Metastability for Kawasaki dynamics at low temperature with two types of particles

This is the first in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and compute the transition time to the full box. This transition is triggered by a \emph{critical droplet} appearing somewhere in the box. We identify the region of parameters for which the system is metastable. For this region, in the limit as the temperature tends to zero, we show that the first entrance distribution on the set of critical droplets is uniform, compute the expected transition time up to a multiplicative factor that tends to one, and prove that the transition time divided by its expectation is exponentially distributed. These results are derived under \emph{three hypotheses} on the energy landscape, which are verified in the second and the third paper for a certain subregion of the metastable region. These hypotheses involve three model-dependent quantities -- the energy, the shape and the number of the critical droplets -- which are identified in the second and the third paper as well.


Introduction and main results
The main motivation behind this work is to understand metastability of multi-type particle systems subject to conservative stochastic dynamics. In the past ten years a good understanding was achieved of the metastable behavior of the lattice gas subject to Kawasaki dynamics, i.e., random hopping of particles of a single type with hardcore repulsion and nearest-neighbor attraction. The analysis was based on a combination of techniques coming from large deviation theory, potential theory, geometry and combinatorics. In particular, a precise description was obtained of the time to nucleation (from the "gas phase" to the "liquid phase"), the critical droplet triggering the nucleation, and the typical nucleation path, i.e., the typical growing and shrinking of droplets. For an overview we refer the reader to two recent papers presented at the 12th Brazilian School of Probability: Gaudillière and Scoppola [14] and Gaudillière [15]. For an overview on metastability and droplet growth in a broader context, we refer the reader to the monograph by Olivieri and Vares [26], and the review papers by Bovier [3], [4], den Hollander [16], Olivieri and Scoppola [25].
It turns out that for systems with two types of particles, as considered in the present paper, the geometry of the energy landscape is much more complex than for one type of particle. Consequently, it is a somewhat delicate matter to capture the proper mechanisms behind the growing and shrinking of droplets. Our proofs in the present paper use potential theory and rely on ideas developed in Bovier, den Hollander and Nardi [7] for Kawasaki dynamics with one type of particle. Our target is to identify the minimal hypotheses that lead to metastable behavior. We will argue that these hypotheses, stated in the context of our specific model, also suffice for Kawasaki dynamics with more than two types of particles and are robust against variations of the interaction.
The model studied in the present paper falls in the class of variations on Ising spins subject to Glauber dynamics and lattice gas particles subject to Kawasaki dynamics. These variations include Blume-Capel, anisotropic interactions, staggered magnetic field, next-nearest-neighbor interactions, and probabilistic cellular automata. In all these models the geometry of the energy landscape is complex and needs to be controlled in order to arrive at a complete description of metastability. For an overview, see the monograph by Olivieri and Vares [26], chapter 7. Section 1.1 defines the model, Section 1.2 introduces basic notation, Section 1.3 identifies the metastable region, while Section 1.4 states the main theorems. Section 1.5 discusses the main theorems, places them in their proper context and provides further motivation. Section 1.6 proves three geometric lemmas that are needed in the proof of the main theorems, which is provided in Section 2.
This dynamics is ergodic and reversible with respect to the Gibbs measure µ β , i.e., µ β (η)c β (η, η ) = µ β (η )c β (η , η) ∀ η, η ∈ X . (1.6) Note that particles are preserved in Λ − , but can be created and annihilated in ∂ − Λ. Think of the particles entering and exiting Λ along non-oriented edges between ∂ − Λ and ∂ + Λ (where we allow only one edge for each site in ∂ − Λ). The pairs (η, η ) with η ∼ η are called communicating configurations, the transitions between them are called allowed moves. Particles in ∂ − Λ do not interact with particles anywhere in Λ (see (1.2)). The dynamics defined by (1.2) and (1.5) models the behavior inside Λ of a lattice gas in Z 2 , consisting of two types of particles subject to random hopping with hard core repulsion and with binding between different neighboring types. We may think of Z 2 \Λ as an infinite reservoir that keeps the particle densities inside Λ fixed at ρ 1 = e −β∆1 and ρ 2 = e −β∆2 . In our model this reservoir is replaced by an open boundary ∂ − Λ, where particles are created and annihilated at a rate that matches these densities. Consequently, our Kawasaki dynamics is a finite-state Markov process.
Note that there is no binding energy between neighboring particles of the same type. Consequently, the model does not reduce to Kawasaki dynamics for one type of particle when ∆ 1 = ∆ 2 . Further note that, whereas Kawasaki dynamics for one type of particle can be interpreted as swaps of occupation numbers along edges, such an interpretation is not possible here.

