (k+1)^{-2})\) in the definition of \(\tau _2\), (32) follows if we show that
$$\begin{aligned} (1-\delta )m \lambda \ge (1+\delta )\exp \left( -\epsilon (k+1)^{-2}\right) \left( m\lambda + 2 \Delta _\upsilon R_k^\mathrm {enc}\right) . \end{aligned}$$
First note that
$$\begin{aligned} 2\Delta _\upsilon R_k^\mathrm {enc}= \frac{2\epsilon }{24 k^2} \cdot \frac{R_k^{\mathrm {outer}}}{3} \le \frac{3\epsilon }{24 k^2} \left( \frac{R_k^{\mathrm {outer}}}{3} - R_k^\mathrm {enc}\right) \le \frac{3\epsilon }{24 k^2} m\lambda = \frac{5\delta }{2} m\lambda , \end{aligned}$$
where the first inequality follows by the definition of \(R_k^{\mathrm {outer}}\) in (15). So now it suffices to show that
$$\begin{aligned} 1-\delta \ge (1+\delta )\exp \left( -\epsilon (k+1)^{-2}\right) \left( 1+ \frac{5\delta }{2}\right) . \end{aligned}$$
Rearranging gives that \(\frac{1-\delta }{(1+\delta )(1+5\delta /2)}\ge \exp \left( -\epsilon (k+1)^{-2}\right) \). The left-hand side is at least \((1-\delta )^2(1-5\delta /2)\ge 1- \frac{9\delta }{2}\). Using that \(e^{-a}\le 1-a+a^2/2\) for all \(a\ge 0\), (32) holds if the following is true
$$\begin{aligned} \frac{9\delta }{2} \le \frac{\epsilon }{(k+1)^2}\left( 1 - \frac{\epsilon }{2(k+1)^2}\right) . \end{aligned}$$
Using the value of \(\delta \), we are left to showing
$$\begin{aligned} \frac{9}{40} \le \frac{k^2}{(k+1)^2}\left( 1 - \frac{\epsilon }{2(k+1)^2}\right) , \end{aligned}$$
which is true since the right-hand side is at least \(\frac{1}{4} \cdot \left( 1-\frac{\epsilon }{8}\right) \ge \frac{1}{4} \cdot \frac{15}{16}\). This establishes (32).
Now we turn to establish (30) and (31), which essentially follow from Proposition 3.1. We start with (30). Scaling the passage times \(\zeta _{k^2}\) by \(\frac{\lambda _k^2}{\lambda }\) we obtain passage times distributed as \(\upsilon \). Hence,
$$\begin{aligned} \mathbb {P}\left( D\Big ( x, \partial ^\mathrm {o}Q_k^\mathrm {enc}(i);\zeta _{k}^2|_{Q_k^{\mathrm {outer}}(i)}\Big ) \le (1-\delta )\tau _1\right)&\le \mathbb {Q}^{\upsilon }\left( S_{\tau _1\lambda _k^2/\lambda }^\delta \right) \\&\le \exp \left( -c' (R_k^{\mathrm {outer}})^{\frac{d+1}{2d+4}}\right) . \end{aligned}$$
The same reasoning holds for (31), which gives
$$\begin{aligned} \mathbb {P}\left( \sup \nolimits _{y\in \partial ^\mathrm {o}Q_k^\mathrm {enc}(i)} D\Big ( x, y;\zeta _{k+1}^2|_{Q_k^{\mathrm {outer}}(i)}\Big ) \ge (1+\delta )\tau _2\right)&\le \mathbb {Q}^{\upsilon }\left( S_{\tau _2\lambda _{k+1}^2/\lambda }^{\delta }\right) \\&\le \exp \left( -c'' (R_k^{\mathrm {outer}})^\frac{d+1}{2d+4}\right) . \end{aligned}$$
Then the lemma follows by taking the union bound over x, and using the fact that \(R_k^{\mathrm {outer}}\) is very large at all scales so that the extra term obtained from the union bound can be absorbed in the constant c. \(\square \)
Proof of Lemma 5.2
Proposition 4.2 gives that \(G_k^\mathrm {enc}(i)\) can be defined so that it is measurable with respect to the passage times inside
$$\begin{aligned} L_ki+\mathcal {B}\left( \alpha R_k \exp \left( \frac{c_1}{1-\lambda _k^{\mathrm {eff}}}\right) \left( \frac{11-\lambda _k^{\mathrm {eff}}}{10}\right) ^2\right)&\subseteq L_ki+\mathcal {B}\left( \alpha R_k \exp \left( \frac{c_1}{2\epsilon }\right) \left( \frac{11}{10}\right) ^2\right) \\&\subseteq L_ki+\mathcal {B}\left( R_k^\mathrm {enc}\right) . \end{aligned}$$
Moreover, Proposition 4.1 gives a constant \(c_2>0\) so that, for all large enough \(L_1\), we have
$$\begin{aligned} \mathbb {P}\left( G_k^\mathrm {enc}(i)\right)&\ge 1- \sum _{x\in B_k(i)} \exp \left( -c_2 \left( \lambda _k^{\mathrm {eff}}\left( 1-\lambda _k^{\mathrm {eff}}\right) \alpha R_k\right) ^\frac{d+1}{2d+4}\right) \\&\ge 1- \exp \left( -c \left( 2\epsilon \lambda \alpha R_k\right) ^\frac{d+1}{2d+4}\right) , \end{aligned}$$
where the last step follows by applying the bounds in (23) and c is a positive constant. By definition, the events \(G_k^1\) and \(G_k^2(i)\) are measurable with respect to the passage times inside \(L_ki+\mathcal {B}\left( R_k^{\mathrm {outer}}\right) \). So the proof is completed by using the bounds in Lemmas 5.3 and 5.4. \(\square \)
Contagious and infected sets
As discussed in the proof overview in Sect. 5.1, for each scale k, we will define a set \(C_k\subset \mathbb {Z}^d\) as the set of contagious vertices at scale k, and also define a set \(I_k\subset \mathbb {Z}^d\) as the set of infected vertices at scale k. The main intuition behind such sets is that \(C_k\) represents the vertices of \(\mathbb {Z}^d\) that need to be handled at scale k or larger, whereas \(I_k\) represents the vertices of \(\mathbb {Z}^d\) that may be taken by \(\eta ^2\) at scale k. In particular, we will show that the vertices of \(\mathbb {Z}^d\) that will be occupied by \(\eta ^2\) are contained in \(\bigcup _{k\ge 1}I_k\).
At scale 1 we set the contagious vertices as those initially taken by \(\eta ^2\); that is,
$$\begin{aligned} C_1 = \eta ^2(0). \end{aligned}$$
All clusters of \(C_1\) that belong to good 1-boxes and that are not too close to contagious clusters from other 1-boxes will be “cured” by the encapsulation process described in the previous section. The other vertices of \(C_1\) will become contagious vertices for scale 2, together with the vertices belonging to bad 1-boxes. Using this, define \(C_k^{\mathrm {bad}}\) as the following subset of the contagious vertices:
$$\begin{aligned} \left\{ x \in C_{k} :\text {for all } i \text { with } x\in Q_k(i) \text { we have } iL_k + \mathcal {B}\left( 3R_k^{\mathrm {outer}}\right) \cap \left( C_k {\setminus } Q_k(i)\right) \ne \emptyset \right\} . \nonumber \\ \end{aligned}$$
(33)
Intuitively, \(C_k^{\mathrm {bad}}\) is the set of contagious vertices that cannot be cured at scale k since they are not far enough from other contagious vertices in other k-boxes. Now for the vertices in \(C_k{\setminus } C_k^{\mathrm {bad}}\), the definition of \(C_k^{\mathrm {bad}}\) gives that we can select a set \(\mathcal {I}_k\subset \mathbb {Z}^d\) representing k-boxes such that for each \(x\in C_k{\setminus } C_k^{\mathrm {bad}}\) there exists a unique \(i\in \mathcal {I}_k\) for which \(x\in Q_k(i)\), and for each pair \(i,j\in \mathcal {I}_k\), we have \(Q_k^{\mathrm {outer}}(i)\cap Q_k^{\mathrm {outer}}(j)=\emptyset \). Then, given \(C_k\), we define \(I_k\) as the set of vertices that can be taken by \(\eta ^2\) during the encapsulation of the good k-box, which is more precisely given by
$$\begin{aligned} I_k = \left\{ Q_k^\mathrm {enc}(i) :Q_k(i) \text { is good and } i \in \mathcal {I}_k\right\} . \end{aligned}$$
We then define inductively
$$\begin{aligned} C_{k+1} = C_{k}^{\mathrm {bad}}\cup \left\{ Q_{k}^{\mathrm {outer}}(i) :i \in \mathcal {I}_k \text { and } Q_{k}(i) \text { is bad}\right\} . \end{aligned}$$
(34)
The lemma below gives that if the contagious sets of scales larger than k are all empty, then \(\eta ^2\) must be contained inside \(\bigcup _{j=1}^{k-1}I_j\).
