Multiparticle diffusion limited aggregation
 564 Downloads
Abstract
We consider a stochastic aggregation model on \(\mathbb {Z}^d\). Start with particles distributed according to the product Bernoulli measure with parameter \(\mu \). In addition, start with an aggregate at the origin. Nonaggregated particles move as continuoustime simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is called multiparticle diffusion limited aggregation. Our main result states that if on \(d>1\) the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms; that is, there exists a constant \(c>0\) so that at time t the aggregate contains a point of distance at least ct from the origin, for all t. The key conceptual element of our analysis is the introduction and study of a new growth process. Consider a first passage percolation process, called type 1, starting from the origin. Whenever type 1 is about to occupy a new vertex, with positive probability, instead of doing it, it gives rise to another first passage percolation process, called type 2, which starts to spread from that vertex. Each vertex gets occupied only by the process that arrives to it first. This process may have three phases: extinction (type 1 gets eventually surrounded by type 2), coexistence (infinite clusters of both types emerge), and strong survival (type 1 produces an infinite cluster which entraps all type 2 clusters). Understanding the various phases of this process is of mathematical interest on its own right. We establish the existence of a strong survival phase, and use this to show our main result.
1 Introduction
In this work we consider one of the classical aggregation processes, introduced in [25] (see also [29]) with the goal of providing an example of “a simple and tractable” mathematical model of dendritic growth, for which theoretical and mathematical concepts and tools could be designed and tested on. Almost four decades later we still encounter tremendous mathematical challenges studying its geometric and dynamic properties, and understanding the driving mechanism lying behind the formation of fractallike structures.
Characterizing the behavior of MDLA is a widely open and challenging problem. Existing mathematical results are limited to one dimension [8, 20]. In this case, it is known that the aggregate has almost surely sublinear growth for any \(\mu \in (0,1)\), having size of order \(\sqrt{t}\) by time t. The main obstacle preventing the aggregate to grow with positive speed is that, from the point of view of the front (i.e., the rightmost point) of the aggregate, the density of particles decreases since the aggregate grows by forming a region of density 1, larger than the initial density of particles.
The main result of this paper is to establish that, unlike in dimension one, in dimensions \(d\ge 2\) MDLA has a phase of linear growth. We actually prove a stronger result, showing that the aggregate grows with positive speed in all directions. For \(t\ge 0\), let \(\mathcal {A}_t\subset \mathbb {Z}^d\) be the set of vertices occupied by the aggregate by time t, and let \(\bar{\mathcal {A}}_t\supseteq \mathcal {A}_t\) be the set of vertices of \(\mathbb {Z}^d\) that are not contained in the infinite component of \(\mathbb {Z}^d{\setminus } \mathcal {A}_t\). Note that \(\bar{\mathcal {A}}_t\) comprises all vertices of \(\mathbb {Z}^d\) that either belong to the aggregate or are separated from infinity by the aggregate. For \(x\in \mathbb {Z}^d\) and \(r\in \mathbb {R}_+\), we denote by B(x, r) the ball of radius r centered at x.
Theorem 1.1
Remark 1.2
It is not difficult to see that the aggregate cannot grow faster than linearly. That is, there exists a constant \(c_3\) such that the probability that \(\bar{\mathcal {A}}_t \subset B(0,c_3 t)\) for all \(t\ge t_0\) goes to 1 with \(t_0\). This is the case because the growth of the aggregate is slower than the growth of a first passage percolation with exponential passage times of rate 1, which has linear growth; see, for example, [1, 21].
We believe that Theorem 1.1 holds in a stronger form, with \(\mathbb {P}\big (\bar{\mathcal {A}}_t \supset B(0,c_1 t) \text { for all } t\ge t_0\big )\) going to 1 with \(t_0\). However, with positive probability, it happens that there is no particle within a large distance to the origin at time 0. In this case, in the initial stages of the process, the aggregate will grow very slowly as if in a system with a small density of particles. We expect that the density of particles near the boundary of the aggregate will become close to \(\mu \) after particles have moved for a large enough time, allowing the aggregate to start having positive speed of growth. However, particles perform a nonequilibrium dynamics due to their interaction with the aggregate, and the behavior and the effect of this initial stage of low density is not yet understood mathematically. This is related to the problem of describing the behavior of MDLA for small values of \(\mu \), which is still far from reach, and raises the challenging question of whether the aggregate has positive speed of growth for any \(\mu >0\). Even in a heuristic level, it is not at all clear what the behavior of the aggregate should be for small \(\mu \). On the one hand, the low density of particles causes the aggregate to grow slowly since particles move diffusively until they are aggregated. On the other hand, since the aggregate is immersed in a dense cloud of particles, this effect of slow growth could be restricted to small scales only, because at very large scales the aggregate could simultaneously grow in many different directions.
We now describe the ideas of the proof of Theorem 1.1. For this we use the language of the dual representation of the exclusion process, where vertices without particles are regarded as hosting another type of particles, called holes, which perform among themselves simple symmetric random walks obeying the exclusion rule. When \(\mu \) is large enough, at the initial stages of the process, the aggregate grows without encountering any hole. The growth of the aggregate is then equivalent to a first passage percolation process with independent exponential passage times. This stage is well understood: it is known that first passage percolation not only grows with positive speed, but also has a limiting shape [9, 24]. However, at some moment, the aggregate will start encountering holes. We can regard the aggregate as a solid wall for holes, as they can neither jump onto the aggregate nor be attached to the aggregate. In one dimension, holes end up accummulating at the boundary of the aggregate, and this is enough to prevent positive speed of growth. The situation is different in dimensions \(d\ge 2\), since the aggregate is able to deviate from any hole it encounters, advancing through the particles that lie in the neighborhood of the hole until it completely surrounds and entraps the hole. The problem is that the aggregate will find regions of holes of arbitrarily large sizes, which require a long time for the aggregate to go around them. When \(\mu \) is large enough, the regions of holes will be typically well spaced out, giving sufficient room for the aggregate to grow inbetween the holes. One needs to show that the delays caused by deviation from holes are not large enough to prevent positive speed. A challenge is that as holes cannot jump onto the aggregate, their motion gets a drift whenever they are neighboring the aggregate. Hence, holes move according to a nonequilibrium dynamics, which creates difficulties in controlling the location of the holes. In order to overcome this problem, we introduce a new process to model the interplay between the aggregate and holes.
First passage percolation in a hostile environment (FPPHE) This is a twotype first passage percolation process. At any time \(t\ge 0\), let \(\eta ^1(t)\) and \(\eta ^2(t)\) denote the vertices of \(\mathbb {Z}^d\) occupied by type 1 and type 2, respectively. We start with \(\eta ^1(0)\) containing only the origin of \(\mathbb {Z}^d\), and \(\eta ^2(0)\) being a random set obtained by selecting each vertex of \(\mathbb {Z}^d{\setminus }\{0\}\) with probability \(p\in (0,1)\), independently of one another. Both type 1 and type 2 are growing processes; i.e., for any times \(t<t'\) we have \(\eta ^1(t)\subseteq \eta ^1(t')\) and \(\eta ^2(t)\subseteq \eta ^2(t')\). Type 1 spreads from time 0 throughout \(\mathbb {Z}^d\) at rate 1. Type 2 does not spread at first, and we denote \(\eta ^2(0)\) as type 2 seeds. Whenever the type 1 process attempts to occupy a vertex hosting a type 2 seed, the occupation is suppressed and that type 2 seed is activated and starts to spread throughout \(\mathbb {Z}^d\) at rate \(\lambda \in (0,1)\). The other type 2 seeds remain inactive until type 1 or already activated type 2 attempts to occupy their location. A vertex of the lattice is only occupied by the type that arrives to it first, so \(\eta ^1(t)\) and \(\eta ^2(t)\) are disjoint sets for all t; this causes the two types to compete with each other for space. Note that type 2 spreads with smaller rate than type 1, but type 2 starts with a density of seeds while type 1 starts only from a single location.
The first phase is the extinction phase, where type 1 stops growing in finite time with probability 1. This occurs, for example, when \(p> 1p_{\mathrm {c}}\), with \(p_{\mathrm {c}}=p_{\mathrm {c}}(d)\) being the critical probability for independent site percolation on \(\mathbb {Z}^d\). In this case, with probability 1, the origin is contained in a finite cluster of vertices not occupied by type 2 seeds, and hence type 1 will eventually stop growing. This extinction phase for type 1 also arises when \(p\le 1p_{\mathrm {c}}\) but \(\lambda \) is close enough to 1 so that type 2 clusters grow quickly enough to surround type 1 and confine it to a finite set.