Notation
To identify the metastable region in Section 1.3 and state our main theorems in Section 1.4, we need some notation.  (c) A droplet is a maximal set of particles connected by active bonds.
(d) is the configuration where Λ is empty, is the configuration where Λ is filled as a checkerboard (see Remark 1.13 below).
and Φ(A, B) is its extension to non-empty sets A, B ⊂ X defined by is the communication level set between η and η defined by (1.10) where I η = {ξ ∈ X : H(ξ) < H(η)} is the set of configurations with energy lower than η. (d) X stab = {η ∈ X : H(η) = min ξ∈X H(ξ)} is the set of stable configurations, i.e., the set of configurations with minimal energy.
(e) X meta = {η ∈ X : V η = max ξ∈X \X stab V ξ } is the set of metastable configurations, i.e., the set of non-stable configurations with maximal stability level.

Metastable region
We want to understand how the system tunnels from to when the former is a local minimum and the latter is a global minimum of H. We begin by identifying the metastable region, i.e., the region in parameter space for which this is the case.
Lemma 1.4. The condition ∆ 1 + ∆ 2 < 4U is necessary and sufficient for to be a local minimum but not a global minimum of H.
Then a straightforward computation gives (1.14) and so Hence, if ∆ 1 + ∆ 2 < 4U , then there exists an¯ ∈ 2N such that H(η) < 0 for all ∈ 2N with ≥¯ . Here, Λ must be taken large enough, so that a droplet of size¯ fits inside Λ − .
Consequently, the "critical droplet" for the transition from to consists of only two free particles, one of type 1 and one of type 2. Therefore this subregion does not exhibit proper metastable behavior.

Main theorems
Theorems 1.7-1.9 below will be proved in the metastable region subject to the following hypotheses: (H1) X stab = .
The third hypothesis consists of three parts characterizing the entrance set of G( , ), the set of critical droplets, and the exit set of G( , ). To formulate this hypothesis some further definitions are needed.

Definition 1.5.
(a) C bd is the minimal set of configurations in G( , ) such that all paths in ( → ) opt enter G( , ) through C bd . (b) P is the set of configurations visited by these paths just prior to their first entrance of G( , ).
(H3-a) Everyη ∈ P consists of a single droplet somewhere in Λ − . This single droplet fits inside an L × L square somewhere in Λ − for some L ∈ N large enough that is independent ofη and Λ. Every η ∈ C bd consists of a single dropletη ∈ P and a free particle of type 2 somewhere in ∂ − Λ. Definition 1.6.
(a) C att is the set of configurations obtained from P by attaching a particle of type 2 to the single droplet, and decomposes as C att = ∪η ∈P C att (η).
(b) C is the set of configurations obtained from P by adding a free particle of type 2 somewhere in Λ, and decomposes as C = ∪η ∈P C (η).
Note that Γ = H(C ) = H(P) + ∆ 2 , and that C consists of precisely those configurations "interpolating" between P and C att : a free particle of type 2 enters ∂ − Λ and moves to the single droplet where it attaches itself via an active bond. Think of P as the set of configurations where the dynamics is "almost over the hill", of C as the set of configurations where the dynamics is "on top of the hill", and of the free particle as "achieving the crossover" before it attaches itself properly to the single droplet (the meaning of the word properly will become clear in Section 2.4).
The set P is referred to as the set of protocritical droplets. We write N to denote the cardinality of P modulo shifts of the droplet. The set C is referred to as the set of critical droplets.
(H3-b) All transitions from C that either add a particle in Λ or increase the number of droplets (by breaking an active bond) lead to energy > Γ . (H3-c) All ω ∈ (C bd → ) opt pass through C att . For everyη ∈ P there exists a ζ ∈ C att (η) such that Φ(ζ, ) < Γ .
We close this section with a few remarks.
Remark 1.11. We will see in Section 1.6 that (H1-H2) imply that (X meta , X stab ) = ( , ), Γ = Γ . (1.18) The reason that is the configuration with lowest energy comes from the "anti-ferromagnetic" nature of the interaction in (1.2). is not the global minimum of H, so that must be the global minumum of H, and hence X stab = according to Definition 1.2(d). One reason why we state (H1)-(H2) as separate hypotheses is that we will later place them in a more general context (see Section 1.5, item 8). Another reason is that they are the key ingredients in the proof of Theorems 1.7-1.9 in Section 2. Remark 1.13. We will see in [19] that, depending on the shape of Λ and the choice of U, ∆ 1 , ∆ 2 , X stab may actually consist of more than the single configuration , namely, it may contain configurations that differ from in ∂ − Λ. Since this boundary effect does not affect our main theorems, we will ignore it here. A precise description of X stab will be given in [19]. Moreover, depending on the choice of U, ∆ 1 , ∆ 2 , large droplets with minimal energy tend to have a shape that is either square-shaped or rhombus-shaped.
Therefore it turns out to be expedient to choose Λ to have the same shape. Details will be given in [19]. Remark 1.14. As we will see in Section 2.4, the value of K is given by a non-trivial variational formula involving the set of all configurations where the dynamics can enter and exit C . This set includes not only the border of the "Γ -valleys" around and , but also the border of "wells inside the energy plateau G( , )" that have energy < Γ but communication height Γ towards both and . This set contains P, C att and possibly more, as we will see in [20] (for Kawasaki dynamics with one type of particle this was shown in Bovier, den Hollander and Nardi [7], Section 2.3.2). As a result of this geometric complexity, for finite Λ only upper and lower bounds are known for K. What (1.17) says is that these bounds merge and simplify in the limit as Λ → Z 2 (after the limit β → ∞ has already been taken), and that for the asymptotics only the simpler quantity N matters rather than the full geometry of critical and near critical droplets. We will see in Section 2.4 that, apart from the uniformity property expressed in Theorem 1.7(b), the reason behind this simplification is the fact that simple random walk (the motion of the free particle) is recurrent on Z 2 .