Lemma 5.5
Let \(A\subset \mathbb {Z}^d\) be arbitrary. Then, for any \(k\ge 1\), either we have that
$$\begin{aligned}&\text {there exists } j> k \text { and } i\in \mathbb {Z}^d \text { with } Q_j^{\mathrm {outer}}(i)\cap A\ne \emptyset \text { for which } \nonumber \\&\quad Q_j^\mathrm {core}(i)\cap C_j \ne \emptyset , \end{aligned}$$
(35)
or
$$\begin{aligned} \eta ^2(t)\cap A \subset \bigcup \nolimits _{j=1}^{k} I_j \quad \text {for all } t\ge 0. \end{aligned}$$
(36)
Proof
We will assume that (35) does not occur; that is,
$$\begin{aligned} \text {the set } \bigcup \nolimits _{j>k}\bigcup \nolimits _{i:Q_j^\mathrm {core}(i)\cap C_j \ne \emptyset } Q_j^{\mathrm {outer}}(i) \text { does not intersect } A. \end{aligned}$$
(37)
The lemma will follow by showing that the above implies (36).
We start with scale 1. Recall that \(C_1\) contains all elements of \(\eta ^2(0)\). Then, all elements of \(C_1{\setminus } C_1^{\mathrm {bad}}\) are handled at scale 1. Let \(i\in \mathcal {I}_1\), so \(Q_1(i)\) intersects \(C_1{\setminus } C_1^{\mathrm {bad}}\). If \(Q_1(i)\) is a good box, the passage times inside \(Q_1^\mathrm {enc}(i)\) are such that \(\eta ^1\) encapsulates \(\eta ^2\) within \(Q_1^\mathrm {enc}(i)\) unless another cluster of \(\eta ^2\) enters \(Q_1^\mathrm {enc}(i)\) from outside. When the encapsulation succeeds, we have that the cluster of \(\eta ^2\) growing inside \(Q_1^\mathrm {enc}(i)\) never exits \(Q_1^\mathrm {enc}(i)\subset I_1\).
Before proceeding to the proof for scales larger than 1, we explain the possibility that the encapsulation above does not succeed because another cluster of \(\eta ^2\) (say, from \(Q_1(j)\)) enters \(Q_1^\mathrm {enc}(i)\) from outside. Note that if \(Q_1^{\mathrm {outer}}(j)\cap Q_1^{\mathrm {outer}}(i) \ne \emptyset \), then the two clusters are not handled at scale 1: they will be handled together at a higher scale. Now assume that \(Q_1^{\mathrm {outer}}(j)\) and \(Q_1^{\mathrm {outer}}(i)\) are disjoint and do not intersect any other region \(Q_1^{\mathrm {outer}}\) from a contagious site. Thus both \(Q_1(i)\) and \(Q_1(j)\) are handled at scale 1. If they are both good, the encapsulations succeed within \(Q_1^\mathrm {enc}(i)\) and \(Q_1^\mathrm {enc}(j)\), and do not interfere with each other. Assume that \(Q_1(i)\) is good, but \(Q_1(j)\) is bad. In this case, we will make \(Q_1^{\mathrm {outer}}(j)\) to be contagious for scale 2, but up to scale 1 this does not interfere with the encapsulation within \(Q_1^\mathrm {enc}(i)\) because these two regions are disjoint. The encapsulation of \(Q_1^{\mathrm {outer}}(j)\) will be treated at scale 2 or higher, and the fact that \(Q_1^{\mathrm {outer}}(j)\cap Q_1^{\mathrm {outer}}(i)=\emptyset \) will be used to allow a coupling argument between scales.
We now explain the analysis for a scale \(j\in \{2,3,\ldots ,k\}\), assuming that we have carried out the analysis until scale \(j-1\). Thus, we have showed that all contagious vertices successfully handled at scale smaller than j are contained inside \(I_1\cup I_2 \cup \cdots \cup I_{j-1}\). Consider a cell \(Q_j(i)\) of scale j with \(i\in \mathcal {I}_j\). During the encapsulation of \(\eta ^2\) inside \(Q_j^\mathrm {enc}(i)\), it may happen that \(\eta ^1\) advances through a cell \(Q_{j-1}(i')\) that was treated at scale \(j-1\); that is, \(i'\in \mathcal {I}_{j-1}\). (For simplicity of the discussion, we assume here that this cell is of scale \(j-1\), but it could be of any scale \(j'\le j-1\).) Note that \(Q_{j-1}(i')\) must be good for scale \(j-1\) because otherwise cell i would not be treated at scale j. The fact that \(Q_{j-1}(i')\) is good implies that the time \(\eta ^1\) takes to go from \(\partial ^\mathrm {i}Q_{j-1}^{{\mathrm {outer}}/3}(i')\) to all points in \(\partial ^\mathrm {i}Q_{j-1}^\mathrm {enc}(i')\), therefore encapsulating \(Q_{j-1}(i')\), is smaller than the time given by the passage times \(\zeta _j^1\). Moreover, \(Q_{j-1}(i')\) being good implies that the time \(\eta ^2\) takes to go from \(\partial ^\mathrm {i}Q_{j-1}^{{\mathrm {outer}}/3}(i')\) to any point in \(\partial ^\mathrm {o}Q_{j-1}^\mathrm {enc}(i')\) is larger than the time given by the passage times \(\zeta _j^2\). This puts us in the context of Proposition 4.2, where the sets \(\{\Pi _\iota \}_\iota \) are given by the clusters of \(\bigcup _{j''=1}^{j-1} \bigcup _{i'' \in \mathcal {I}_{j''}} Q_{j''}^{\mathrm {outer}}(i'')\), and for each \(\iota \), the set \(\Pi _\iota '\subset \Pi _\iota \) is given by the union of \(Q_{j''}^\mathrm {enc}(i'')\) over all \(j'',i''\) for which \(Q_{j''}^{\mathrm {outer}}(i'')\subset \Pi _\iota \). Therefore, under the event that all the cells involved in the definition of \(\{\Pi _\iota \}_\iota \) are good, Proposition 4.2 gives \(\eta ^2\) cannot escape the set \(\bigcup _{\iota =1}^j I_\iota \), after all contagious vertices of scale at most j have been analyzed. Therefore, inductively we obtain that \(\eta ^2(t)\subset \bigcup _{\iota =1}^\infty I_\iota \).
For scales larger than k, we will use that (37) holds. Since for any scale j and any \(i\in \mathbb {Z}^d\) we have that
$$\begin{aligned} \bigcup \nolimits _{i': Q_j(i')\cap Q^\mathrm {core}_j(i)\ne \emptyset } Q_j^\mathrm {enc}(i')\subset Q_j^{\mathrm {outer}}(i), \end{aligned}$$
we obtain
$$\begin{aligned} \bigcup \nolimits _{j>k}I_j = \bigcup \nolimits _{j>k}\bigcup \nolimits _{i \in \mathcal {I}_j}Q_j^\mathrm {enc}(i) \subset \bigcup \nolimits _{j>k}\bigcup \nolimits _{i:Q_j^\mathrm {core}(i)\cap C_j \ne \emptyset }Q_j^{\mathrm {outer}}(i). \end{aligned}$$
This and (37) give that \(\bigcup \nolimits _{j>k}I_j\) does not intersect A, hence \(\eta ^2(t)\cap A\subset \bigcup _{\iota =1}^k I_\iota \). \(\square \)
Recursion
Define
$$\begin{aligned} \rho _k(i) = \mathbb {P}\left( Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) {.} \end{aligned}$$
Recall that \(Q_k^\mathrm {core}(i)\) are disjoint for different \(i\in \mathbb {Z}^d\), as defined in (22). Define also
$$\begin{aligned} q_k = \exp \left( -c\left( R_k^{\mathrm {outer}}\right) ^\frac{d+1}{2d+4}\right) , \end{aligned}$$
where c is the constant in Lemma 5.2 so that for any \(k\in \mathbb {N}\) and \(i\in \mathbb {Z}^d\), we have \(\mathbb {P}\left( G_k(i)\right) \ge 1-q_k\).
