# Nucleation for One-Dimensional Long-Range Ising Models

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## Abstract

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration \(\mathbf {-1}\) is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet might be *macroscopic* or *mesoscopic*, according to the value of the external magnetic field.

## Keywords

Metastability Long-range Ising model Nucleation## 1 Introduction

Metastability is a dynamical phenomenon observed in many different contexts, such as physics, chemistry, biology, climatology, economics. Despite the variety of scientific areas, the common feature of all these situations is the existence of multiple, well-separated *time scales*. On short time scales the system is in a quasi-equilibrium within a single region, while on long time scales it undergoes rapid transitions between quasi-equilibria in different regions. A rigorous description of metastability in the setting of stochastic dynamics is relatively recent, dating back to the pioneering paper [9], and has experienced substantial progress in the last decades. See [1, 4, 5, 27, 28] for reviews and for a list of the most important papers on this subject.

One of the big challenges in rigorous study of metastability is understanding the dependence of the metastable behaviour and of the nucleation process of the stable phase on the dynamics. The nucleation process of the critical droplet, i.e. the configuration triggering the crossover, has been indeed studied in different dynamical regimes: sequential [6, 13] vs. parallel dynamics [2, 11, 14]; non-conservative [6, 13] vs. conservative dynamics [19, 20, 21]; finite [7] vs. infinite volumes [8]; competition [15, 16, 23, 29] vs. non-competition of metastable phases [12, 17]. All previous studies assumed that the microscopic interaction is of short-range type.

The present paper pushes further this investigation, studying the dependence of the metastability scenario on the *range* of the interaction of the model. Long-range Ising models in low dimensions are known to behave like higher-dimensional short-range models. For instance in [10, 22] (and later generalized by [3, 24]) it was shown that long-range Ising models undergo a phase transition already in one dimension, and this transition persists in fast enough decaying fields. Furthermore, Dobrushin interfaces are rigid already in two dimensions for anisotropic long-range Ising models, see [18].

We consider the question: does indeed a *long-range* interaction change substantially the nucleation process? Are we able to define in this framework a critical configuration triggering the crossover towards the stable phase? In [26] the author already considered the *Dyson-like* long-range models, i.e. the one-dimensional lattice model of Ising spins with interaction decaying with a power \(\alpha \), in a external magnetic field. Despite the long-range potential, the author showed, by *instanton* arguments, that the system has a finite-sized critical droplet.

In this manuscript we want to make rigorous this claim for a general long-range interaction, showing as well that the long-range interaction completely changes the metastability scenario: in the short–range one-dimensional Ising model a droplet of size one, already nucleates the stable phase. We show instead that for a given external field *h*, and pair long-range potential *J*(*n*), we can define a nucleation droplet which gets larger for smaller *h*. For \(d=1\) finite-range interactions, inserting a minus interval of size \(\ell \) in the plus phase costs a finite energy, which is uniform in the length of the interval, the same is almost true for a fast decaying interaction, as there is a uniform bound on the energy an interval costs. Thus, for low temperature, there is a diverging timescale and we will talk also in this case (maybe by abuse of terminology) of metastability. The spatial scale of a nucleating interval, however, defined as an interval which lowers its energy when growing, is finite for finite-range interactions, but diverges as \(h\rightarrow 0\) for infinite-range. The Dyson model has energy and spatial scale of the nucleating droplet diverging as *h* goes to zero. We will show that, depending on the value of *h*, the critical droplet can be *macroscopic* or *mesoscopic*. Roughly speaking, an interval of minuses of length \(\ell \) which grows to \(\ell +1\) gains energy 2*h*, but loses \(E_\ell = \sum _{ n=\ell }^{\infty } J(n)\). \(E_\ell \) converges to zero as \(\ell \rightarrow \infty \), but the smaller *h* is, the larger the size of the critical droplet. Moreover, by taking *h* volume-dependent, going to zero with *N* as \( N^{-\delta }\), one can make the nucleation interval mesoscopic (e.g. \(O(N^\delta )\), with \(\delta \in (0,1)\)) or macroscopic (i.e. *O*(*N*)).