Discussion
1. Theorem 1.7(a) says that C is a gate for the nucleation, i.e., on its way from to the dynamics passes through C . Theorem 1.7(b) says that all protocritical droplets and all locations of the free particle in ∂ − Λ are equally likely to be seen upon first entrance in G( , ). Theorem 1.8 says that the average nucleation time is asymptotic to Ke Γβ , which is the classical Arrhenius law, and it identifies the asymptotics of the prefactor K in the limit as Λ becomes large. Theorem 1.9, finally, says that the nucleation time is exponentially distributed on the scale of its average. same form for a large class of stochastic dynamics in a finite box at low temperature exhibiting metastable behavior. So far this universality has been verified for only a handful of examples, including Kawasaki dynamics with one type of particle (see also item 4 below). In Section 2 we will see that (H1)-(H3) are the minimal hypotheses needed for metastable behavior, in the sense that any relative of Kawasaki dynamics for which Theorems 1.7-1.9 hold must satisfy appropriate analogues of (H1)-(H3) (including multi-type Kawasaki dynamics).
The model-dependent ingredient of Theorems 1.7-1.9 is the triple This triple depends on the parameters U, ∆ 1 , ∆ 2 in a manner that will be identified in [19] and [20]. The set C also depends on Λ, but in such a way that |C | ∼ N |Λ| as Λ → Z 2 , with the error coming from boundary effects. Clearly, Λ must be taken large enough so that critical droplets fit inside (i.e., Λ must contain an L × L square with L as in (H3-a)). Figure 2: Subregion of the proper metastable region considered in [19] and [20].