By the definition of \(C_k\) from (34), in order to have \(Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \) it must happen that either
$$\begin{aligned} \exists j\in \mathbb {Z}^d :Q_{k-1}(j) \text { is bad and } Q_{k-1}^{\mathrm {outer}}(j)\cap Q_k^\mathrm {core}(i)\ne \emptyset \end{aligned}$$
(38)
or
$$\begin{aligned} \exists x,y\in \mathbb {Z}^d:x,y \in C_{k-1},\; x\in Q_{k}^\mathrm {core}(i),\; y \not \in Q_{k-1}(\iota ),\; y \in \iota L_{k-1} + \mathcal {B}\left( 3R_{k-1}^{\mathrm {outer}}\right) , \end{aligned}$$
(39)
where \(\iota \) is the unique number such that \(x\in Q_{k-1}^\mathrm {core}(\iota )\). The condition above holds by the following. If (38) does not happen, then there must exist a \(x\in C_{k-1}\cap Q_k^\mathrm {core}(i)\) that was not treated at scale \(k-1\); that is, \(x\in C_{k-1}^{\mathrm {bad}}\). Then, by the definition of \(C^{\mathrm {bad}}_{k-1}\), it must be the case that there exists a y satisfying the conditions in (39). The values x, y as in (39) must satisfy
$$\begin{aligned} \frac{(10d^2-1/2)}{C_\mathrm {FPP}'}L_{k-1}\le |x-y|\le \frac{L_{k-1}}{2 C_\mathrm {FPP}} + 3 R_{k-1}^{\mathrm {outer}}. \end{aligned}$$
(40)
Lemma 5.6
For any \(k\ge 2\) and any \(i\in \mathbb {Z}^d\), define the super cell
$$\begin{aligned} Q_k^\mathrm {super}(i) = \bigcup _{x\in Q_k^\mathrm {core}(i)} \left( x+\mathcal {B}\left( 3R_{k-1}^{\mathrm {outer}}+R_{k-1}\right) \right) ; \end{aligned}$$
(41)
for \(k=1\), set \(Q_1^\mathrm {super}(i) = Q_1^\mathrm {core}(i)\). Then the event \(\left\{ Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right\} \) is measurable with respect to \(Q_k^\mathrm {super}(i)\).
Proof
The theorem is true for \(k=1\) since \(\left\{ Q_1^\mathrm {core}(i)\cap C_1\ne \emptyset \right\} \) is equivalent to \(\left\{ Q_1^\mathrm {core}(i)\cap \eta ^2(0)\ne \emptyset \right\} \). Our goal is to apply an induction argument to establish the lemma for \(k>1\). First note that, since the event that a box of scale \(k-1\) is good is measurable with respect to passage times inside a ball of diameter \(R_{k-1}^{\mathrm {outer}}\), we have that condition (38) is measurable with respect to the passage times inside
$$\begin{aligned} \bigcup _{x\in Q_k^\mathrm {core}(i)} \left( x+\mathcal {B}\left( 2R_{k-1}^{\mathrm {outer}}\right) \right) . \end{aligned}$$
It remains to establish the measurability result for condition (39). Note that condition (39) gives the existence of a point y in \(\bigcup _{x\in Q_k^\mathrm {core}(i)} \left( x+\mathcal {B}\left( 3R_{k-1}^{\mathrm {outer}}\right) \right) \) such that \(y\in C_{k-1}\). Let j be the integer such that \(y\in Q_{k-1}^\mathrm {core}(j)\). Then the induction hypothesis gives that \(\{Q_{k-1}^\mathrm {core}(j) \cap C_{k-1}\ne \emptyset \}\) is measurable with respect to the passage times inside
$$\begin{aligned} \bigcup _{x\in Q_{k-1}^\mathrm {core}(j)} \left( x+\mathcal {B}\left( 3R_{k-2}^{\mathrm {outer}}+R_{k-2}\right) \right) \subset y+\mathcal {B}\left( R_{k-1}\right) \subset Q_k^\mathrm {super}(i). \end{aligned}$$
Therefore, condition (39) is measurable with respect to the passage times inside \(Q_k^\mathrm {super}(i)\). \(\square \)
Lemma 5.7
There exists a constant \(c=c(d,\epsilon ,\alpha ,\upsilon )>0\) such that, for all \(k\in \mathbb {N}\) and all \(i\in \mathbb {Z}^d\), we have
$$\begin{aligned} \rho _k(i) \le c k^{2d(d+2)}\sup _{j}\rho _{k-1}^2(j)+c k^{d(d+2)}q_{k-1}. \end{aligned}$$
Proof
From the discussion above, we have that \(\rho _k(i)\) is bounded above by the probability that condition (38) occurs plus the probability that condition (39) occurs. We start with condition (38). Note that \(Q_{k-1}^\mathrm {core}(j)\), for j defined as in (38), must be contained inside
$$\begin{aligned} \bigcup _{x\in Q_k^\mathrm {core}(i)} \mathcal {B}\left( R_{k-1}^{\mathrm {outer}}+R_{k-1}\right) . \end{aligned}$$
Therefore, there is a constant \(c_0\) depending only on d such that the number of options for the value of j is at most
$$\begin{aligned} c_0 \left( \frac{R_k+R_{k-1}^{\mathrm {outer}}+R_{k-1}}{R_{k-1}}\right) ^d \le c_0 \left( \frac{2 R_k}{R_{k-1}}\right) ^d \le c_1 k^{d(d+2)}, \end{aligned}$$
for some constant \(c_1\), where the first inequality comes from (21) and the last inequality follows from (19). Then, taking the union bound on the value of j, we obtain that the probability that condition (38) occurs is at most
$$\begin{aligned} c_1 k^{d(d+2)} q_{k-1}. \end{aligned}$$
Now we bound the probability that condition (39) happens. For any \(z\in \mathbb {Z}^d\), let \(\varphi (z)\in \mathbb {Z}^d\) be such that \(z\in Q_{k-1}^\mathrm {core}(\varphi (z))\). We will need to estimate the number of different values that \(\varphi (x)\) and \(\varphi (y)\) can assume. Since \(x\in Q_k^\mathrm {core}(i)\), we have that \(\varphi (x)\) can assume at most \(\left( \frac{L_k+2L_{k-1}}{L_{k-1}}\right) ^d\) values. For \(\varphi (y)\), first note that any point in \(Q_{k-1}^\mathrm {core}(\varphi (y))\) must be contained inside \(\varphi (x)L_{k-1} + [-3C_\mathrm {FPP}' R_{k-1}^{\mathrm {outer}}- L_{k-1},3C_\mathrm {FPP}' R_{k-1}^{\mathrm {outer}}+ L_{k-1}]^d\). Therefore, \(Q_{k-1}^\mathrm {core}(\varphi (y))\) must be contained inside a cube of side length
$$\begin{aligned} L_k + 6C_\mathrm {FPP}' R_{k-1}^{\mathrm {outer}}+ 2L_{k-1}, \end{aligned}$$
and consequently there are at most
$$\begin{aligned} \left( \frac{L_k + 6C_\mathrm {FPP}' R_{k-1}^{\mathrm {outer}}+ 2L_{k-1}}{L_{k-1}}\right) ^d \end{aligned}$$
possible values for \(\varphi (y)\). Letting \(A_k\) be the number of ways of choosing the \(Q_{k-1}^\mathrm {core}\) boxes containing x, y according to condition (39), we obtain
$$\begin{aligned} A_k \le \left( \frac{L_k+2L_{k-1}}{L_{k-1}}\right) ^d\left( \frac{L_k + 6C_\mathrm {FPP}' R_{k-1}^{\mathrm {outer}}+ 2L_{k-1}}{L_{k-1}}\right) ^d \le c_2 k^{2d(d+2)}, \end{aligned}$$
for some constant \(c_2=c_2(d,\epsilon ,\alpha ,\upsilon )>0\), where the inequality follows from (19). Now, given x, y, we want to give an upper bound for
$$\begin{aligned} \mathbb {P}\left( \left\{ Q_{k-1}^\mathrm {core}(\varphi (x)) \cap C_{k-1}\ne \emptyset \right\} \cap \left\{ Q_{k-1}^\mathrm {core}(\varphi (y)) \cap C_{k-1}\ne \emptyset \right\} \right) . \end{aligned}$$
From Lemma 5.6 we have that the event \(\left\{ Q_{k-1}^\mathrm {core}(\varphi (x)) \cap C_{k-1}\right\} \) is measurable with respect to the passage times inside \(Q_{k-1}^\mathrm {super}(\varphi (x))\). By the definition of \(Q_{k-1}^\mathrm {super}\) in (41), for any \(z\in Q_{k-1}^\mathrm {super}(\varphi (x))\) we have
$$\begin{aligned} |x-z|&\le |x-\varphi (x)L_{k-1}| + \left( 3R_{k-2}^{\mathrm {outer}}+R_{k-2}\right) \\&\le \frac{L_{k-1}}{2C_\mathrm {FPP}} + 4R_{k-2}^{\mathrm {outer}}\\&\le \frac{R_{k-1}}{20d^2} + \frac{R_{k-1}}{500d^2(k-1)^d},\\&\le \frac{R_{k-1}}{19d^2}, \end{aligned}$$
where we related \(R_{k-2}^{\mathrm {outer}}\) and \(R_{k-1}\) via (21). Since by (40) and (18) we have
$$\begin{aligned} |x-y|> & {} \frac{(10d^2-1/2)}{C'_\mathrm {FPP}}L_{k-1} \ge \left( \frac{10d^2-1/2}{C'_\mathrm {FPP}}\right) \left( \frac{C^2_\mathrm {FPP}R_{k-1}}{10d^2C'_\mathrm {FPP}}\right) \\\ge & {} \left( 1-\frac{1}{20d^2}\right) \left( \frac{R_{k-1}}{d^2}\right) , \end{aligned}$$
where the last inequality follows from (12), we then obtain \(Q_{k-1}^\mathrm {super}(\varphi (x)) \cap Q_{k-1}^\mathrm {super}(\varphi (y))= \emptyset \). This gives that the events \(\left\{ Q_{k-1}^\mathrm {core}(\varphi (x)) \cap C_{k-1}\right\} \) and \(\left\{ Q_{k-1}^\mathrm {core}(\varphi (y)) \cap C_{k-1}\right\} \) are independent, yielding
$$\begin{aligned} \rho _k(i) \le c_1 k^{d(d+2)} q_{k-1} + A_k \sup _j \rho _{k-1}^2(j). \end{aligned}$$
\(\square \)
In the lemma below, recall that \(\eta ^2(0)\) is given by adding each vertex of \(\mathbb {Z}^d\) with probability p, independently of one another. Also let \({\bar{\rho }}\) be such that \({\bar{\rho }}\ge \sup _{j}\rho _{1}(j)\).