We show in this work that another phase exists, called the strong survival phase, and which is characterized by a positive probability of appearance of an infinite cluster of type 1, while type 2 is confined to form only finite clusters. Note that type 1 cannot form an infinite cluster with probability 1, since with positive probability all neighbors of the origin contain seeds of type 2. Unlike the extinction phase, whose existence is quite trivial to show, the existence of a strong survival phase for some value of p and \(\lambda \) is far from obvious. Here we not only establish the existence of this phase, but we show that such a phase exists for any\(\lambda <1\) provided that p is small enough. We also show that type 1 has positive speed of growth. 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 \(\eta ^1(t)\). The theorem below will be proved in Sect. 5, as a consequence of a more general theorem, Theorem 5.1.
Theorem 1.3
There is a third possible regime, which we call the coexistence phase, and is characterized by type 1 and type 2 simultaneously forming infinite clusters with positive probability. (We regard the coexistence phase as a regime of weak survival for type 1, in the sense that type 1 survives but leaves enough space for type 2 to produce at least one infinite cluster.) Whether this regime actually occurs for some value of p and \(\lambda \) is an open problem, and even simulations do not seem to give good evidence of the existence of this regime. For example, in the rightmost picture of Fig. 3, we observe a regime where \(\eta ^1\) survives, while \(\eta ^2\) seems to produce only finite clusters, but of quite long sizes. This also seems to be the behavior of the central picture in Fig. 3, though it is not as clear whether each cluster of \(\eta ^2\) will be eventually confined to a finite set. However, the behavior in the leftmost picture of Fig. 3 is not at all clear. The cluster of \(\eta ^1\) has survived until the simulation was stopped, but produced a very thin set. It is not clear whether coexistence will happen in this situation, whether \(\eta ^1\) will eventually stop growing, or even whether after a much longer time the “arms” produced by \(\eta ^1\) will eventually find one another, constraining \(\eta ^2\) to produce only finite clusters.
Establishing whether a coexistence phase exists for some value of p and \(\lambda \) is an interesting open problem. We can establish that a coexistence phase occurs in a particular example of FPPHE, where type 1 and type 2 have deterministic passage times, with all randomness coming from the locations of the seeds. In this example, all three phases occur. We discuss this in Sect. 2. See also the recent paper [6], where coexistence is established when \(\mathbb {Z}^d\) is replaced by a hyperbolic nonamenable graph.
Historical remarks and related works MDLA belongs to a class of models, introduced firstly in the physics and chemistry literature (see [15] and references therein), and later in the mathematics literature as well, with the goal of studying geometric and dynamic properties of static formations produced by aggregating randomly moving colloidal particles. Some numerically established quantities, such as fractal dimension, showed striking similarities between clusters produced by aggregating particles and clusters produced in other growth processes of entirely different nature, such as dielectric breakdown cascades and Laplacian growth models (in particular, HeleShaw cell [26]). These similarities were further investigated by the introduction of the Hastings–Levitov growth model [13], which is represented as a sequence of conformal mappings. Nonetheless, it is still debated in the physics literature whether some of these models belong to the same universality class or not [4].
In the mathematics literature, the diffusion limited aggregation model (DLA), introduced in [14] following the introduction of MDLA in [25], became a paradigm object of study among aggregation models driven by diffusive particles. However, progress on understanding DLA and MDLA mathematically has been relatively modest. The main results known about DLA are bounds on its rate of growth, derived by Kesten [16, 17] (see also [2]), but several variants have been introduced and studied [3, 5, 7, 10, 23, 27]. Regarding MDLA, it was rigorously studied only in the onedimensional case [8, 19, 20], for which sublinear growth has been proved for all densities \(p\in (0,1)\) in [20].
Structure of the paper We start in Sect. 2 with a discussion of an example of FPPHE where the passage times are deterministic, and show that this process has a coexistence phase. Then, in preparation for the proof of strong survival of FPPHE (Theorem 1.3), we state in Sect. 3 existing results on first passage percolation, and discuss in Sect. 4 a result due to Häggstrom and Pemantle regarding noncoexistence of a twotype first passage percolation process. This result plays a fundamental role in our analysis of FPPHE. Then, in Sect. 5, we state and prove Theorem 5.1, which is a more general version of Theorem 1.3. In Sect. 6 we relate FPPHE with MDLA, giving the proof of Theorem 1.1.
2 Example of coexistence in FPPHE
Our theorem below establishes the existence of a coexistence phase. We note that here the phase for survival for \(\eta ^1\) is stronger than that shown in Theorem 1.3. Here we show that for some small enough p, \(\eta ^1\) survives for any\(\lambda <1\). The actual value of \(\lambda \) plays a role only on determining whether coexistence happens. In the theorem below and its proof, a directed path in \(\mathbb {Z}^d\) is defined to be a path whose jumps are only along the positive direction of the coordinates.
Theorem 2.1
Proof
A different situation occurs when \(\eta ^1\) finds a vertex of \(\eta ^2(0)\) in the axis, as with the yellow vertex of Fig. 6c. Note that, in a directed percolation process, all vertices below the yellow seed will not be reachable from the origin. In our twotype process, something similar occurs, but only for a finite number of steps. When \(\eta ^1\) activates the yellow seed at \(x\in \eta ^2(0)\), \(\eta ^1\) cannot immediately go around x as explained above. For \(\lambda \) close enough to 1, \(\eta ^2\) occupies the successive vertex in the axis before \(\eta ^1\) can go around x. This continues for some steps, with \(\eta ^2\) being able to grow along the axis; see Fig. 6d, e. However, at each step \(\eta ^1\) will be \(1\lambda \) faster than \(\eta ^2\). This will accumulate for roughly \(\frac{1}{1\lambda }\) steps, when \(\eta ^1\) will finally be able to go around \(\eta ^2\); as in Fig. 6f. This happens unless \(\eta ^2(0)\) happens to have a seed at a vertex neighboring one of the vertices on the axis occupied by the growth of \(\eta ^2\). This is illustrated by the green vertices of Fig. 6f–h. When the first green vertex out of the axis is activated, \(\eta ^1\) will not be able to occupy the vertex to the right of the green vertex, and will encounter the next green seed before it can go around the first green seed found at the axis. The crucial fact to observe is that the clusters of \(\eta ^2\) that start to grow after the activation of each green seed can only occupy vertices located to the right of the seeds, and at the same vertical coordinate. This is a subset of the vertices that are shaded by the green seeds in a directed percolation process. Therefore, (1) follows since the vertices occupied by \(\eta ^2\) are a subset of the following set: take the union of all triangles obtained from sets of consecutive seeds away from the axis (as with the pink, red and blue seeds in Fig. 6), and take the union of semilines starting at seeds located at the axis or at seeds neighboring semilines starting from seeds of smaller \(\ell _1\) distance to the origin (as with the yellow and green seeds in Fig. 6). This set is exactly the set of vertices not reached by a directed path from the origin.
Now we turn to (2). First notice that, from the first part, we have that \(\eta ^1(t)\supseteq C_t\) for all p and \(\lambda \). Since \(C_t\) does not depend on \(\lambda \), once we fix \(p\in (0,1p_{\mathrm {c}}^{\mathrm {dir}})\), we can take \(\lambda \) as close to 1 as we want, and \(\eta ^1\) will still produce an infinite component. Now we consider one of the axis. For example, the one containing the green vertices in Fig. 6. Let (x, 0) be the first vertex occupied by \(\eta ^2\) in that axis. For each integer k, we will define \(X_k\) as the smallest nonnegative integer such that \((k,X_k)\) will be occupied by \(\eta ^1\). Similarly, \(Y_k\) is the smallest nonnegative integer such that \((k,Y_k)\) will be occupied by \(\eta ^1\). Now we analyze the evolution of \(X_k\); the one of \(Y_k\) will be analogous. Assume that \(X_1,X_2,\ldots ,X_{k1}=0\). Then, with probability at least p we have that \(X_{k+1}\ge 1\). When this happens, \(\eta ^1\) will need to do at least \(\frac{1}{1\lambda }\) steps before being able to occupy the axis again. However, for each \(s\ge 2\), the probability that \(X_{k+s}> X_{k+1}\) is at least p. This gives that the probability that the random variable X reaches value above 1 before going back to zero is at least \(1(1p)^{\frac{1}{1\lambda }}\). Once we have fixed p, by setting \(\lambda \) close enough to 1 we can make this probability very close to 1. This gives that \(X_k\) has a drift upwards. Since the downwards jumps of \(X_k\) are of size at most 1, this implies that at some time \(X_k\) will depart from 0 and will never return to it. A similar behavior happens for \(Y_k\), establishing (2). \(\square \)
3 Preliminaries on first passage percolation
Let \(\upsilon \) be a probability distribution on \((0,\infty )\) with no atoms and with a finite exponential moment. Consider a first passage percolation process \(\{\xi (t)\}_t\), which starts from the origin and spreads according to \(\upsilon \). More precisely, for each pair of neighboring vertices \(x,y\in \mathbb {Z}^d\), let \(\zeta _{x,y}\) be an independent random variables of distribution \(\upsilon \). The value \(\zeta _{x,y}\) is regarded as the time that \(\xi \) needs to spread throughout the edge (x, y). Note that \(\zeta \) defines a random metric on \(\mathbb {Z}^d\), where the distance between two vertices is the length of the shortest path between them, and the length of a path is the sum of the values of \(\zeta \) over the edges of the path. Hence, given any initial configuration \(\xi (0)\subset \mathbb {Z}^d\), the set \(\xi (t)\) comprises all vertices of \(\mathbb {Z}^d\) that are within distance t from \(\xi (0)\) according to the metric \(\zeta \). We assume throughout the paper that \(d\ge 2\).