3.
In [19] and [20], we will prove (H1)-(H3), identify (Γ , C , N ) and derive an upper bound on V in the subregion of the proper metastable region given by (see Fig. 2) (1.20) More precisely, in [19] we will prove (H1), identify Γ , show that V ≤ 10U − ∆ 1 , and conclude that (H2) holds as soon as Γ > 10U − ∆ 1 , which poses further restrictions on U, ∆ 1 , ∆ 2 on top of (1.20). In [19] we will also see that it would be possible to show that V ≤ 4U + ∆ 1 provided certain boundary effects (arising when a droplet sits close to ∂ − Λ or when two or more droplets are close to each other) could be controlled. Since it will turn out that Γ > 4U + ∆ 1 throughout the region (1.20), this upper bound would settle (H2) without further restrictions on U, ∆ 1 , ∆ 2 . In [20] we will prove (H3) and identify C , N . The simplifying features of (1.20) are the following: ∆ 1 < U implies that each time a particle of type 1 enters Λ and attaches itself to a particle of type 2 in the droplet the energy goes down, while ∆ 2 − ∆ 1 > 2U implies that no particle of type 2 sits on the boundary of a droplet that has minimal energy given the number of particles of type 2 in the droplet. We conjecture that (H1)-(H3) hold throughout the proper metastable region (see Fig. 1). However, as we will see in [19] and [20], (Γ , C , N ) is different when ∆ 1 > U compared to when ∆ 1 < U (because the critical droplets are squareshaped, respectively, rhombus-shaped).
4. Theorems 1.7-1.9 generalize what was obtained for Kawasaki dynamics with one type of particle in den Hollander, Olivieri and Scoppola [18], and Bovier, den Hollander and Nardi [7]. In these papers, the analogues of (H1)-(H3) were proved, (Γ , C , N ) was identified, and bounds on K were derived that become sharp in the limit as Λ → Z 2 .
What makes the model with one type of particle more tractable is that the stochastic dynamics follows a skeleton of subcritical droplets that are squares or quasi-squares, as a result of a standard isoperimetric inequality for two-dimensional droplets. For the model with two types of particles this tool is no longer applicable and the geometry is much harder, as will become clear in [19] and [20].
Similar results hold for Ising spins subject to Glauber dynamics, as shown in Neves and Schonmann [24], and Bovier and Manzo [9]. For this system, K has a simple explicit form. Theorems 1.7-1.9 are close in spirit to the extension for Glauber dynamics of Ising spins when an alternating external field is included, as carried out in Nardi and Olivieri [22], for Kawakasi dynamics of lattice gases with one type of particle when the interaction between particles is different in the horizontal and the vertical direction, as carried out in Nardi, Olivieri and Scoppola [23], and for Glauber dynamics with threestate spins (Blume-Capel model), as carried out in Cirillo and Olivieri [10] Our results can in principle be extended from Z 2 to Z 3 . For one type of particle this extension was achieved in den Hollander, Nardi, Olivieri and Scoppola [17], and Bovier, den Hollander and Nardi [7]. For one type of particle the geometry of the critical droplet is more complex in Z 3 than in Z 2 . This will also be the case for two types of particles, and hence it will be hard to identify C and N . Again, only upper and lower bounds can be derived for K. Moreover, since simple random walk on Z 3 is transient, these bounds do not merge in the limit as Λ → Z 3 . For Glauber dynamics the extension from Z 2 to Z 3 was achieved in Ben Arous and Cerf [1], and Bovier and Manzo [9], and K again has a simple explicit form.

5.
In Gaudillière, den Hollander, Nardi, Olivieri and Scoppola [11], [12], [13], and Bovier, den Hollander and Spitoni [8], the result for Kawasaki dynamics (with one type of particle) on a finite box with an open boundary obtained in den Hollander, Olivieri and Scoppola [18] and Bovier, den Hollander and Nardi [7] have been extended to Kawasaki dynamics (with one type of particle) on a large box Λ = Λ β with a closed boundary. The volume of Λ β grows exponentially fast with β, so that Λ β itself acts as a gas reservoir for the growing and shrinking of subcritical droplets. The focus is on the time of the first appearance of a critical droplet anywhere in Λ β . It turns out that the nucleation time in Λ β roughly equals the nucleation time in a finite box Λ divided by the volume of Λ β , i.e., spatial entropy enters into the game. A challenge is to derive a similar result for Kawasaki dynamics with two types of particles.

6.
The model in the present paper can be extended by introducing three binding energies U 11 , U 22 , U 12 < 0 for the three different pairs of types that can occur in a pair of neighboring particles. Clearly, this will further complicate the analysis, and consequently both (X meta , X stab ) and (Γ , C , N ) will in general be different. The model is interesting even when ∆ 1 , ∆ 2 < 0 and U < 0, since this corresponds to a situation where the infinite gas reservoir is very dense and tends to push particles into the box.
When ∆ 1 < ∆ 2 , particles of type 1 tend to fill Λ before particles of type 2 appear, but this is not the configuration of lowest energy. Indeed, if ∆ 2 − ∆ 1 < 4U , then the binding energy is strong enough to still favor configurations with a checkerboard structure (modulo boundary effects). Identifying (Γ , C , N ) seems a complicated task.