Lemma 5.8
Fix any positive constant a. We can set \(L_1\) large enough and then p small enough, both depending on \(a,\alpha ,\epsilon , d\) and \(\upsilon \), such that for all \(k\in \mathbb {N}\) and all \(i\in \mathbb {Z}^d\), we have
$$\begin{aligned} \rho _k(i) \le \exp \left( - a 2^{k}\right) . \end{aligned}$$
Proof
For \(k=1\), then \(\rho _k(i)\) is bounded above by the probability that \(\eta ^2(0)\) intersects \(Q_k^\mathrm {core}(i)\). Once \(L_1\) has been fixed, this probability can be made arbitrarily small by setting p small enough.
Now we assume that \(k\ge 2\). We will expand the recursion in Lemma 5.7. Using the same constant c as in Lemma 5.7, define
$$\begin{aligned} {\bar{q}}_{k-1} = c k^{d(d+2)}q_{k-1} \quad \text { for all } k\ge 2. \end{aligned}$$
Now fix k, set \(A_{-1}=1\), and define for \(\ell =0,1,\ldots ,k-1\)
$$\begin{aligned} A_\ell = A_{\ell -1} 2^{2^\ell -1} \left( c(k-\ell )^{2d(d+2)}\right) ^{2^{\ell }} = \prod _{m=0}^\ell 2^{2^m-1} \left( c(k-m)^{2d(d+2)}\right) ^{2^{m}}. \nonumber \\ \end{aligned}$$
(42)
With this, the recursion in Lemma 5.7 can be written as
$$\begin{aligned} \rho _k(i)&\le A_0 \sup _{j}\rho _{k-1}^2(j)+{\bar{q}}_{k-1}\\&\le A_0\,2\left( \left( c (k-1)^{2d(d+2)}\right) ^2 \sup _{j}\rho _{k-2}^4(j)+{\bar{q}}_{k-2}^2\right) +{\bar{q}}_{k-1}\\&= A_1\sup _{j}\rho _{k-2}^4(j)+2A_0 q_{k-2}^2+{\bar{q}}_{k-1}, \end{aligned}$$
where in the second inequality we used that \((x+y)^m \le 2^{m-1}\left( x^m+y^m\right) \) for all \(x,y\in \mathbb {R}\) and \(m\in \mathbb {N}\). Iterating the above inequality, we obtain
$$\begin{aligned} \rho _k(i)&\le A_{k-2} \sup _{j}\rho _{1}^{2^{k-1}}(j)+\sum _{\ell =1}^{k-1}2^{2^{\ell -1}-1}A_{\ell -2}\bar{q}_{k-\ell }^{2^{\ell -1}} \nonumber \\&= A_{k-2} \bar{\rho }^{2^{k-1}}+\sum _{\ell =1}^{k-1}2^{2^{\ell -1}-1}A_{\ell -2}\bar{q}_{k-\ell }^{2^{\ell -1}}. \end{aligned}$$
(43)
We now claim that
$$\begin{aligned} A_\ell \le \left( 4 c^2\, \left( 3(k-\ell )\right) ^{5d(d+2)}\right) ^{2^\ell } \quad \text {for all } \ell =0,1,\ldots ,k-1. \end{aligned}$$
(44)
We can prove (44) by induction on \(\ell \). Note that \(A_0\) does satisfy the above inequality. Then, using the induction hypothesis and the recursive definition of \(A_\ell \) in (42), we have
$$\begin{aligned} A_\ell&= A_{\ell -1} 2^{2^\ell -1} \left( c(k-\ell )^{2d(d+2)}\right) ^{2^{\ell }}\\&\le \left( 4c^2 \left( 3(k-\ell +1)\right) ^{5d(d+2)}\right) ^{2^{\ell -1}}2^{2^\ell -1} \left( c(k-\ell )^{2d(d+2)}\right) ^{2^{\ell }}\\&= \frac{1}{2}\left( 2c \left( 3(k-\ell +1)\right) ^{5d(d+2)/2}\, 2\,c(k-\ell )^{2d(d+2)}\right) ^{2^{\ell }}. \end{aligned}$$
Now we use that \((x+1)^{5/2}\le 6x^{3}\) for all \(x\ge 1\), which yields
$$\begin{aligned} A_\ell&\le \frac{1}{2}\left( 2c \left( 3^{5/2}6(k-\ell )^3\right) ^{d(d+2)}\, 2\,c(k-\ell )^{2d(d+2)}\right) ^{2^{\ell }}\\&= \frac{1}{2}\left( 4c^2 \left( 3^{1/2}6^{1/5}(k-\ell )\right) ^{5d(d+2)}\right) ^{2^{\ell }} \le \frac{1}{2}\left( 4c^2 \left( 3(k-\ell )\right) ^{5d(d+2)}\right) ^{2^{\ell }}, \end{aligned}$$
establishing (44). Plugging (44) into (43), we obtain
$$\begin{aligned} \rho _k(i)&\le \left( 4c^2\,6^{5d(d+2)}\right) ^{2^{k-2}}{\bar{\rho }}^{2^{k-1}} +\sum _{\ell =1}^{k-1}2^{2^{\ell -1}-1}\left( 4c^2\left( 3(k-\ell +2)\right) ^{5d(d+2)}\right) ^{2^{\ell -2}}{\bar{q}}_{k-\ell }^{2^{\ell -1}}\nonumber \\&\le \left( 2c\,6^{5d(d+2)/2}{\bar{\rho }}\right) ^{2^{k-1}} +\frac{1}{2}\sum _{\ell =1}^{k-1}\left( 4c\left( 3(k-\ell +2)\right) ^{5d(d+2)/2}\bar{q}_{k-\ell }\right) ^{2^{\ell -1}}. \end{aligned}$$
(45)
Given a value of \(L_1\), for all small enough p we obtain that \({\bar{\rho }}\) is sufficiently small to yield
$$\begin{aligned} \left( 2c\,6^{5d(d+2)/2}{\bar{\rho }}\right) ^{2^{k-1}} \le \frac{1}{2}\exp \left( -a 2^{k}\right) . \end{aligned}$$
Now we turn to the second term in (45). Note that for small enough \(\epsilon \), we have \(\epsilon \lambda R_k^\mathrm {enc}\ge \epsilon \lambda \exp \left( \frac{1+c_1}{2\epsilon }\right) R_k> R_k\). Thus, from Lemma 5.2, we have that \(q_{k-\ell }\le \exp \left( -c R_{k-\ell }^\frac{d+1}{2d+4}\right) \), for some constant \(c=c(\alpha ,\epsilon ,d,\upsilon )>0\). We have from the relations (19) and (20) that \(R_j\le c_1 c_2^j (j!)^{d}L_1\) for positive constants \(c_1,c_2\). Therefore, for any \(k\ge \ell \), we have that
$$\begin{aligned}&\left( 4c\left( 3(k-\ell +2)\right) ^{5d(d+2)/2}{\bar{q}}_{k-\ell }\right) ^{2^{\ell -1}}\\&\quad \le \exp \left( -c_3 \left( (k-\ell )!\right) ^\frac{d(d+1)}{2d+4}2^{\ell -1}c_2^\frac{(k-\ell )(d+1)}{2d+4}L_1^\frac{d+1}{2d+4}\right) \\&\quad \le \exp \left( -c_3 2^{k}L_1^\frac{d+1}{2d+4}\right) , \end{aligned}$$
where in the last step we use that \(c_2^\frac{(d+1)}{2d+4}\ge c_2^{1/3}\ge 2\). Hence, for sufficiently large \(L_1\) we obtain
$$\begin{aligned} \rho _k(i) \le \frac{1}{2}\exp \left( -a 2^{k}\right) +\frac{k}{2}\exp \left( -c_3 2^{k}L_1^\frac{d+1}{2d+4}\right) \le \exp \left( -a 2^{k}\right) . \end{aligned}$$
\(\square \)
Multiscale paths of infected sets
Let \(x\in \mathbb {Z}^d\) be a fixed vertex. We say that \(\Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\) is a multi-scale path from x if \(x\in Q_{k_1}^\mathrm {enc}(i_1)\), and for each \(j\in \{2,3,\ldots ,\ell \}\) we have \(Q_{k_j}^\mathrm {enc}(i_j)\cap Q_{k_{j-1}}^\mathrm {enc}(i_{j-1})\ne \emptyset \). Hence,