For \(X\subset \mathbb {Z}^d\), let \(\mathbb {Q}_X^\upsilon \) be the probability measure induced by the process \(\xi \) when \(\xi (0)=X\). When the value of \(\xi (0)\) is not important, we will simply write \(\mathbb {Q}^\upsilon \), and when \(\upsilon \) is the exponential distribution of rate 1, we write \(\mathbb {Q}\).
Proposition 3.1
Proof
First we establish (7). Note that the event \(\big \{\inf _{x\in \partial ^\mathrm {i}\mathcal {B}_\upsilon \left( (1+\delta )t\right) }D(0,x; \zeta )\le t\big \}\) is measurable with respect to the passage times inside \(\mathcal {B}_\upsilon \left( (1+\delta )t\right) \). Then, if this event does not hold, that is under \(\left\{ \inf _{x\in \partial ^\mathrm {i}\mathcal {B}_\upsilon \left( (1+\delta )t\right) }D(0,x; \zeta )>t\right\} \), the event \(\left\{ \sup _{x\in \partial ^\mathrm {o}\mathcal {B}_\upsilon \left( (1\delta )t\right) }D(0,x; \zeta )\le t\right\} \) is also measurable with respect to the passage times inside \(\mathcal {B}_\upsilon \left( (1+\delta )t\right) \), establishing (7). The bound in (6) follows directly from Kesten’s result [18, Theorem 2]. \(\square \)
4 Encapsulation of competing first passage percolation
Here we consider two first passage percolation processes that compete for space as they grow through \(\mathbb {Z}^d\). One of the processes spreads throughout \(\mathbb {Z}^d\) at rate 1, while the other spreads according to a distribution \(\upsilon \) such that its limit shape is contained in \(\mathcal {B}\left( \lambda \right) \) for some \(\lambda <1\), with \(\lambda \) being a parameter of the system. We will say that \(\lambda \) is the rate of spread of the second process. We assume that the starting configuration of each process comprises only a finite set of vertices. In this case, one expects that both processes cannot simultanenously grow indefinitely; that is, one of the processes will eventually surround the other, confining it to a finite subset of \(\mathbb {Z}^d\). This was studied by Häggström and Pemantle [12]. In the proof of our main result, we will employ a refined version of a result in their paper. In particular, we will give a lower bound on the probability that the faster process surrounds the slower one within some fixed time.
First we define the processes precisely. Let \(\xi ^1\) denote the faster process so that, for each time \(t\ge 0\), \(\xi ^1(t)\) gives the set of vertices occupied by the faster process at time t. Similarly, let \(\xi ^2\) denote the slower process. For each neighbors \(x,y \in \mathbb {Z}^d\), let \(\zeta ^1_{x,y}\) be an independent exponential random variables of rate 1, and let \(\zeta ^2_{x,y}\) be an independent random variable of distribution \(\upsilon \). For \(i\in \{1,2\}\), \(\zeta ^i_{x,y}\) represents the passage time of process \(\xi ^i\) through the edge (x, y).
The proposition below is a more refined version of a result of Häggström and Pemantle [12, Proposition 2.2]. It establishes that if \(\xi ^2\) starts from inside \(\mathcal {B}\left( r\right) \) for some \(r\in \mathbb {R}_+\), and \(\xi ^1\) starts from a single vertex outside of a larger ball \(\mathcal {B}\left( \alpha r\right) \), for some \(\alpha >1\), then there is initially a large separation between \(\xi ^1\) and \(\xi ^2\), allowing \(\xi ^1\) to surround \(\xi ^2\) with high probability. Moreover, we obtain that \(\xi ^1\) will eventually confine \(\xi ^2\) to some set \(\mathcal {B}\left( R\right) \) for some given R, and the probability that this happens goes to 1 with \(\alpha \). We need to state this result in a high level of detail, as we will apply it at various scales later in our proofs. We say that an event is increasing (resp., decreasing) with respect to some passage times \(\zeta \) if whenever the event holds for \(\zeta \) it also holds for any passage times \(\zeta '\) that satisfies \(\zeta '_{x,y}\ge \zeta _{x,y}\) (resp., \(\zeta '_{x,y}\le \zeta _{x,y}\)) for all neighboring \(x,y\in \mathbb {Z}^d\).
Proposition 4.1
We defer the proof of the proposition above to “Appendix A”. The proof will follow along the lines of [12, Proposition 2.2], but we need to perform some steps with more care, as we need to obtain bounds on the probability that F occurs, to establish bounds on R and T, to derive that F is increasing with respect to \(\eta ^2\) and decreasing with respect to \(\zeta ^1\), and to obtain the measurability constraints on F.
We will need to apply the above proposition in a more complex setting. For this, it is important to keep in mind the process FPPHE defined in Sect. 1, where a cluster of type 2 starts spreading from each type 2 seed when that seed is activated, and type 2 seeds are initially distributed in \(\mathbb {Z}^d\) as a product measure. We will apply the encapsulation procedure of Proposition 4.1 for each different cluster of type 2 growing out of its seed. This means that we will apply Proposition 4.1 at several scales (that is, with different values of r) and at several places of \(\mathbb {Z}^d\). The encapsulation happening in one place may end up interfering in the spread of type 1 and type 2 in the other places.
In order to have a version of Proposition 4.1 that can handle this situation, we will focus in one such encapsulation. For that encapsulation, we represent type 1 as \(\xi ^1\), and assume that \(\xi ^1(0)\) contains at least one vertex from \(\partial ^\mathrm {o}\mathcal {B}\left( \alpha r\right) \). For the cluster of type 2 whose encapsulation we are considering, we let it start from \(\xi ^2(0)\subseteq \mathcal {B}\left( r\right) \). Here \(\xi ^2\) will only represent the cluster of type 2 that spreads from \(\xi ^2(0)\). For the other clusters of type 2, we will not refer to them as \(\xi ^2\) but simply as type 2.
 (P1)
For each \(\iota \), if \(\xi ^2\) does not enter \(\Pi _\iota \) from \(\xi ^2(0)\), then \(\Pi _\iota {\setminus } \Pi _\iota '\) becomes entirely occupied by \(\xi ^1\).
 (P2)
For any \(\iota ,x\in \partial ^\mathrm {i}\Pi _\iota ,\) and \(y\in \Pi _\iota \), either the time that \(\xi ^1\) takes to spread from x to y within \(\Pi _\iota \) is smaller than that given by the passage times \(\zeta ^1\), or y is occupied by type 2.
 (P3)
For any \(\iota ,x\in \partial ^\mathrm {i}\Pi _\iota ,\) and \(y\in \Pi _\iota \), either the time that \(\xi ^2\) takes to spread from x to y within \(\Pi _\iota \) is larger than that given by the passage times \(\zeta ^2\), or y is occupied by type 1.
The goal of the proposition below, which is a refinement of Proposition 4.1, is to argue that with high probability the passage times \(\zeta ^1,\zeta ^2\) are such that \(\xi ^1\) encapsulates \(\xi ^2\) inside a ball surrounding \(\mathcal {B}\left( r\right) \) unless there exists a set \(\Pi _\iota \) that does not satisfy one of the properties (P1)–(P3). Given three sets \(S_1,S_2,S_3\subset \mathbb {Z}^d\), we say that \(S_1\) separates \(S_2\) from \(S_3\) if any path in \(\mathbb {Z}^d\) from \(S_2\) to \(S_3\) intersects \(S_1\).
Proposition 4.2
5 Proof of Theorem 1.3
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.
5.1 Proof overview
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 nonencapsulated 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.
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 topright 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 multiscale 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 multiscale 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.
5.2 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
5.3 Multiscale setup
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.
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.
5.4 Definition of good boxes
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.
5.5 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
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
Proof
The next lemma shows that \(G_k^2(i)\) occurs with high probability.
Lemma 5.4
Proof
Proof of Lemma 5.2
5.6 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 \(\bigcup _{k\ge 1}I_k\).
Lemma 5.5
Proof
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 \(j1\). Thus, we have showed that all contagious vertices successfully handled at scale smaller than j are contained inside \(I_1\cup I_2 \cup \cdots \cup I_{j1}\). 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 \(Q_{j1}(i')\) that was treated at scale \(j1\); that is, \(i'\in \mathcal {I}_{j1}\). (For simplicity of the discussion, we assume here that this cell is of scale \(j1\), but it could be of any scale \(j'\le j1\).) Note that \(Q_{j1}(i')\) must be good for scale \(j1\) because otherwise cell i would not be treated at scale j. The fact that \(Q_{j1}(i')\) is good implies that the time \(\eta ^1\) takes to go from \(\partial ^\mathrm {i}Q_{j1}^{{\mathrm {outer}}/3}(i')\) to all points in \(\partial ^\mathrm {i}Q_{j1}^\mathrm {enc}(i')\), therefore encapsulating \(Q_{j1}(i')\), is smaller than the time given by the passage times \(\zeta _j^1\). Moreover, \(Q_{j1}(i')\) being good implies that the time \(\eta ^2\) takes to go from \(\partial ^\mathrm {i}Q_{j1}^{{\mathrm {outer}}/3}(i')\) to any point in \(\partial ^\mathrm {o}Q_{j1}^\mathrm {enc}(i')\) 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 \(\bigcup _{j''=1}^{j1} \bigcup _{i'' \in \mathcal {I}_{j''}} Q_{j''}^{\mathrm {outer}}(i'')\), 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 \(\eta ^2(t)\subset \bigcup _{\iota =1}^\infty I_\iota \).
5.7 Recursion
Lemma 5.6
Proof
Lemma 5.7
Proof
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
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.
5.8 Multiscale paths of infected sets
The idea behind the definitions above is that we will look at “paths” of multiscale 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
Proof
Now we define the type of multiscale paths we will consider.
Definition 5.10
 (i)