7.
We will see in Section 2 that (H1)-(H2) alone are enough to prove Theorems 1.7-1.9, with the exception of the uniform entrance distribution of C bd and the scaling of K in (1.17). The latter require (H3) and come out of a closer analysis of the energy landscape near C , respectively, a variational formula for 1/K that is derived on the basis of (H1)-(H2) alone. In Manzo, Nardi, Olivieri and Scoppola [21] an "axiomatic approach" to metastability similar to the one in the present paper was put forward, but the results that were obtained (for a general dynamics) based on hypotheses similar to (H1)-(H2) were cruder, e.g. the nucleation time was shown to be exp[βΓ + o(β)], which fails to capture the fine asymptotics in (1.16) and consequently also the scaling in (1.17). Also the uniform entrance distribution was not established. These finer details come out of the potentialtheoretic approach to metastability developed in Bovier, Eckhoff, Gayrard and Klein [5] explained in Section 2.
8. Hypotheses (H1)-(H3) are the minimal hypotheses in the following sense. If we consider Kawasaki dynamics with more than two types of particles and/or change the details of the interaction (e.g. by adding to (1.2) also interactions between particles of different type), then all that changes is that and are replaced by different configurations, while (H1)-(H2) remain the same for their new counterparts and (H3) remains the same for the analogues of P, C , C bd and C att . The proof in Section 2 will show that Theorems 1.7-1.9 continue to hold under (H1)-(H3) in the new setting. For further reading we refer the reader to the monograph in progress by Bovier and den Hollander [6].
Proof. Let η be any configuration in C att . Then, by (H3-a) and Definition 1.6(a), there is a configuration ξ, consisting of a single protocritical droplet, say, D and a free particle (of type 2) next to the border of D, such that η is obtained from ξ in a single move: the free particle attaches itself somewhere to D. Now, consider any path starting at η, ending at , and not exceeding energy level Γ . The reverse of this path, starting at and ending at η, can be extended by the single move from η to ξ to obtain a path from to ξ that is also not exceeding energy level Γ . Moreover, this path can be further extended from ξ to without exceeding energy level Γ as well. To see the latter, note that, by (H3-c), there is some location x on the border of D such that the configuration ζ ∈ C att consisting of D with the free particle attached at x is such that there is a path from ζ to that stays below energy level Γ . Furthermore, we can move from ξ (with H(ξ) = Γ ) to ζ (with H(ζ) < Γ ) at constant energy level Γ , dropping below Γ only at ζ, simply by moving the free particle to x without letting it hit ∂ − Λ. (By (H3-a), there is room for the free particle to do so because D fits inside an L × L square somewhere in Λ − . Even when D touches ∂ − Λ the free particle can still avoid ∂ − Λ, because x can never be in ∂ − Λ: particles in ∂ − Λ do not interact with particles in Λ − .) The resulting path from to (via η, ξ and ζ) is a path in ( → ) opt . However, by Definition 1.5(a), any path in ( → ) opt must hit C bd . Hence, the piece of the path from η to must hit C bd , because the piece of the path from η to (via ξ and ζ) does not.
We will follow the potential-theoretic argument that was used in Bovier, den Hollander and Nardi [7] for Kawasaki dynamics with one type of particle. In fact, we will see that (H1)-(H3) are the minimal assumptions needed to prove Theorems 1.7-1.9.

Dirichlet form and capacity
The key ingredient of the potential-theoretic approach to metastability is the Dirichlet form where µ β is the Gibbs measure defined in (1.4) and c β is the kernel of transition rates defined in (1.5). Given a pair of non-empty disjoint sets A, B ⊂ X , the capacity of the pair A, B is defined by where h| A ≡ 1 means that h(η) = 1 for all η ∈ A and h| B ≡ 0 means that h(η) = 0 for all η ∈ B. The unique minimizer h A,B of (2.2), called the equilibrium potential of the pair A, B, is given by and is the solution of the equation with c β (η, X \η) = η ∈X \η c β (η, η ) the rate of moving out of η. This rate enters because τ A is the first hitting time of A after the initial configuration is left (recall Defini-  The following lemma establishes bounds on the capacity of two disjoint sets. These bounds are referred to as a priori estimates and will serve as the starting point for more refined estimates later on. Lemma 2.1. For every pair of non-empty disjoint sets A, B ⊂ X there exist constants 0 < C 1 ≤ C 2 < ∞ (depending on Λ and A, B) such that Proof. The proof is given in [7], Lemma 3.1.1. We repeat it here, because it uses basic properties of communication heights that provide useful insight.
Upper bound: The upper bound is obtained from (2.2) by picking h = 1 K(A,B) with The key observation is that if η ∼ η with η ∈ K(A, B) and η ∈ X \K(A, B), then   To see (2), note that (1) implies the reverse of (2.10):  (2.12) where the equalities come from (2.11), while the second inequality uses the fact that η ∈ X \K (A, B).
Combining (2.9) with (1.4-1.5) and using reversibility, we find that (2.14) Lower bound: The lower bound is obtained by picking any path ω = (ω 0 , ω 1 , . . . , ω L ) that realizes the minimax in Φ(A, B) and ignoring all the transitions not in this path, i.e., where the Dirichlet form E ω β is defined as E β in (2.1) but with X replaced by ω. Due to the one-dimensional nature of the set ω, the variational problem in the right-hand side  with C 1 = 1/L.