$$\begin{aligned} \bigcup \nolimits _{(k,i)\in \Gamma } Q_{k}^\mathrm {enc}(i) \text { is a connected subset of } \mathbb {Z}^d \text { and contains } x. \end{aligned}$$
Given such a path, we say that the reach of \(\Gamma \) is given by \(\sup \big \{|z-x| :z \in \bigcup \nolimits _{(k,i)\in \Gamma } Q_{k}^\mathrm {enc}(i)\big \}\), that is, the distance between x and the furthest away point of \(\Gamma \). We will only consider paths such that \(Q_{k_j}^\mathrm {enc}(i_j)\subset I_{k_j}\). Recall the way the sets \(I_\kappa \) are constructed from \(C_\kappa {\setminus } C_\kappa ^{\mathrm {bad}}\), which is defined in (33). Then for any two \((k,i),(k',i')\in \Gamma \) with \(k=k'\) we have \(Q_{k}^\mathrm {enc}(i)\cap Q_{k'}^\mathrm {enc}(i')=\emptyset \). Therefore, we impose the additional restriction that on any multi-scale path \(\Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\) we have \(k_j\ne k_{j-1}\) for all \(j\in \{2,3,\ldots ,\ell \}\).
Now we introduce a subset \({\tilde{\Gamma }}\) of \(\Gamma \) as follows. For each \(k\in \mathbb {N}\) and \(i\in \mathbb {Z}^d\), define
$$\begin{aligned} Q_k^\mathrm {neigh}(i) = i L_k + \mathcal {B}\left( \tfrac{11}{10}R_k^{\mathrm {outer}}\right) \quad \text {and}\quad Q_k^\mathrm {neigh2}(i) = i L_k + \mathcal {B}\left( \tfrac{6}{5}R_k^{\mathrm {outer}}\right) . \end{aligned}$$
Note that \(Q_k^{\mathrm {outer}}(i)\subset Q_k^\mathrm {neigh}(i)\subset Q_k^\mathrm {neigh2}(i)\). Let \(\kappa _1> \kappa _2>\cdots \) be an ordered list of the scales that appear in cells of \(\Gamma \). The set \({\tilde{\Gamma }}\) will be constructed in steps, one step for each scale. First, add to \({\tilde{\Gamma }}\) all cells of \(\Gamma \) of scale \(\kappa _1\). Then, for each \(j\ge 2\), after having decided which cells of \(\Gamma \) of scale at least \(\kappa _{j-1}\) we add to \({\tilde{\Gamma }}\), we add to \({\tilde{\Gamma }}\) all cells \((k,i)\in \Gamma \) of scale \(k=\kappa _j\) such that \(Q_k^\mathrm {neigh}(i)\) does not intersect \(Q_{k'}^\mathrm {neigh}(i')\) for each \((k',i')\) already added to \({\tilde{\Gamma }}\). Recall that, from the definition of \(C_k^{\mathrm {bad}}\) in (33), two cells \((k,j),(k,j')\) of the same scale that are part of \(I_k\) must be such that
$$\begin{aligned} Q_k(j') \not \subset jL_k + \mathcal {B}\left( 3R_k^{\mathrm {outer}}\right) . \end{aligned}$$
This gives that \(|jL_k-j'L_k| \ge 3 R_k^{\mathrm {outer}}-R_k\), which implies that \(Q_k^\mathrm {neigh2}(j)\) and \(Q_k^\mathrm {neigh2}(j')\) do not intersect.
The idea behind the definitions above is that we will look at “paths” of multi-scale cells such that two neighboring cells in the path are such that their \(Q^\mathrm {neigh2}\) regions intersect, and any two cells in the path have disjoint \(Q^\mathrm {neigh}\) regions. The first property limits the number of cells that can be a neighbor of a given cell, allowing us to control the number of such paths, while the second property allows us to argue that the encapsulation procedure behaves more or less independently for different cells of the path.
Lemma 5.9
Let \(\Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\) be a multi-scale path starting from x. Then, the subset \({\tilde{\Gamma }}\) defined above is such that
$$\begin{aligned} \bigcup \nolimits _{(k,i)\in {\tilde{\Gamma }}}Q_{k}^\mathrm {neigh2}(i) \text { is a connected subset of } \mathbb {Z}^d. \end{aligned}$$
Furthermore, any point \(y\in \bigcup _{(k,i)\in \Gamma }Q_k^\mathrm {enc}(i)\) must belong to \(\bigcup _{(k,i)\in {\tilde{\Gamma }}}Q_k^\mathrm {neigh2}(i)\).
Proof
Let \(\Upsilon \) be an arbitrary subset of \({\tilde{\Gamma }}\) with \(\Upsilon \ne {\tilde{\Gamma }}\). The first part of the lemma follows by showing that
$$\begin{aligned} \text {there exists } (\kappa ,\iota )\!\in \! {\tilde{\Gamma }}{\setminus } \Upsilon \text { such that } Q_\kappa ^\mathrm {neigh2}(\iota )\text { intersects } \bigcup \nolimits _{(k,i)\in \Upsilon }Q_k^\mathrm {neigh2}(i). \nonumber \\ \end{aligned}$$
(46)
Define
$$\begin{aligned} \Upsilon ^\mathrm {neigh}= \left\{ (k,i)\in \Gamma :Q_{k}^\mathrm {neigh}(i) \text { intersects }Q_{k'}^\mathrm {neigh}(i')\text { for some } (k',i')\in \Upsilon \right\} . \end{aligned}$$
Clearly, \(\Upsilon ^\mathrm {neigh}\supset \Upsilon \), and since \(\{Q_k^\mathrm {neigh}(i) :(k,i)\in {\tilde{\Gamma }}\}\) is by definition a collection of disjoint sets, we have that
$$\begin{aligned} \text {all elements of } \Upsilon ^\mathrm {neigh}{\setminus } \Upsilon \text { do not belong to } {\tilde{\Gamma }}. \end{aligned}$$
Recall that \({\tilde{\Gamma }}\ne \Upsilon \), and since no element of \({\tilde{\Gamma }}{\setminus }\Upsilon \) was added to \(\Upsilon ^\mathrm {neigh}\), we have that \(\Upsilon ^\mathrm {neigh}\ne \Gamma \). Using that \(\bigcup _{(k,i)\in \Gamma } Q_{k}^\mathrm {enc}(i)\) is a connected set, we obtain a value
$$\begin{aligned}&(k,i)\in \Gamma {\setminus } \Upsilon ^\mathrm {neigh}\text { for which } Q_{k}^\mathrm {enc}(i)\text { intersects } \\&\quad Q_{k'}^\mathrm {enc}(i')\subset Q_{k'}^\mathrm {neigh}(i')\text { for some }(k',i')\in \Upsilon ^\mathrm {neigh}. \end{aligned}$$
Refer to Fig. 9 for a schematic view of the definitions in this proof.