\(x \in Q_{k_1}^\mathrm {neigh2}(i_1),\)
 (ii)

\(\text {for any } j\in \{2,3,\ldots ,\ell \}\text { we have that } Q_{k_j}^\mathrm {neigh2}(i_j)\text { intersects } Q_{k_{j1}}^\mathrm {neigh2}(i_{j1}),\)
 (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)

\(\text {for any } j\in \{2,3,\ldots ,\ell \} ,\text { we have }k_j\ne k_{j1},\)
 (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 multiscale 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
Proposition 5.12
Proof
5.9 Completing the proof of Theorem 5.1
Proof of Theorem 5.1
6 From MDLA to FPPHE
Here we show how to use the proof scheme for FPPHE from Sect. 5 to establish Theorem 1.1. The relation between FPPHE and MDLA is very delicate, and we will need to introduce another process, which we call the hprocess. For the sake of clarity, this section is split into a few subsections.
6.1 Dual representation and Poisson clocks
We start by recalling the dual representation of the exclusion process. In this dual representation, vertices without particles are regarded as hosting another type of particle, called holes, while vertices hosting an original particle are seen as unoccupied. Using the terminology of the dual representation, in MDLA, holes perform a simple exclusion process among themselves, where they move as simple symmetric random walks obeying the exclusion rule (jumps to vertices already occupied by a hole or by the aggregate are suppressed). Then the growth of the aggregate is equivalent to a first passage percolation process which expands along its boundary edges at rate 1, but with the additional feature that the aggregate does not occupy vertices that are occupied by holes.
To be more precise, we now define MDLA in terms of Poisson clocks. A Poisson clock of rate \(\nu \) is a clock that rings infinitely many times, and such that the time until the first ring, as well as the time between any two consecutive rings, are given by independent exponential random variables of rate \(\nu \). Even though edges of \(\mathbb {Z}^d\) have so far always been considered as undirected, we will need to assign an independent Poisson clock of rate 1 to each oriented edge \((x \rightarrow y)\). Then the evolution of MDLA is as follows. When the clock of \((x \rightarrow y)\) rings, if x is occupied by a hole and y is unoccupied, the hole jumps from x to y. If x is occupied by the aggregate and y is unoccupied, then the aggregate occupies y. In any other case, nothing is done. Henceforth, the Poisson clocks used to construct MDLA will be referred to as the MDLAclocks.
6.2 MDLA with discovery of holes
We give a different representation of MDLA, which we refer to as MDLA with discovery of holes. Each vertex of \(\mathbb {Z}^d\) will either be occupied by the aggregate, be occupied by a hole, or be unoccupied. As before, the aggregate starts from the origin. However, unlike before, each vertex of \(\mathbb {Z}^d{\setminus }\{0\}\) is initially unoccupied, and is assigned a nonnegative integer value, which is given by an independent random variable having value \(i\ge 0\) with probability \((1\mu )^i\mu \). This value represents the number of holes that can be born at that vertex.

If x hosts a hole and y is unoccupied, the hole jumps from x to y.

If x belongs to the aggregate, y is unoccupied and the value of y is 0, then the aggregate occupies y.

If x belongs to the aggregate, y is unoccupied and the value of y is \(i\ge 1\), then the value of y is changed to \(i1\), a hole is born at y (so y becomes occupied), and the aggregate does not occupy y.

In any other case, nothing happens.
6.3 Backtracking jumps and overall strategy
 1.
The seeds of \(\eta _2\) are the vertices of value at least 1 in MDLA.
 2.
The aggregate contains \(\eta _1\) at all times.
 3.
For all \(t\ge 0\), the holes that have been discovered by time t are contained inside \(\eta _2(t)\).
To go around the above issue, we will employ a coupling argument to show that MDLA stochastically dominates FPPHE locally. In particular, we will use that coupling to show that the encapsulation procedure used for FPPHE (via Proposition 4.2) works as well for MDLA. Then, the multiscale machinery developed in Sect. 5 can be used to obtain that each cluster of holes get encapsulated by the aggregate at some finite (possibly large) scale. For this, we will use the fact that the encapsulation procedure we did for FPPHE in Proposition 4.2 is implied by the occurrence of a monotone event F, which is increasing with respect to the passage times of type 2 and decreasing with respect to the passage times of type 1.
6.4 Coupling of the initial configuration
We now formalize the coupling of the initial configurations of MDLA and FPPHE, as suggested in the previous section. For each vertex \(x\in \mathbb {Z}^d{\setminus }\{0\}\), note that the probability that x is assigned a value at least 1 in MDLA with discovery of holes is \(\sum _{i=1}^{\infty }(1\mu )^i \mu = 1\mu \). Then, we set \(p=1\mu \) so that we can couple the vertices with value at least 1 with the location of the type2 seeds of FPPHE. From now on, for each vertex of value at least 1, we will refer to it as a seed, regardless of whether we are talking about MDLA or FPPHE.
6.5 The hprocess
We will not actually couple MDLA with FPPHE, but we will couple MDLA with another process \(\{h_t\}_t\), which will be a growing subset of \(\mathbb {Z}^d\). We call this process the hprocess. The hprocess will be constructed using the MDLAclocks and the seeds, where the seeds have been coupled with MDLA as described in Sect. 6.4.
When a vertex x belongs to \(h_t\), we will say that x is infected. To avoid confusion, we will not say that x is occupied by \(h_t\) since, as we explain later, a vertex that is occupied by the aggregate can also be infected. Our goal with the hprocess is to obtain that the holes that have already been discovered at time t are contained in \(h_t \cup \partial ^\mathrm {o}h_t\), and the ones in \(\partial ^\mathrm {o}h_t\) are the holes that will jump back to \(h_t\) (in a backtracking jump).

Birth. If at time t a hole is discovered by the aggregate inside a cluster \(C\subset \mathbb {Z}^d\) of seeds,^{2} then we infect C; that is, we add to \(h_t\) the whole cluster C of seeds.

Expansion. For each unoriented edge (x, y), we will define a passage time \(\tau _{x,y}\) (which we will specify later on and will depend on the evolution of MDLA). So if x gets infected at time t, then x infects y at time \(t+\tau _{x,y}\); note that y could get infected before \(t+\tau _{x,y}\) if a neighbor of y different than x infects y.