Graph structure of the energy landscape
View X as a graph whose vertices are the configurations and whose edges connect communicating configurations, i.e., (η, η ) is an edge if and only if η ∼ η . Define -X is the subgraph of X obtained by removing all vertices η with H(η) > Γ and all edges incident to these vertices; -X is the subgraph of X obtained by removing all vertices η with H(η) = Γ and all edges incident to these vertices; Metastability for Kawasaki dynamics at low temperature with two types of particles -X and X are the connected components of X containing and , respectively.
Proof. By Definition 1.2(f), all paths connecting and reach energy level ≥ Γ . Therefore X and X are disconnected in X (because X does not contain vertices with energy ≥ Γ ).
We now have all the geometric ingredients that are necessary for the proof of Theorems 1.7-1.9 along the lines of [7], Section 3. Our hypotheses (H1)-(H3) replace the somewhat delicate and model-dependent geometric analysis for Kawasaki dynamics with one type of particle that was carried out in [7], Section 2. They are the minimal hypotheses that are necessary to carry out the proof below. Their verification for our specific model will be given in [19] and [20].

Metastable set, link between average nucleation time and capacity
Bovier, Eckhoff, Gayrard and Klein [5] define metastable sets in terms of capacities: (2.20) The following key lemma, relying on hypotheses (H1)-(H2) and Definition 1.2(d)-(e), allows us to apply the theory in [5]. Proof. By (1.4), Lemma 1.16 and the lower bound in (2.7), the numerator is bounded from above by e V β /C 1 = e (Γ −δ)β /C 1 for some δ > 0. By (1.4), the definition of Γ and the upper bound in (2.7), the denominator is bounded from below by e Γ β /C 2 (with the minimum being attained at ).

Lemma 2.4 has an important consequence:
Proof. According to [5], Theorem 1.3(i), we have  Hence, for β large enough, X ⊂ R ⊂ X \X . for the derivation of these bounds.

Proof of Theorem 1.8
Our starting point is Lemma 2.5. Recalling (2.1-2.3), our task is to show that (2.26) and to identify the constant Θ, since (2.26) will imply (1.16) with Θ = 1/K. This is done in four steps, organized in Sections 2.4.1-2.4.4.

2.4.1
Step 1: Triviality of h , on X , X and X \(X ∪ X ) For all η ∈ X \X we have H(η) > Γ , and so there exists a δ > 0 such that Therefore, we can replace X by X in the sum in (2.26) at the cost of a prefactor 1 + O(e −δβ ). Moreover, we have the following analogue of [7], Lemma 3.3.1.
Lemma 2.6. There exist C < ∞ and δ > 0 such that Proof. A standard renewal argument gives the relations, valid for η / ∈ { , }, . (2.28) and, with the help of (2.5) and Lemma 2.1, which proves the first claim with C = max η∈X \ C(η). Note that h , ( ) is a convex combination of h , (η) with η ∈ X \ , and so the claim includes η = .
For η ∈ X \ , we estimate and, with the help of (2.5) and Lemma 2.1, which proves the second claim with C = max η∈X \ C(η).
In view of Lemma 2.7, the contribution to the sum in (2.26) of the transitions inside a well can also be put into the prefactor 1 + o(1). Thus, only the transitions in and out of wells contribute.

2.4.2
Step 2: Variational formula for K By Step 1, the estimation of Z β CAP β ( , ) reduces to the study of a simpler variational problem. The following is the analogue of [7], Proposition 3.3.3.
(2.41) Next, with the help of Lemmas 2.6-2.7, we get
To check the latter, note that there are no allowed moves between these sets, so that either H(η) = Γ > H(η ) or H(η) < Γ = H(η ) for allowed moves in and out of these sets.
Combining Lemmas 2.5 and 2.8, we see that we have completed the proof of (1.16) with K = 1/Θ. The variational formula for Θ = Θ(Λ; U, ∆ 1 , ∆ 2 ) is non-trivial because it depends on the geometry of the wells X i , i = 1, . . . , I.