Let \((k',i')\) be the cell of \(\Upsilon ^\mathrm {neigh}\) for which \(Q_k^\mathrm {enc}(i)\) intersects \(Q_{k'}^\mathrm {enc}(i')\). Since \((k',i')\in \Upsilon ^\mathrm {neigh}\), let \((k'',i'')\) be the element of \(\Upsilon \) of largest scale for which \(Q_{k'}^\mathrm {neigh}(i')\) intersects \(Q_{k''}^\mathrm {neigh}(i'')\); if \((k',i')\in \Upsilon \), then \((k'',i'')=(k',i')\). We obtain that
$$\begin{aligned}&\text {the distance according to } |\cdot |\text { between } Q_{k''}^\mathrm {neigh}(i'')\text { and } Q_k^\mathrm {enc}(i) \nonumber \\&\quad \le R_{k'}^\mathrm {enc}+\tfrac{11}{10}R_{k'}^{\mathrm {outer}}. \end{aligned}$$
(47)
By the construction of \({\tilde{\Gamma }}\), and the fact that \((k'',i'')\) was set as the element of largest scale satisfying \(Q_{k''}^\mathrm {neigh}(i'')\cap Q_{k'}^\mathrm {neigh}(i')\ne \emptyset \), we must have that
$$\begin{aligned} k''> k' \quad \text {or}\quad (k'',i'')=(k',i'). \end{aligned}$$
In the former case, the distance in (47) is bounded above by \(2R_{k''-1}^{\mathrm {outer}}\), while in the latter case the distance is zero. So we assume that the distance between \(Q_{k''}^\mathrm {neigh}(i'')\) and \(Q_k^\mathrm {enc}(i)\) is at most \(2 R_{k''-1}^{\mathrm {outer}}\), which yields that \(Q_k^\mathrm {enc}(i)\) intersects \(Q_{k''}^\mathrm {neigh2}(i'')\). Therefore, if \((k,i)\in {\tilde{\Gamma }}\), we have (46) and we are done. When \((k,i)\not \in {\tilde{\Gamma }}\), take the cell \((k''',i''')\in {\tilde{\Gamma }}\) of largest scale such that \(Q_k^\mathrm {neigh}(i)\) intersects \(Q_{k'''}^\mathrm {neigh}(i''')\) and, by the construction of \({\tilde{\Gamma }}\), we have
$$\begin{aligned} k'''>k. \end{aligned}$$
We obtain that \((k''',i''')\not \in \Upsilon \), otherwise it would imply that \((k,i)\in \Upsilon ^\mathrm {neigh}\) violating the definition of (k, i). The distance between \(Q_{k'''}^\mathrm {neigh}(i''')\) and \(Q_{k''}^\mathrm {neigh}(i'')\) is at most
$$\begin{aligned} \tfrac{6}{5}R_{k}^{\mathrm {outer}}+R_{k}^\mathrm {enc}+2R_{k''-1}^{\mathrm {outer}}\le 2 \left( R_{k'''-1}^{\mathrm {outer}}+R_{k''-1}^{\mathrm {outer}}\right) \le \frac{1}{20}\left( R_{k''}^{\mathrm {outer}}+R_{k'''}^{\mathrm {outer}}\right) . \end{aligned}$$
Therefore, we have that \(Q_{k''}^\mathrm {neigh2}(i'')\) intersects \(Q_{k'''}^\mathrm {neigh2}(i''')\), establishing (46) and concluding the first part of the proof.
For the second part, take y to be a point of \(Q_{k}^\mathrm {enc}(i)\) with \((k,i)\in \Gamma \). If \((k,i)\in {\tilde{\Gamma }}\), then the lemma follows. Otherwise, let \((\kappa ,\iota )\) be the cell of largest scale in \({\tilde{\Gamma }}\) such that \(Q_{\kappa }^\mathrm {neigh}(\iota )\cap Q_{k}^\mathrm {neigh}(i)\ne \emptyset \). By the construction of \({\tilde{\Gamma }}\), we have that \(\kappa >k\). The distance between y and \(Q_{\kappa }^\mathrm {neigh}(\iota )\) is at most
$$\begin{aligned} R_{k}^\mathrm {enc}+ R_{k}^\mathrm {neigh}\le 2R_{\kappa -1}^\mathrm {neigh}\le \frac{1}{10} R_\kappa , \end{aligned}$$
which gives that \(y\in Q_\kappa ^\mathrm {neigh2}(\iota )\). \(\square \)
Now we define the type of multi-scale paths we will consider.
Definition 5.10
Given \(x\in \mathbb {Z}^d\) and \(m>0\), we say that \(\Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\) is a well separated path of reach m starting from x if all the following hold:
- (i):
-
\(x \in Q_{k_1}^\mathrm {neigh2}(i_1),\)
- (ii):
-
\(\text {for any } j\in \{2,3,\ldots ,\ell \}\text { we have that } Q_{k_j}^\mathrm {neigh2}(i_j)\text { intersects } Q_{k_{j-1}}^\mathrm {neigh2}(i_{j-1}),\)
- (iii):
-
\(\text {for any } j,\iota \in \{1,2,\ldots ,\ell \}\text { with } |j-\iota |\ge 2\text { we have } Q_{k_j}^\mathrm {neigh2}(i_j)\text { does not} \text {intersect } Q_{k_\iota }^\mathrm {neigh2}(i_\iota ),\)
- (iv):
-
\(\text {for any distinct } j,\iota \in \{1,2,\ldots ,\ell \}\text { we have that } Q_{k_j}^\mathrm {neigh}(i_j)\text { does not} \text {intersect } Q_{k_\iota }^\mathrm {neigh}(i_\iota ),\)
- (v):
-
\(\text {for any } j\in \{2,3,\ldots ,\ell \} ,\text { we have }k_j\ne k_{j-1},\)
- (vi):
-
\(\text {and the point of } Q_{k_\ell }^\mathrm {neigh2}(i_\ell )\text { that is furthest away from } x\text { is of distance } m\text { from } x.\)
We say that a well separated path \(\Gamma \) is infected if for all \((k,i)\in \Gamma \) we have \(Q_{k}^\mathrm {enc}(i)\subset I_k\). If the origin is separated from infinity by \(\eta ^2\), then there must exist a multi-scale path for which the union of the \(Q_k^\mathrm {enc}(i)\) over the cells (k, i) in the path contains the set occupied by \(\eta ^2\) that separates the origin from infinity. Then Lemma 5.9 gives the existence of a well separated path for which the union of the \(Q_k^\mathrm {neigh2}(i)\) over (k, i) in the path separates the origin from infinity.
Lemma 5.11
Fix any positive constant c. We can set \(L_1\) large enough and then p small enough, both depending only on \(c,\alpha ,d, \epsilon \) and \(\upsilon \), so that the following holds. For any integer \(\ell \ge 1\), any given collection of (not necessarily distinct) integer numbers \(k_1,k_2,\ldots ,k_\ell \), and any vertex \(x\in \mathbb {Z}^d\), we have \(\mathbb {P}\big (\exists \text { a well separated path } \Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\text { from } x\text { that is} \text {infected}\big ) \le \exp \left( -c \sum \nolimits _{j=1}^\ell 2^{k_j}\right) \).