Halting upon encapsulation. The hprocess is allowed to infect vertices that are occupied by the aggregate. However, if at some moment a cluster C of \(h_t\) is separated from infinity by the aggregate, which means that any path from C to infinity intersects \(\mathcal {A}_t\), then \(h_t\) will not infect any vertex of \(\partial ^\mathrm {o}C\) that already belongs to the aggregate.^{3} This is to guarantee that a cluster of the hprocess is confined to a finite set when it gets encapsulated by the aggregate.
6.6 Evolution of the hprocess
Here we will describe how the hprocess uses the MDLAclocks to evolve. Our description here will be precise, but will not enter in the details needed to define the passage times of the hprocess. This will be carried out in Sect. 6.7.
Start from time 0, where we have \(h_0=\emptyset \) and \(\mathcal {H}(t)=\emptyset \). From this time, we let MDLA evolve using its MDLAclocks. If at some time t MDLA tries to occupy a seed (that is, MDLA discovers a hole) inside some cluster \(C\subset \mathbb {Z}^d\) of seeds, the hprocess undergoes a birth operation and we set \(h_t=C\). At this moment, we continue to let MDLA evolve using its MDLAclocks. If new holes are discovered, new births take place and clusters are added to the hprocess (that is, new clusters are infected).
We will now discuss all possibilities that could happen for the expansion of the hprocess. In all cases below, we will assume that the expansion of the hcluster is happening in an infected cluster that is not disconnected from infinity by the aggregate. Otherwise, the halting upon encapsulation would already have happened to that cluster, which would prevent it from expanding.
The hprocess only expands when an edge at the boundary of the hprocess or an edge from \(\mathcal {H}_B\) rings. (The edges in \(\mathcal {H}_B\) are needed to verify backtracking jumps, and “B” in the subscript actually stands for backtracking.) If an edge that is internal to the hprocess (that is, both of its endpoints are already infected) rings, then the hprocess does not change, even if that causes new holes to be discovered. Similarly, if an edge that is external to the hprocess (that is, both endpoints are not infected) and does not belong to \(\mathcal {H}_B\) rings, and no new hole get discovered by this operation, then the hprocess does not change.
Now we assume that an edge \((x\rightarrow y)\) from the boundary of the hprocess or from \(\mathcal {H}_B\) rings at time t, and discuss what occurs with the hprocess at this time. We split our discussion into three cases, and at the end explain two particular situations. Let \(s<t\) be the last time before t that the hprocess changed.
6.6.1 Case 1: A hole jumps out of the hprocess
This corresponds to \((x\rightarrow y)\in \partial ^\mathrm {o}h_s\) with a hole at \(x\in h_s\) and \(y\not \in h_s\) unoccupied at time \(t\). Thus, the ring of \((x\rightarrow y)\) causes the hole to jump from x to y.
It could be the case that there is already an edge \((y'\rightarrow y)\in \mathcal {H}(t)\) with \(y'\ne x\). If this is the case, we simply do nothing; this case will be better discussed in Sect. 6.6.5. Otherwise, if there is no such edge, we add \((x\rightarrow y)\) to \(\mathcal {H}(t)\) to verify whether the hole will do a backtracking jump to x. Moreover, we draw (independently from previous values that this random variable could have assumed) the Bernoulli random variable \(\mathfrak {B}_{x\rightarrow y}\) of parameter 1 / M. If \(\mathfrak {B}_{x\rightarrow y}=0\), which means that the hole will not backtrack to x, then we infect y at time t.
6.6.2 Case 2: Verification of backtracking jumps
This corresponds to \((x\rightarrow y)\in \mathcal {H}_B(s)\). Let \((u\rightarrow v)\) be the edge from \(\mathcal {H}(s)\) such that \((x\rightarrow y)\in \mathcal {E}_{u\rightarrow v}\subset \mathcal {H}_B(s)\), and assume that \((u\rightarrow v)\) was added to \(\mathcal {H}\) at time \(t'\le s\). Note that, from Sect. 6.6.1, this was done because a hole jumped from u to v at time \(t'\). Then while no clock from \(\mathcal {E}_{u \rightarrow v}\) rings, u will remain unoccupied and the hole will remain at v.
When the clock of an edge \((x\rightarrow y)\) from \(\mathcal {E}_{u \rightarrow v}\) rings at time t, then the first thing we do is to remove \((u\rightarrow v)\) from \(\mathcal {H}(t)\). We will say that the possibility of a backtracking jump through \((u\rightarrow v)\) has been verified. We then couple the value of \(\mathfrak {B}_{u\rightarrow v}\) with the MDLAclocks so that if \(\mathfrak {B}_{u\rightarrow v}=1\), we have that the first clock to ring among the ones from \(\mathcal {E}_{u \rightarrow v}\) is \((v \rightarrow u)\). If this happens, then \((x\rightarrow y)=(v\rightarrow u)\) so, at time t, the hole backtracks to u. Note that both u and v are already infected. In this case, nothing else needs to be done.
Finally, if \((x \rightarrow y)\in \mathcal {E}_v^\rightarrow \), then the hole at \(v=x\) may jump to y, if y is unoccupied. If, in addition, we have that \(y\not \in h_s\), then the hole jumped out of the hprocess, so we perform the steps described in Sect. 6.6.1 to \((x\rightarrow y)\) so that we can later verify the possibility of a backtracking jump through \((x\rightarrow y)\). In particular, we add \((x \rightarrow y)\) to \(\mathcal {H}(t)\), sample \(\mathfrak {B}_{x\rightarrow y}\), and infect y if \(\mathfrak {B}_{x\rightarrow y}=0\).
6.6.3 Case 3: Expansion without jump of holes

\(x\in h_{s}\) is unoccupied at time \(t\).

x is occupied by the aggregate at time \(t\).

x is occupied by a hole, but y is occupied by either a hole or the aggregate at time \(t\) (preventing the hole from x to jump to y).
6.6.4 Edges in \(\mathcal {H}^*_B\)
To solve this, for each \((x\rightarrow y)\in \mathcal {H}_B^*(t)\), we will consider two Poisson clocks: the actual MDLAclock, which will be associated to the backtracking jump of \((y \rightarrow y')\), and a fakeclock, which will be associated to the backtracking jump of \((x' \rightarrow x)\). So, if \(\mathfrak {B}_{x' \rightarrow x}=1\), it means that the clock of \((x \rightarrow x')\) will ring before the MDLAclocks of \(\mathcal {E}_{x'\rightarrow x}{\setminus } (x\rightarrow y)\) and before the fakeclock of \((x \rightarrow y)\). Similarly, if \(\mathfrak {B}_{y \rightarrow y'}=1\), it means that the clock of \((y' \rightarrow y)\) will ring before the MDLAclocks of \(\mathcal {E}_{y \rightarrow y'}\). The evolution of MDLA simply ignores the fakeclocks. Since the fakeclocks and the MDLAclocks are independent, there is no conflict with the independence of \(\mathfrak {B}_{x' \rightarrow x}\) and \(\mathfrak {B}_{y \rightarrow y'}\). Now we explain why this does not create other problems.
If the MDLAclock of \((x \rightarrow y)\) rings, we say that a clock from \(\mathcal {E}_{y\rightarrow y'}\) rings, whereas when the fakeclock of \((x \rightarrow y)\) rings, we say that a clock from \(\mathcal {E}_{x'\rightarrow x}\) rings. Assume that the first clock to ring among the MDLAclocks and fakeclocks of \(\mathcal {E}_{x\rightarrow y}\) is the MDLAclock of \((x \rightarrow y)\). Let \(s>t\) be the time at which that clock rings, and assume that this is the first clock to ring among the clocks of \(\mathcal {E}_{x'\rightarrow x}\) and \(\mathcal {E}_{y\rightarrow y'}\). Note that in this case we have \(\mathfrak {B}_{y \rightarrow y'}=0\). Then, the hole that is in x jumps to y, and we perform the steps described in Sect. 6.6.2 for the backtracking jump of \((y \rightarrow y')\) when \(\mathfrak {B}_{y \rightarrow y'}=0\). No action is taken with regards to the backtracking jump of \((x'\rightarrow x)\). In particular, we have that \((y\rightarrow y') \not \in \mathcal {H}(s)\) and, more crucially, we have that \((x'\rightarrow x) \in \mathcal {H}(s)\) even if there is no hole at x. (This is the reason why in (53) we have not required y to host a hole.)
The fact that \((x'\rightarrow x)\) remained in \(\mathcal {H}(s)\) will not cause problems because the hole that was in x jumped inside \(h_s\) (because \(y\in h_t \subseteq h_s\)). So, in some sense, that hole did backtrack to the hprocess. We will later still process the backtracking jump of \((x'\rightarrow x)\) even if there may not be a hole at x (which just means that no hole will jump, but the hprocess may still be updated according to the decision of a backtracking jump). For example, if \(\mathfrak {B}_{x' \rightarrow x}=1\), we will assume that there is a backtracking jump over \((x'\rightarrow x)\) causing x not to be added to the hprocess, which remains to be true even if x does not host a hole.
6.6.5 Holes revisiting uninfected vertices
Consider the setting in the previous section, where \((x'\rightarrow x)\in \mathcal {H}(s)\) but there is no hole at x. Note that if \(\mathfrak {B}_{x'\rightarrow x}=1\) (so that \(x\not \in h_{s}\)), before the backtracking jump of \((x'\rightarrow x)\) is processed (that is, before the MDLAclocks of \(\mathcal {E}_{x'\rightarrow x}{\setminus } (x\rightarrow y)\) and the fakeclock of \((x\rightarrow y)\) ring), it could happen that a hole jumps from a vertex \(x''\) to x. Note that \(x'' \ne x'\), because \(x'\) remained unoccupied from the time the hole jumped from \(x'\) to x since \((x'\rightarrow x)\) is still in \(\mathcal {H}\). Since \((x''\rightarrow x)\not \in \mathcal {E}_{x'\rightarrow x}\), this jump does not affect the backtracking jump of \((x'\rightarrow x)\). But since x is not infected, the hole just jumped out of the hprocess. Suppose that this happens at some time \(s''\). This is the situation explained in Sect. 6.6.1, when nothing needs to be done to the hprocess. The reason is that this hole just occupied the place of the yet to be verified backtracking jump of \((x'\rightarrow x)\). Thus we only need to wait that backtracking to be processed.
Note that it can also happen that much later x is still not infected and a hole jumps from \(x'\) to x again. But, as we explained above, this can only happen after the initial backtracking jump of \((x'\rightarrow x)\) has been proceed. Since we had that \(\mathfrak {B}_{x'\rightarrow x}=1\) (so that x remained uninfected), the second jump of a hole from \(x'\) to x can only happen after the clock of the edge \((x\rightarrow x')\) rings (because this happens before any edge from \(\mathcal {E}_{x'}^{\leftarrow }\) rings, so \(x'\) remained unoccupied). When the edge \((x\rightarrow x')\) rings, the memoryless property of exponential random variables guarantees that the rings of the clocks in \(\mathcal {E}_{x'\rightarrow x}\) are from this moment independent of the past. Note that at this time it also happens that the value of \(\mathfrak {B}_{x'\rightarrow x}\) is redrawn independently, so this new hole jumping to x will have no correlation with the previous backtracking jump through \((x'\rightarrow x)\).
6.7 Construction of the passage times of the hprocess
In the previous section we described how the hprocess uses the MDLAclocks to evolve. Here we will use the discussion from the previous section to describe the construction of the passage times of the hprocess.
 1.
A birth operation takes place (see the definition in Sect. 6.5). Note that in this case new vertices are added to the hprocess, so \(h_{t+W}\ne h_t\).
 2.
A clock (regardless of whether it is a MDLAclock or a fakeclock) from \(\partial ^\mathrm {e}h_t \cup \mathcal {H}_B(t)\) rings. In this case, we will not yet observe which of these clocks rang. We will call this case a potential expansion operation.
6.7.1 Birth operation or addition of vertices to the hprocess
6.7.2 Potential expansion operation of the hprocess
Suppose that at time \(t+W\) an edge from \(\partial ^\mathrm {e}h_t \cup \mathcal {H}_B(t)\) rings. The hprocess will expand according to the description in Sect. 6.6, but we elaborate a little bit more here. We do not immediately observe which is the edge that rings, this edge is still random and could also correspond to a fakeclock from an edge of \(\mathcal {H}_B^*(t)\).
We can now state precisely how \((x\rightarrow y)\) is chosen. Recall that if \((x' \rightarrow y')\in \mathcal {H}_B^*(t)\) with \((x'\rightarrow y')\in \mathcal {E}_{x''\rightarrow x'}\cap \mathcal {E}_{y' \rightarrow y''}\), then we say that \((x' \rightarrow y')\in \mathcal {E}_{x'' \rightarrow x'}\) only if it was the fakeclock of \((x'\rightarrow y')\) that rang, and say that \((x' \rightarrow y')\in \mathcal {E}_{y' \rightarrow y''}\) if it was the MDLAclock of \((x'\rightarrow y')\) that rang. Then, if \((x' \rightarrow y')\in \mathcal {E}_{u\rightarrow v}\) for some \((u\rightarrow v)\in \mathcal {H}(t)\) with \(\mathfrak {B}_{u\rightarrow v}=1\), set \((x\rightarrow y)=(v \rightarrow u)\). If \((x' \rightarrow y')=(v\rightarrow u)\) for some \((u\rightarrow v)\in \mathcal {H}(t)\) with \(\mathfrak {B}_{u\rightarrow v}=0\), then \((x\rightarrow y)\) is chosen uniformly at random from \(\mathcal {E}_{u\rightarrow v}{\setminus } (v\rightarrow u)\). In any other case, \((x\rightarrow y)=(x'\rightarrow y')\).