Step 3: Bounds on K in terms of capacities of simple random walk
So far we have only used (H1)-(H2). In the remainder of the proof we use (H3) to prove (1.17). The intuition behind (1.17) is the following. When the free particle attaches itself to the protocritial droplet, the dynamics enters the set C att . The entrance configurations of C att are either in X or in one of the X i 's. In the former case the path can reach while staying below Γ in energy, in the latter case it cannot. By Lemma 1.18, if the path exits an X i , then for it to return to X it must pass through C bd , i.e., it must go through a series of configurations consisting of a single protocritical droplet and a free particle moving away from that protocritical droplet towards ∂ − Λ. Now, this backward motion has a small probability because simple random walk in Z 2 is recurrent, namely, the probability is [1 + o(1)] 4π/ log |Λ| as Λ → Z 2 (see [7], Equation (3.4.5)). Therefore, the free particle is likely to re-attach itself to the protocritical droplet before it manages to reach ∂ − Λ. Consequently, with a probability tending to 1 as Λ → Z 2 , before the free particle manages to reach ∂ − Λ it will re-attach itself to the protocritical droplet in all possible ways, which must include a way such that the dynamics enters X . In other words, after entering C att the path is likely to reach X before it returns to X , i.e., it "goes over the hill". Thus, in the limit as Λ → Z 2 , the X i 's become irrelevant, and the dominant role is played by the transitions in and out of X and by the simple random walk performed by the free particle. Remark 2.9. The protocritical droplet may change each time the path enters and exits an X i . There are X i 's from which the path can reach without going back to C and without exceeding Γ in energy (see the proof of [7], Theorem 1.4.3, where this is shown for Kawasaki dynamics with one type of particle).
In order to make the above intuition precise, we need some further notation. (c) For η ∈ C ∪ C att , write η = (η, x) withη ∈ P the protocritical droplet and x ∈ Λ the location of the free/attached particle of type 2. (d) Forη ∈ P, A(η) = {x ∈ ∂ + supp(η) : H(η, x) < Γ } is the set of sites where the free particle of type 2 can attach itself to a particle of type 1 in ∂ − supp(η) to form an active bond. Note that x ∈ A(η) if and only if η = (η, x) ∈ C att , and that for every η ∈ C att either η ∈ X or η ∈ X i for some i = 1, . . . , I. be called the set of good sites, respectively, bad sites. Note that (η, x) may be in the same X i for different x ∈ B(η).
Note that Definitions 2.10(c-d) rely on (H3-a), and that G(η) = ∅ for allη ∈ P by (H3-c) and Lemma 2.2. For the argument below it is of no relevance whether B(η) = ∅ for some or allη ∈ P.
The following lemma is the analogue of [7], Proposition 3.3.4.
Proof. The variational problem in (2.40) decomposes into disjoint variational problems for the maximally connected components of X . Only those components that contain X or X contribute, since for the other components the minimum is achieved by picking h constant.
Θ ≥ Θ 1 : A lower bound is obtained from (2.40) by removing all transitions that do not involve a fixed protocritical droplet and a move of the free/attached particle of type 2.
Before we prove (2.54), note that the capacity in the right-hand side of (2.54) includes more transitions than the capacity in the left-hand side, namely, all transitions from supp(η) to B(η). Let (2.55) Below we will show that g Λ + \supp(η) ∂ + Λ,G(η) (x) ≤ C/ log |Λ| for all x ∈ B(η) and some C < ∞.
Since in the Dirichlet form in (2.49) the equilibrium potential appears squared, the error made by adding to the capacity in the left-hand side of (2.54) the transitions from supp(η) to B(η) therefore is of order [1/ log |Λ|] 2 times |B(η)|, which explains how (2.54) arises.
Formally, let Pη x be the law of the simple random walk that starts at x ∈ B(η) and is forbidden to visit the sites in supp(η). Let y ∈ G(η). Using a renewal argument similar to the one used in the proof of Lemma 2.6, and recalling the probabilistic interpretation of the equilibrium potential in (2.3) and of the capacity in (2.5), we get g Λ + \supp(η) CAP Λ + \supp(η) (x, y) . (2.56) The denominator of (2.56) can be bounded from below by some C > 0 that is independent of x, y and supp(η). To see why, pick a path from x to y that avoids supp(η) but stays inside an L × L square aroundη (recall (H3-a)), and argue as in the proof of the lower bound of Lemma 2.1. On the other hand, the numerator is bounded from above by CAP Λ + (∂ + Λ, G(η)), i.e., by the capacity of the same sets for a random walk that is not forbidden to visit supp(η), since the Dirichlet problem associated to the latter has the same boundary conditions, but includes more transitions. In the proof of Lemma 2.12 below, we will see that CAP Λ + (∂ + Λ, G(η)) decays like C / log |Λ| for some C < ∞ (see (2.63-2.64) below). We therefore conclude that indeed g supp(η) The upper bound is obtained from (2.40) by picking C i = 0, i = 1, . . . , I, and • By Lemma 2.2, P ⊂ X . Therefore the first line of (2.57) implies that h(η) = 1 for η = (η, x) withη ∈ P and x ∈ ∂ + Λ, which is consistent with the boundary condition g| ∂ + Λ ≡ 1 in (2.49).
Further note that: • By Definitions 1.5-1.6 and (H3-b), the only transitions in X between X and C ++ are those where a free particle enters ∂ − Λ.
• The only transitions in X between C ++ and X \[X ∪ C ++ ] are those where the free particle moves from distance 2 to distance 1 of the protocritical droplet. All other transitions either involve a detachment of a particle from the protocritical droplet (which raises the number of droplets) or an increase in the number of particles in Λ. By (H3-b), such transitions lead to energy > Γ , which is not possible in X .
• There are no transitions between X and X \[X ∪ C ++ ].
The latter show that (2.49) includes all the transitions in (2.40).