Proof
For any j, since the path is infected we have \(Q_{k_j}^\mathrm {enc}(i_j)\subset I_{k_j}\). This gives that there exists \({\tilde{i}}_j\) such that \(Q_{k_j}^\mathrm {core}({\tilde{i}}_j)\cap Q_{k_j}(i_j)\cap C_{k_j}\ne \emptyset \). From Lemma 5.6, we have that the event \(\left\{ Q_{k_j}^\mathrm {core}({\tilde{i}}_j)\cap C_{k_j}\ne \emptyset \right\} \) is measurable with respect to the passage times inside \(Q_{k_j}^\mathrm {super}({\tilde{i}}_j)\subset Q_{k_j}^\mathrm {neigh}(i_j)\). Also, the number of choices for \({\tilde{i}}_j\) is at most some constant \(c_1\), depending only on d. Since \(\{Q_{k_j}^\mathrm {neigh}(i_j)\}_{j=1,\ldots ,\ell }\) is a collection of disjoint sets, if we fix the path \(\Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\), and take the union bound over the choices of \({\tilde{i}}_1, \tilde{i}_2,\ldots ,{\tilde{i}}_\ell \), we have from Lemma 5.8 that
$$\begin{aligned} \mathbb {P}\left( \Gamma \text { is an infected path}\right) \le c_1^\ell \exp \left( -a \sum \nolimits _{j=1}^\ell 2^{k_j}\right) , \end{aligned}$$
where a can be made as large as we want by properly setting \(L_1\) and p. It remains to bound the number of well separated paths that exist starting from x. Since \(x \in Q_{k_1}^\mathrm {neigh2}(i_1)\), the number of ways to choose the first cell is at most \(\left( \frac{12}{5}C_\mathrm {FPP}' R_{k_1}^{\mathrm {outer}}\right) ^d\). Consider a j such that \(k_j>k_{j+1}\). We have that \(Q_{k_{j}}^\mathrm {neigh2}(i_{j})\) must intersect \(Q_{k_{j+1}}^\mathrm {neigh2}(i_{j+1})\), which gives that
$$\begin{aligned} Q_{k_{j+1}}^\mathrm {core}(i_{j+1})&\subset i_j L_{k_j}+\mathcal {B}\left( \tfrac{6}{5}R_{k_j}^{\mathrm {outer}}+\tfrac{6}{5}R_{k_{j+1}}^{\mathrm {outer}}+R_{k_{j+1}}\right) \\&\subset i_j L_{k_j}+\mathcal {B}\left( \tfrac{7}{5}R_{k_j}^{\mathrm {outer}}\right) \\&\subset i_j L_{k_j}+\left[ -\tfrac{7C_\mathrm {FPP}'}{5}R_{k_j}^{\mathrm {outer}},\tfrac{7C_\mathrm {FPP}'}{5}R_{k_j}^{\mathrm {outer}}\right] ^d. \end{aligned}$$
Hence, the number of ways to choose \(i_{j+1}\) given \((k_j,i_j)\) and \(k_{j+1}\) is at most
$$\begin{aligned} \left( \frac{14C_\mathrm {FPP}'}{5}\frac{R_{k_j}^{\mathrm {outer}}}{L_{k_{j+1}}}\right) ^d. \end{aligned}$$
Therefore, we have that \(\mathbb {P}\big (\exists \text { a well separated path } \Gamma =(k_1,i_1),(k_2,i_2),\ldots ,(k_\ell ,i_\ell )\text { from } x\text { that is infected}\big )\) is at most
$$\begin{aligned}&c_1^\ell \left( \frac{12}{5}C_\mathrm {FPP}' R_{k_1}^{\mathrm {outer}}\right) ^d \prod _{j=1}^\ell \left( \frac{14C_\mathrm {FPP}'}{5}\frac{R_{k_j}^{\mathrm {outer}}}{L_{k_{j+1}}}\right) ^{2d}\exp \left( -a2^{k_j}\right) \\&\quad \le c_1^\ell \prod _{j=1}^\ell \exp \left( c_2 k_j\log (k_j) -a2^{k_j}\right) \\&\quad \le \prod _{j=1}^\ell \exp \left( -a2^{k_j-1}\right) , \end{aligned}$$
where the second inequality follows for some \(c_2=c_2(d,\alpha ,\epsilon ,\upsilon )\) by the value of \(R_k^{\mathrm {outer}}\) from (20), and the last inequality follows by setting a large enough and such that \(a\ge 2c\). \(\square \)
For the lemma below, define the event
$$\begin{aligned} E_{\kappa ,r} = \{&\text {there exists a well separated path from the origin that is infected,}\nonumber \\&\text {has only cells of scale smaller than } \kappa ,\text { and has reach at least } C_\mathrm {FPP}r\}. \end{aligned}$$
(48)
Let \(E_{\infty ,r}\) be the above event without the restriction that all scales must be smaller than \(\kappa \). Below we restrict to \(r>3\) just to ensure that \(\log \log r >0\).
Proposition 5.12
Fix any positive constant c, any \(r>3\) and any time \(t\ge 0\). We can set \(L_1\) large enough and then p small enough, both depending only on \(c,\alpha ,d,\epsilon \) and \(\upsilon \), so that there exists a positive constant \(c'\) depending only on d for which
$$\begin{aligned} \mathbb {P}\left( E_{\infty ,r}\right) \le \exp \left( -c r^\frac{c'}{\log \log r}\right) . \end{aligned}$$
Proof
Let \(A_r\) be the set of vertices of \(\mathbb {Z}^d\) of distance at most \(C_\mathrm {FPP}r\) from the origin. Set \(\delta _r = \frac{1}{(d+3)\log \log r}\) and \(\kappa = \delta _r \log r\). For any large enough a depending on \(L_1\) and p, we have
$$\begin{aligned}&\mathbb {P}\left( \exists (k,i) :k\ge \kappa \text { and } Q_k^\mathrm {neigh2}(i)\cap A_r \ne \emptyset \text { and } Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) \\&\le \sum _{k=\kappa }^\infty \sum _{i:Q_k^\mathrm {neigh2}(i)\cap A_r \ne \emptyset } \mathbb {P}\left( Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) \\&\le \sum _{k=\kappa }^\infty \left( \frac{2C_\mathrm {FPP}' \left( r+\frac{6}{5}R_k^{\mathrm {outer}}+R_k\right) }{L_k}\right) ^d \exp \left( -a 2^k\right) , \end{aligned}$$
where in the last inequality we use Lemma 5.8. Since a above can be chosen as large as needed (by requiring that \(L_1\) is large enough and p is small enough), we can choose a large enough a so that \(\left( \frac{2C_\mathrm {FPP}' \left( r+\frac{6}{5}R_k^{\mathrm {outer}}+R_k\right) }{L_k}\right) ^d\le \exp \left( a2^{k-1}\right) \) for all k, yielding
$$\begin{aligned}&\mathbb {P}\left( \exists (k,i) :k\ge \kappa \text { and } Q_k^\mathrm {neigh2}(i)\cap A_r \ne \emptyset \text { and } Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) \\&\le \sum _{k=\kappa }^\infty \exp \left( -a 2^{k-1}\right) \le 2\exp \left( -a 2^{\kappa -1}\right) . \end{aligned}$$
If the event above does not happen, then Lemma 5.5 gives that \(\eta ^2(t)\cap A_r \subset \bigcup _{j=1}^{\kappa -1}I_j\). Hence,
$$\begin{aligned} \mathbb {P}\left( E_{\infty ,r}\right) \le 2\exp \left( -a 2^{\kappa -1}\right) + \mathbb {P}\left( E_{\kappa ,r}\right) . \end{aligned}$$
Let \(\Gamma \) be a well separated path from the origin, with all cells of scale smaller than \(\kappa \), and which has reach at least r. Define \(m_k(\Gamma )\) to be the number of cells of scale k in \(\Gamma \). Since \(\Gamma \) must contain at least one cell for which its \(Q^\mathrm {neigh2}\) region is not contained in \(A_r\), we have
$$\begin{aligned} C_\mathrm {FPP}r \le \sum _{k=1}^{\kappa -1} m_k(\Gamma ) \frac{12}{5} R_k^{\mathrm {outer}}. \end{aligned}$$
Because of the type of bounds derived in Lemma 5.11, it will be convenient to rewrite the inequality above so that the term \(\sum _{k=1}^{\kappa -1}2^k\) appears. Note that using (20) we can set a constant \(c_0\ge 2\) such that \(R_j^{\mathrm {outer}}\le c_0^j (j!)^{d+2}L_1\) for all \(j\ge 1\), which gives
$$\begin{aligned} C_\mathrm {FPP}r \le \sum _{k=1}^{\kappa -1} m_k(\Gamma ) 2^k \frac{12 (c_0-2)^k(k!)^{d+2}L_1}{5} \le \frac{12 (c_0-2)^\kappa (\kappa !)^{d+2}L_1}{5} \sum _{k=1}^{\kappa -1} m_k(\Gamma ) 2^k. \end{aligned}$$
For any \(\Gamma \), define \(\varphi (\Gamma )=\sum _{k=1}^{\kappa -1} m_k(\Gamma ) 2^k\). We can then split the sum over all paths according to the value of \(\varphi (\Gamma )\) of the path. Using this, Lemma 5.11, and the fact that \(\varphi (\Gamma )\ge \frac{5 C_\mathrm {FPP}r}{12 c' (\kappa !)^{d+2}L_1}\), we have