\((x\rightarrow y) \in \partial ^\mathrm {e}h_t {\setminus } \mathcal {H}_B(t)\), with x not hosting a hole in MDLA or y being occupied in MDLA. This was described in Sect. 6.6.3, where y gets infected with probability \(\frac{M1}{M}\).

\((x\rightarrow y) \in \partial ^\mathrm {e}h_t {\setminus } \mathcal {H}_B(t)\), with x hosting a hole in MDLA, y unoccupied, there exists no other edge \((\cdot \rightarrow y)\in \mathcal {H}(t)\), and \(\mathfrak {B}_{x\rightarrow y}=0\). This is the case that a hole jumps out of \(h_t\) (from x to y) and does not backtrack to x. This was described in Sects. 6.6.1 and 6.6.2.

\((x\rightarrow y) \in \mathcal {E}_{u\rightarrow v}\) for some \((u\rightarrow v)\in \mathcal {H}(t)\) with \(\mathfrak {B}_{u\rightarrow v}=0\) (so \(v\in h_t\) and \((x\rightarrow y)\ne (v\rightarrow u)\)). As mentioned in (52), the probability that \((x\rightarrow y)\) is equal to a given \((w\rightarrow z)\in \mathcal {E}_{u\rightarrow v}{\setminus } (v\rightarrow u)\) is \(\frac{1}{M1}\). If, in addition, we have that \((x\rightarrow y) \in \mathcal {E}_v^\rightarrow \), so \(x=v\), we may infect y if there is still a hole at \(x=v\) and \(\mathfrak {B}_{x\rightarrow y}=0\). (Note that this actually falls into the setting of the previous case, but we chose to highlight it here since a given edge from \(\mathcal {E}_v^\rightarrow \) rings at rate \(\frac{M}{M1}\) instead of rate 1, due to the conditioning on \(\mathfrak {B}_{x\rightarrow y}=0\).)
6.8 Properties of the passage times
Before establishing properties of the passage times, we establishe (53).
Lemma 6.1
For any \(t\ge 0\), (53) holds.
Proof
Clearly (53) holds at time 0 since \(\mathcal {H}(0)=\emptyset \). Now assume that it holds during [0, t), and that at time t a hole jumps out of \(h_{t}\), from x to y; so \((x\rightarrow y)\) is added to \(\mathcal {H}(t)\) via Case 1 (cf. Sect. 6.6.1) and \(x\in h_t\). Then, at time t, x is not occupied by a hole or the aggregate in MDLA and belongs to \(h_t\), while y hosts a hole in MDLA. If \(\mathfrak {B}_{x\rightarrow y}=0\), y gets infected at time t and (53) continues to hold. If \(\mathfrak {B}_{x\rightarrow y}=1\), y remains uninfected, but x remains unoccupied until at a time \(s>t\) the clock of an edge from \(\mathcal {E}_{x}^\leftarrow \) rings, but that edge must be \((y\rightarrow x)\) since \(\mathfrak {B}_{x\rightarrow y}=1\). So the hole gets back to an infected vertex.
It remains to show that the edges in \(\mathcal {H}(t)\) are disjoint. Assume that this is not the case; that is, there are edges \((x\rightarrow y),(u\rightarrow v)\in \mathcal {H}(t)\) with \(\{u,v\}\cap \{x,y\}=1\). Assume that \((x\rightarrow y)\) and \((u\rightarrow v)\) were added to \(\mathcal {H}\) at times \(t_x\) and \(t_u\), respectively, with \(t_x>t_u\). If \(u=x\), then at some time during \((t_u,t_x)\) a hole jumped into \(u=x\) in order to go to y at time \(t_x\). But this would cause \((u\rightarrow v)\) to be removed from \(\mathcal {H}\) by Case 2, which would imply that \((u\rightarrow v)\not \in \mathcal {H}(t_x)\). If \(v=y\), then at time \(t_x\) we have \(y\not \in h_{t_x}\); otherwise we would not add \((x\rightarrow y)\) to \(\mathcal {H}(t_x)\). In this case, since y does not host a hole at time \(t_x\), Case 1 gives that \((x\rightarrow y)\) is not added to \(\mathcal {H}(t_x)\) because there is already an edge \((\cdot \rightarrow y) \in \mathcal {H}(t_x)\). The case \(u=y\) cannot happen, because \((x\rightarrow y)\in \mathcal {E}_{u}^\leftarrow \), so Case 2 would remove \((u\rightarrow v)\) from \(\mathcal {H}(t_x)\). The case \(v=x\) is similar, since \((x\rightarrow y)\in \mathcal {E}_{v}^\rightarrow \), so Case 2 would remove \((u\rightarrow v)\) from \(\mathcal {H}(t_x)\). . \(\square \)
Lemma 6.2
For any \(M> 0\), \(Z^{(M)}\) stochastically dominates (strictly) an exponential random variable of rate 1. Moreover, \(Z^{(M)}\) is stochastically dominated by an exponential random variable of rate \(\frac{M1}{M}\).
Proof
For the first part, note that if we take (62) and replace \(Z''\) with \({\widehat{Z}}\), where \({\widehat{Z}}\) is an exponential random variable of rate 1, then Lemma B.3 with \(k=M\), \(W=Z'\) and \(X_1={\widehat{Z}}\) (the values of \(X_2,X_3,\ldots ,X_M\) being irrelevant) gives that \(Z' + {{\mathfrak {1}}}\left( Q=1\right) {\widehat{Z}}\) is an exponential random variable of rate 1. Since \(Z''\) stochastically dominates \({\widehat{Z}}\), we obtain the first part of the lemma.
For the second statement, we know from Lemma B.3 and the first part that \(Z' + {{\mathfrak {1}}}\left( Q=1\right) {\widehat{Z}}\) is an exponential random variable of rate 1. Then, using Lemma B.2, we have that \(\frac{M}{M1}Z' + {{\mathfrak {1}}}\left( Q=1\right) \frac{M}{M1}{\widehat{Z}}\) is an exponential random variable of rate \(\frac{M1}{M}\). But that variable has the same distribution as \(\frac{M}{M1}Z' + {{\mathfrak {1}}}\left( Q=1\right) Z''\ge Z^{(M)}\). \(\square \)
Now we show that the passage times \(\tau _{x,y}\) stochastically dominate i.i.d. random variables distributed as \(Z^{(M)}\).
Lemma 6.3
The collection of passage times \(\tau _{x,z}\), for each edge (x, y), stochastically dominates an i.i.d. collection of random variables distributed as \(Z^{(M)}\).
Proof
If an edge (w, z) is such that its passage time was not completed during the procedure above, then we know that its passage time stochastically dominates an independent, exponential random variable of rate \(\frac{M1}{M}\), which in turn stochastically dominates \(Z^{(M)}\) by the second part of Lemma 6.2.
So now we consider a given edge (w, z), whose passage time was completely constructed during the procedure described in Sects. 6.6 and 6.7. Let \(t_w<t_z\) be the times such that w is infected at time \(t_w\) and z is infected at time \(t_z\). The passage time of (w, z) is then completed at time \(t_z\) and is \(t_zt_w\).
Now we split the proof into two parts. In the first part, assume that w does not host a hole at time \(t_w\) (for example, because it was infected according to Case 3, Sect. 6.6.3). The crucial property we will use in this case is that \((w\rightarrow z)\) cannot belong to \(\mathcal {H}_B\) during \([t_w,t_z)\) since an edge \((z\rightarrow \cdot )\) cannot be in \(\mathcal {H}\) as z is not infected and an edge \((\cdot \rightarrow w)\) cannot be in \(\mathcal {H}\) since w was infected without a hole. As the hprocess evolves from \(t_w\), at each step, we add W to \(\Lambda _{w,z}\), until we find a step where the clock that rings is the one of \((w\rightarrow z)\). This is the procedure described in Lemma B.3 for the construction of independent exponential random variables of rate 1. Hence, at the time the clock of \((w \rightarrow z)\) rings, call this time s, we have that \(\Lambda _{w,z}\) is an exponential random variable of rate 1, which is then added to \(\Pi _{w,z}\). If at time s we fall into the setting of Case 3 (Sect. 6.6.3), then we only infect z with probability \(\frac{M1}{M}\); otherwise we wait for the next time the clock of \((w\rightarrow z)\) rings, adding another exponential random variable of rate 1 to the list \(\Pi _{w,z}\), and iterating this procedure. If at time s we fall into the setting of Case 1 (Sect. 6.6.1), we only infect z if \(\mathfrak {B}_{w \rightarrow z}=0\), which occurs with probability \(\frac{M1}{M}\); otherwise we iterate this procedure since the hole will jump back to w (or to the infected set before \((z\rightarrow w)\) rings). Therefore, each element of the list \(\Pi _{w,z}\) is an independent random variable of rate 1, and the number of elements is given by a geometric random variable of success probability \(\frac{M1}{M}\). Lemma B.1 gives that \(\tau _{w,z}\) is in this case an exponential random variable of rate \(\frac{M1}{M}\). Since this stochastically dominates \(Z^{(M)}\) by Lemma 6.2, this first part is completed.