Step 4: Sharp asymptotics for capacities of simple random walk
With Lemma 2.11 we have obtained upper and lower bounds on Θ in terms of capacities for simple random walk on Z 2 of the pairs of sets ∂ + Λ and CS(η), respectively, CS ++ (η), withη summed over P. The transition rates of the simple random walk are 1 between neighboring pairs of sites. Lemma 2.12 below, which is the analogue of [7], Lemma 3.4.1, shows that, in the limit as Λ → Z 2 , each of these capacities has the same asymptotic behavior, namely, [1 + o(1)] 4π/ log |Λ|, irrespective of the location and shape of the protocritical droplet (provided it is not too close to ∂ + Λ, which is a negligible fraction of the possible locations). In what follows we pretend that Λ = B M = [−M, +M ] 2 ∩ Z 2 for some M ∈ N large enough. It is straightforward to extend the proof to other shapes of Λ (see van den Berg [2] for relevant estimates). where d(∂ + B M , supp(η)) = min{|x − y| : x ∈ ∂ + B M , y ∈ supp(η)}.

Proof of Theorem 1.9
Proof. The proof is immediate from Lemma 2.4 and Bovier, Eckhoff, Gayrard and Klein [5], Theorem 1.3(iv), and relies on (H1)-(H2) only. The main idea is that, each time the dynamics reaches the critical droplet but "fails to go over the hill and falls back into the valley around ", it has a probability exponentially close to 1 to return to (because, by (H2), lies at the bottom of its valley (recall (2.3) and (2.27))) and to "start from scratch". Thus, the dynamics manages to grow a critical droplet and go over the hill to only after a number of unsuccessful attempts that tends to infinity as β → ∞, each having a small probability that tends to zero as β → ∞. Consequently, the time to go over the hill is exponentially distributed on the scale of its average.
(2.75) Moreover (recall (2.3-2.4)), (2.77) Because P ⊂ X by Lemma 2.2 and C bd ⊂ G( , ) by Definition 1.5(a), for all η ∈ P we have Φ(η , C bd ) − Φ(η , ) = Γ − Φ(η , ) ≥ δ > 0. Therefore, as in the proof of Lemma 2.6, it follows that Moreover, lettingC be the set of configurations that can be reached from C bd via an allowed move that does not return to P, we have max η ∈C h ,C bd (η ) ≤ Ce −δβ . (2.79) Indeed, h ,C bd (η ) = 0 for η ∈ C bd , while we have the following: Any path fromC \C bd to that avoids C bd must reach an energy level > Γ . (2.80) To obtain (2.79) from (2.80), we can do an estimate similar to (2.29-2.30) for η ∈C \C bd . To prove (2.80) we argue as follows. Let ζ ∈C , and let η be the configuration in C bd from which ζ is obtained in a single transition. If ζ ∈ C bd , then any path from ζ to already starts from C bd and there is nothing to prove. Therefore, let ζ ∈C \C bd . Note that, by (H3-a), η consists of a single (protocritical) droplet in Λ − plus a particle of type 2 in ∂ − Λ. Recalling that particles in ∂ − Λ do not interact with other particles, we see that any configuration obtained from η by detaching a particle from the (protocritical) droplet increases the number of droplets and, by (H3-b), raises the energy above Γ . Therefore, ζ can only be obtained from η by moving the free particle from ∂ − Λ to Λ − . Only two cases are possible: either ζ ∈ C att or ζ ∈ C \C bd . In the former case, the claim follows via Lemma 1.18. In the latter case, we must show that if there is a path ω : ζ → that avoids C bd such that max σ∈ω H(σ) ≤ Γ , then a contradiction occurs.
Indeed, if ω is such a path, then the reversed path ω is a path from → ζ such that max σ∈ω H(σ) ≤ Γ . But ω can be extended by the single move from ζ to η to obtain a path ω : → η such that max σ∈ω H(σ) ≤ Γ . Moreover, since η ∈ C bd , there exists a path γ : η → such that max σ∈γ H(σ) ≤ Γ . But then the path obtained by joining ω and γ is a path in ( → ) opt such that the configuration ζ visited just before η ∈ C bd belongs to C \C bd ⊂ C . However, by Definitions 1.5-1.6, this implies that ζ ∈ P, which is impossible because P ∩ C = ∅.