$$\begin{aligned} \mathbb {P}\left( E_{\kappa ,r}\right) \le \sum _{m\ge \frac{5 C_\mathrm {FPP}r}{12 c' (\kappa !)^{d+2}L_1}}^\infty \exp \left( -c'' m\right) A_m, \end{aligned}$$
where \(A_m\) is the number of ways to fix \(\ell \) and set \(k_1,k_2,\ldots ,k_\ell \) such that \(\varphi (\Gamma )=\sum _{j=1}^\ell 2^{k_j}=m\), and \(c''\) is the constant in Lemma 5.11. For each choice of \(\ell ,k_1,k_2,\ldots ,k_\ell \), we can define a string from \(\{0,1\}^m\) by taking \(2^{k_1}\) consecutive 0s, \(2^{k_2}\) consecutive 1s, \(2^{k_3}\) consecutive 0s, and so on and so forth. Note that each string is mapped to at most one choice of \(\ell ,k_1,k_2,\ldots ,k_\ell \). Therefore, \(A_m\le 2^m\), the number of strings in \(\{0,1\}^m\). The proof is completed since \(c''\) can be made arbitrarily large by setting \(L_1\) large enough and then p small enough, and \(\frac{5 C_\mathrm {FPP}r}{12 c' (\kappa !)^{d+2}L_1}\ge \frac{5 C_\mathrm {FPP}r}{12 c' \kappa ^{(d+2)\kappa }L_1}\ge \frac{5 C_\mathrm {FPP}r^\frac{1}{d+3}}{12 c' L_1}\). \(\square \)
Completing the proof of Theorem 5.1
Proof of Theorem 5.1
We start showing that \(\eta ^1\) grows indefinitely with positive probability. Let \(e_1=(1,0,0,\ldots ,0)\in \mathbb {Z}^d\). Any set of vertices that separates the origin from infinity must contain a vertex of the form \(b e_1\) for some nonnegative integer b. For any b and \(t\ge 0\), let
$$\begin{aligned} f_b(t) = \mathbb {P}\left( \eta ^2(t) \text { contains } b e_1 \text { and separates the origin from infinity}\right) . \end{aligned}$$
For the moment, we assume that b is larger than some fixed, large enough value \(b_0\). Recall that \(\mathcal {B}\left( r\right) \subseteq [- C_\mathrm {FPP}' r, C_\mathrm {FPP}' r]^d\), which gives that \(b e_1 + \mathcal {B}\left( \frac{b}{2C'_\mathrm {FPP}}\right) \) does not contain the origin. Hence, in order for \(\eta _2(t)\) to contain \(b e_1\) and separate the origin from infinity, \(\eta _2(t)\) must contain at least a vertex of distance (according to the norm \(|\cdot |\)) greater than \(\frac{b}{2C'_\mathrm {FPP}}\) from \(b e_1\). When \(\eta _2(t)\) separates the origin from infinity, it must contain a set of sites that form a connected component according to the \(\ell _\infty \) norm, which itself separates the origin from infinity and contains a vertex of distance (now according to the norm \(|\cdot |\)) greater than \(\frac{b}{2C'_\mathrm {FPP}}\) from \(b e_1\). This connected component implies the occurence of the event in Proposition 5.12, hence
$$\begin{aligned} f_b(t) \le \exp \left( -c \left( \frac{b}{2C'_\mathrm {FPP}}\right) ^\frac{c'}{\log \log b}\right) . \end{aligned}$$
Note that, as needed, the bound above does not depend on t; this will allows us to derive a bound for the survival of \(\eta ^1\) that is uniformly bounded away from 0 as t grows to infinity. Note also that the constant c from Proposition 5.12 can be made arbitrarily large by setting \(L_1\) and p properly. Therefore, \(\sum _{b=b_0}^\infty f_b(t)\) can be made smaller than 1, and in fact goes to zero with \(b_0\). Regarding the case \(b\le b_0\), for each \(k\ge 1\) let \(\mathcal {K}_k\) be the set of (k, i) such that \(R_k^{\mathrm {outer}}(i)\cap \{e_1,2e_1,\ldots ,b_0 e_1\}\). Note that there exists a constant \(c_b\) depending on \(b_0\) such that the cardinality of \(\mathcal {K}_k\) is at most \(c_b\) for all k. Then, using Lemma 5.8, we have
$$\begin{aligned} \mathbb {P}\left( \exists k :\left( \bigcup \nolimits _{(k,i)\in \mathcal {K}_k}Q_k^\mathrm {core}(i)\right) \cap C_k \ne \emptyset \right) \le \sum _{k\ge 1} c_b \exp \left( -a 2^k\right) , \end{aligned}$$
which can be made arbitrarily small since a can be made large enough by choosing \(L_1\) large and p small. This concludes this part of the proof, since
$$\begin{aligned} \mathbb {P}\left( \eta ^1 \text { grows indefinitely}\right) \ge 1 - \sum _{k\ge 1} c_b \exp \left( -a 2^k\right) - \limsup _{t\rightarrow \infty }\sum _{b=b_0}^\infty f_b(t). \end{aligned}$$
Now we turn to the proof of positive speed of growth for \(\eta ^1\). Note that \(\eta ^1\cup \eta ^2\) is stochastically dominated by a first passage percolation process where the passage times are at least i.i.d. exponential random variables of rate 2, because \(\eta ^2\) is slower than a first passage percolation of exponential passage times of rate 1. Then, by the shape theorem we have that there exists a constant \(c>0\) large enough such that
$$\begin{aligned} \mathbb {P}\left( \eta ^1(t)\cup \eta ^2(t) \subset [-ct,ct]^d\right) \rightarrow 1 \quad \text {as } t\rightarrow \infty . \end{aligned}$$
Now fix any t, take c as above, and set \(\kappa = 1+\frac{\log t}{\left( \log \log t\right) ^2}\). For any large enough a depending on \(L_1\) and p, we have
$$\begin{aligned}&\mathbb {P}\left( \exists (k,i) :k\ge \kappa \text { and } Q_k^\mathrm {neigh2}(i)\cap [-ct,ct]^d \ne \emptyset \text { and } Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) \\&\le \sum _{k=\kappa }^\infty \sum _{i:Q_k^\mathrm {neigh2}(i)\cap [-ct,ct]^d \ne \emptyset } \mathbb {P}\left( Q_k^\mathrm {core}(i)\cap C_k\ne \emptyset \right) \\&\le \sum _{k=\kappa }^\infty \left( \frac{2C_\mathrm {FPP}' \left( ct+\frac{6}{5}R_k^{\mathrm {outer}}+R_k\right) }{L_k}\right) ^d \exp \left( -a 2^k\right) \\&\le \sum _{k=\kappa }^\infty \exp \left( -a 2^{k-1}\right) \le 2\exp \left( -a 2^{\kappa -1}\right) , \end{aligned}$$
where in the second inequality we use Lemma 5.8, and the third inequality follows because a can be chosen large enough in Lemma 5.8. The above derivation allows us to restrict to cells of scale smaller than \(\kappa \). Note that since there are no contagious set of scale \(\kappa \) or larger intersecting \([-ct,ct]^d\), the spread of \(\eta ^1(t)\) inside \([-ct,ct]^d\) stochastically dominates a first passage percolation process of rate \(\lambda ^1_\kappa \). Thus, disregarding regions taken by \(\eta ^2\), we can set a sufficiently small constant \(c'>0\) so that, at time t, \({\bar{\eta }}^1\) will contain a ball of radius \(2c't\) around the origin with probability at least \(1-\exp \left( -c'' t^\frac{d+1}{2d+4}\right) \) for some constant \(c''\), by Proposition 3.1. The only caveat is that, at time t, there may be regions of scale smaller than \(\kappa \) that are taken by \(\eta ^2\) and intersects the boundary of \(\mathcal {B}\left( 2c't\right) \). If we show that such regions cannot intersect \(\partial ^\mathrm {i}\mathcal {B}\left( c't\right) \), then we have that the probability that \(\eta ^1\) survives up to time t but \({\bar{\eta }}^1(t)\) does not contain a ball of radius \(c't\) around the origin is at most \(1-2\exp \left( -a 2^{\kappa -1}\right) -\exp \left( -c'' t^\frac{d+1}{2d+4}\right) \). This is indeed the case, since we can take a constant \(c'''\) such that any cell of scale smaller than \(\kappa \) has diameter at most
$$\begin{aligned} c'''(\kappa !)^{d+2} \le c''' \exp \left( (d+2)\kappa \log \kappa \right) \le c'''\exp \left( 2(d+2)\frac{\log t}{\log \log t}\right) <c't, \end{aligned}$$
where the inequalities above hold for all large enough t, completing the proof. \(\square \)