Now, for the second part, assume that w hosts a hole at time \(t_w\). Assume that that hole jumped to w from a vertex \(w'\). The first situation to imagine is that the hole jumped from \(w'\) to w at time \(t_w\), which causes \((w' \rightarrow w)\) to be added to \(\mathcal {H}(t_w)\); thus \(\mathfrak {B}_{w'\rightarrow w}=0\). But, it could also be the case that \(\mathfrak {B}_{w'\rightarrow w}=1\). For this to happen, the hole must have jumped from \(w'\) to w at a time \(t_w'<t_w\), which caused \((w' \rightarrow w)\) to be added to \(\mathcal {H}(t_w')\) and w not to be added to \(h_{t_{w'}}\). Then, in a time \(t_w\), the clock of an edge \((w'' \rightarrow w)\) (which is not part of \(\mathcal {E}_{w'\rightarrow w}\)) rings at time \(t_w\) when w is still occupied by a hole, triggering Case 3, which decides to infect w. Regardless of which of the two situations above occurs, we know that \((w \rightarrow z)\in \mathcal {E}_{w'\rightarrow w}\).
If \(\mathfrak {B}_{w'\rightarrow w}=1\), then we know the hole will do a backtracking jump from w to \(w'\) before the clock of \((w\rightarrow z)\) rings, which will cause \((w'\rightarrow w)\) to be removed from \(\mathcal {H}\). At this point, w will not have a hole anymore and we may proceed as in the first part of the proof, which implies that the passage time \(\tau _{w,z}\) stochastically dominates an exponential random variable of rate \(\frac{M1}{M}\).
The most delicate case is when \(\mathfrak {B}_{w'\rightarrow w}=0\). Suppose that at time s a clock from \(\mathcal {E}_{w'\rightarrow w}\) rings. Note that, since \(\mathcal {E}_{w'\rightarrow w}=M\), we will have that \(\Lambda _{w,z}=st_w\) is distributed as an exponential random variable of rate M. Let \((x\rightarrow y)\) be the edge whose clock rang at time s. Then a few cases may happen.
Case A: \((x \rightarrow y)=(w\rightarrow z)\). This happens with probability \(\frac{1}{M1}\). When this is the case, we will have to do the steps of Case 2 plus Case 1, which causes \((w\rightarrow z)\) to be added to \(\mathcal {H}(s)\). Two subcases can then happen.
Case A.1: With probability \(\frac{M1}{M}\) we have \(\mathfrak {B}_{w\rightarrow z}=0\), so the hole at z will not do a backtracking jump to w; hence we infect z at time s. Therefore, with overall probability \(\frac{1}{M1} \times \frac{M1}{M}= \frac{1}{M}\) we obtain that the passage time of (w, z) is completed at time s, which implies that \(\tau _{w,z}\) is an exponential random variable of rate M.
Case A.2: With probability \(\frac{1}{M}\) we have \(\mathfrak {B}_{w\rightarrow z}=1\), so the hole will do a backtracking jump to w and we will not infect z. From this time onwards, the passage time of (w, z) is computed exactly as in the first part of the lemma. Therefore, from s, we will wait a time that is distributed according to an exponential random variable of rate \(\frac{M1}{M}\) to complete the passage time of (w, z). This gives that \(\tau _{w,z}\) is the sum of an exponential random variable of rate M plus an exponential random variable of rate \(\frac{M1}{M}\).
Case B: \((x\rightarrow y)\ne (w\rightarrow z)\). This happens with probability \(\frac{M2}{M1}\) and concludes the processing of the backtracking jump of \((w' \rightarrow w)\). From this time onwards, the passage time of (w, z) is computed exactly as in the first part of the proof. This gives that \(\tau _{w,z}\) is the sum of an exponential random variable of rate M plus an exponential random variable of rate \(\frac{M1}{M}\).
This concludes all cases in this second part. Then, the probability that \(\tau _{w,z}\) is given by the sum of an exponential random variable of rate M and an exponential random variable of rate \(\frac{M1}{M}\) is the probability that case A.1 does not happen, which is \(1\frac{1}{M}\). Therefore, \(\tau _{w,z}\) is distributed as \(Z^{(M)}\) from (62).
The independence of \(\tau \) across different edges follows from the fact that elements are added to each list \(\Pi _{x,y}\) only when the edge (x, y) rings, and edges ring independently of one another. \(\square \)
6.9 Concluding the proof of Theorem 1.1
The proof will rely on a result by van den Berg and Kesten [28] about strict inequalities for first passage percolation. We will state here a version that is adapted to our needs. In the one hand, it is a special case of the main result in [28], as we explain in Remark 6.5 below. On the other hand, the result we need does not follow from the theorems in [28], but follows from the proof there, without changing a single word. We will not repeat the full proof of [28] here, but will give a description of the main steps.
Recall that \(\mathcal {B}_\upsilon =\mathcal {B}_\upsilon \left( 1\right) \) denotes the ball of radius 1 according to the norm given by a first passage percolation with passage times given by i.i.d. random variables of distribution \(\upsilon \). Recall also that we drop the subscript \(\upsilon \) when \(\upsilon \) is the exponential distribution of rate 1.
Proposition 6.4
Proof
The proof in [28] goes via a renormalization argument. First define a fixed, but large value \(\ell \) and partition \(\mathbb {Z}^d\) into cubes of sidelength \(\ell \). Then they say that a cube R is good if a certain “good” event happens. The good event is such that for any given path P of \(\mathrm {FPP}_\upsilon \) inside P, there is a positive probability (uniformly over P and R) such that can find an alternative path \(P'\) which differs very little from P and such that the time \(\mathrm {FPP}_1\) takes to traverse \(P'\) is at most the time that \(\mathrm {FPP}_\upsilon \) takes to traverse P minus a fixed value \(\delta >0\). Then a percolation argument (which is by now quite standard) gives that the set of good cubes percolates on \(\mathbb {Z}^d\). This means that any long enough path on \(\mathbb {Z}^d\), say of size n, must pass through a number of good cubes of order n. (Here \(\ell \) and \(\delta \) are fixed, while n can grow.) Then, we can consider the geodesic path P that \(\mathrm {FPP}_\upsilon \) takes to go from 0 to \(x_n\), and using the above reasoning we obtain an order of n good cubes R such that P has a long piece inside R. Then, for each good cube, with positive probability we can replace the long pieces of P within the good cube with the alternative path provided by the definition of good cubes, which produces a path from 0 to \(x_n\) whose passage time in \(\mathrm {FPP}_1\) is faster than that of \(\mathrm {FPP}_\upsilon \) by a factor of order n. This implies, for example, that one obtains a value \(\delta '>0\), depending only on \(\upsilon \), for which \(\nu \le (1\delta ) \nu _\upsilon \). This is the argument in [28].
Then one can obtain two continuous functions \({\bar{\nu }},{\bar{\nu }}_\upsilon \) from \(\mathbb {R}^d\) to \(\mathbb {R}_+\) by taking the unique continuous extension of \(\nu ,\nu _\upsilon \). Since \(\delta ''>0\) uniformly on the choice of q, we have the existence of \(\epsilon >0\) for which \(\mathcal {B}_\upsilon \subset \mathcal {B}(1\epsilon )\). \(\square \)
Remark 6.5
The result in [28] is in some sense more general than stated above since instead of requiring stochastic domination, it just requires that \(\upsilon \) is less variable than an exponential random variable of rate 1; but we will not require such level of generality.
Proof of Theorem 1.1
Using Lemma 6.3, we have that the hprocess grows slower than if the red seeds grows clusters of first passage percolation with passage times distributed as \(Z^{(M)}\). Let \(\upsilon \) be the distribution of \(Z^{(M)}\). Then, by Proposition 6.4, we have the existence of \(\lambda <1\) such that \(\mathcal {B}_\upsilon \subseteq \mathcal {B}(\lambda )\). Then, performing the whole multiscale procedure described in Sect. 5 and using the encapsulation procedure of Sect. 4, we obtain that with positive probability the set of sites that are not infected by the hprocess and that are occupied by the aggregate grows indefinitely and contains a ball (up to regions separated from infinity by this set of sites).
Note that, in the hprocess, whole clusters of seeds are added when the aggregate tries to occupy a seed, while in FPPHE seeds are activated one by one. This is not a problem. In fact, the same proof works if in FPPHE the activation of a seed implies the activation of its whole cluster of seeds. The reason is that, in the encapsulation procedure of Proposition 4.2, we already assumed that \(\xi ^2(0)\) starts from time 0 from any subset of a ball \(\mathcal {B}\left( r\right) \) of radius r. \(\square \)
Footnotes
 1.
 2.
For the sake of clarify, given any set of vertices \(X\subset \mathbb {Z}^d\), when we say that C is a cluster of X, we mean that \(C\subset X\) and \(\partial ^\mathrm {o}C \cap X = \emptyset \).
 3.
If the aggregate decides to occupy a site x at the same time as the hprocess decides to infect x, then we assume that the aggregate occupies x immediately before the hprocess infects x. This is just a convenience to take care of the following situation. Assume that x is a neighbor of a cluster of infected sites which is disconnected from infinity by the aggregate but x itself is not disconnected from infinity by the aggregate. This implies that the aggregate occupies the neighbors of x that belong to the infected cluster. If at this time the aggregate and the hprocess both decide to occupy x, we obtain that the aggregate does so first, and then the hprocess does not infect x due to the halting upon encapsulation operation.
Notes
References
 1.Auffinger, A., Damron, M., Hanson, J.: 50 years of first passage percolation. American Mathematical Society, New York (2017)zbMATHGoogle Scholar
 2.Barlow, M.T.: Fractals, and diffusionlimited aggregation. Bull. Sci. Math. 117(1), 161–169 (1993)MathSciNetzbMATHGoogle Scholar
 3.Barlow, M.T., Pemantle, R., Perkins, E.A.: Diffusionlimited aggregation on a tree. Probab. Theory Relat. Fields 107(1), 1–60 (1997)MathSciNetzbMATHGoogle Scholar
 4.Barra, F., Davidovitch, B., Levermann, A., Procaccia, I.: Laplacian growth and diffusion limited aggregation: different universality classes. Phys. Rev. Lett. 87(13), 134501 (2001)Google Scholar
 5.Benjamini, I., Yadin, A.: Diffusion limited aggregation on a cylinder. Commun. Math. Phys. 279(1), 187–223 (2008)MathSciNetzbMATHGoogle Scholar
 6.Candellero, E., Stauffer, A.: Coexistence of competing first passage percolation on hyperbolic graphs (2018). arXiv:1810.04593
 7.Carleson, L., Makarov, N.: Aggregation in the plane and Loewner’s equation. Commun. Math. Phys. 216(3), 583–607 (2001)MathSciNetzbMATHGoogle Scholar
 8.Chayes, L., Swindle, G.: Hydrodynamic limits for onedimensional particle systems with moving boundaries. Ann. Probab. 24(2), 559–598 (1996)MathSciNetzbMATHGoogle Scholar
 9.Cox, J.T., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)MathSciNetzbMATHGoogle Scholar
 10.Eldan, R.: Diffusionlimited aggregation on the hyperbolic plane. Ann. Probab. 43(4), 2084–2118 (2015)MathSciNetzbMATHGoogle Scholar
 11.Grimmett, Geoffrey, Kesten, Harry: Firstpassage percolation, network flows and electrical resistances. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 66(3), 335–366 (1984)MathSciNetzbMATHGoogle Scholar
 12.Häggström, O., Pemantle, R.: Absence of mutual unbounded growth for almost all parameter values in the twotype Richardson model. Stoch. Process. Appl. 90(2), 207–222 (2000)MathSciNetzbMATHGoogle Scholar
 13.Hastings, M.B., Levitov, L.S.: Laplacian growth as onedimensional turbulence. Phys. D 116, 244–252 (1998)zbMATHGoogle Scholar
 14.Witten Jr., T.A., Sander, L.M.: Diffusionlimited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47(19), 1400–1403 (1981)Google Scholar
 15.Kassner, K.: Pattern Formation in DiffusionLimited Crystal Growth. World Scientific, Singapore (1996)Google Scholar
 16.Kesten, H.: How long are the arms in DLA? J. Phys. A 20(1), L29–L33 (1987)MathSciNetGoogle Scholar
 17.Kesten, H.: Upper bounds for the growth rate of DLA. Phys. A 168(1), 529–535 (1990)MathSciNetGoogle Scholar
 18.Kesten, H.: On the speed of convergence in firstpassage percolation. Ann. Appl. Probab. 3(2), 296–338 (1993)MathSciNetzbMATHGoogle Scholar
 19.Kesten, H., Sidoravicius, V.: Positive recurrence of a onedimensional variant of diffusion limited aggregation. In and Out of Equilibrium 2. Progress in Probability, vol. 60, pp. 429–461. Birkhäuser, Basel (2008)Google Scholar
 20.Kesten, H., Sidoravicius, V.: A problem in onedimensional diffusionlimited aggregation (DLA) and positive recurrence of Markov chains. Ann. Probab. 36(5), 1838–1879 (2008)MathSciNetzbMATHGoogle Scholar
 21.Kingman, J.F.C.: Subadditive ergodic theory. Ann. Probab. 1(6), 883–899 (1973)MathSciNetzbMATHGoogle Scholar
 22.Marchand, R.: Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12(3), 1001–1038 (2002)MathSciNetzbMATHGoogle Scholar
 23.Martineau, Sébastien: Directed diffusionlimited aggregation. ALEA Lat. Am. J. Probab. Math. Stat. 14(1), 249–270 (2017)MathSciNetzbMATHGoogle Scholar
 24.Richardson, D.: Random growth in a tessellation. Proc. Camb. Philos. Soc. 74, 515–528 (1973)MathSciNetzbMATHGoogle Scholar
 25.Rosenstock, H., Marquardt, C.: Cluster formation in twodimensional random walks: application to photolysis of silver halides. Phys. Rev. B 22(12), 5797–5809 (1980)MathSciNetGoogle Scholar
 26.Saffman, P.G., Taylor, G.I.: The penetration of a fluid into a porous medium or HeleShaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329 (1958)MathSciNetzbMATHGoogle Scholar
 27.Silvestri, Vittoria: Fluctuation results for Hastings–Levitov planar growth. Probab. Theory Relat. Fields 167(1), 417–460 (2017)MathSciNetzbMATHGoogle Scholar
 28.van den Berg, J., Kesten, H.: Inequalities for the time constant in firstpassage percolation. Ann. Appl. Probab. 3(1), 56–80 (1993)MathSciNetzbMATHGoogle Scholar
 29.Voss, R.: Multiparticle fractal aggregation. J. Stat. Phys. 36(5/6), 861–872 (1984)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.