The paper is organised as follows. In Sect. 2 we describe the lattice model and we give the main definitions; in Sect. 3 the main results of the paper are stated, while in Sects. 4 and 5 the proofs of the model-dependent results are given.

## 2 The Model and Main Definitions

*h*a positive external field. Given a configuration \(\sigma \) in \(\Omega _{\Lambda } = \{-1,1\}^{\Lambda }\), we define the

*Hamiltonian*with free boundary condition by

*pair interaction*, is assumed to be positive and decreasing. The interactions that we want to include in the present analysis are of

*long-range type*, for instance,

- 1.
exponential decay: \(J(|i-j|)= J \cdot \lambda ^{-|i-j|}\) with constants \(J>0\) and \(\lambda >1\);

- 2.
polynomial decay: \(J(|i-j|)= J \cdot |i-j|^{-\alpha }\), where \(\alpha >0\) is a parameter.

*finite-volume Gibbs measure*will be denoted by

*ground states*\(\mathscr {X}^{s}\) is defined as \(\mathscr {X}^{s}: = argmin _{\sigma \in \Omega _\Lambda } H_{\Lambda , h}(\sigma )\). Note that for the class of interactions considered \(\mathscr {X}^{s} = \{\mathbf {+1}\}\), where \(\mathbf {+1}\) stands for the configuration with all spins equal to \(+1\).

*k*positive spins, and we define the configurations \(L^{(k)}\) and \(R^{(k)}\) as follows. Let

*k*positive spins on

*left*side of the interval and on the

*right*one. We will show that \(L^{(k)}\) and \(R^{(k)}\) are the minimizers of the energy function \(H_{\Lambda ,h}\) on \(\mathscr {M}_k\) (see Proposition 4.1). Let us denote by \(\mathscr {P}^{(k)}\) the set \(\mathscr {P}^{(k)}:= \{L^{(k)}, R^{(k)}\}\) consisting of the minimizers of the energy on \(\mathscr {M}_k\). With abuse of notation we will indicate with \(H_{\Lambda ,h}(\mathscr {P}^{(k)})\) the energy of the elements of the set, that is, \(H_{\Lambda ,h}(\mathscr {P}^{(k)}):=H_{\Lambda ,h}({L}^{(k)})=H_{\Lambda , h}({R}^{(k)})\).

*hitting time*\(\tau _{\eta }^{\sigma }\) of a configuration \(\eta \) of the chain

*X*started at \(\sigma \) as

*n*, a sequence \(\gamma = (\sigma ^{(1)}, \dots , \sigma ^{(n)})\) such that \(\sigma ^{(i)}\in \Omega _\Lambda \) and \(c(\sigma ^{(i)},\sigma ^{(i+1)})>0\) for all \(i=1,\dots ,n-1\) is called a

*path*joining \(\sigma ^{(1)}\) to \(\sigma ^{(n)}\); we also say that

*n*is the length of the path. For any path \(\gamma \) of length

*n*, we let

*height*of the path. We also define the

*communication height*between \(\sigma \) and \(\eta \) by

*communication cost*of passing from \(\sigma \) to \(\eta \) is given by the quantity \(\Phi (\sigma ,\eta )-H_{\Lambda , h}(\sigma )\). Moreover, if we define \(\mathscr {I}_\sigma \) as the set of all states \(\eta \) in \(\Omega _\Lambda \) such that \(H_{\Lambda , h}(\eta )< H_{\Lambda , h}(\sigma )\), then the

*stability level*of any \(\sigma \in \Omega _\Lambda {\setminus } \mathscr {X}^{s}\) is given by

*maximal stability level*. Assuming that \(\Omega _\Lambda {\setminus } \mathscr {X}^{s}\ne \emptyset \), we let the

*maximal stability level*be

### Definition 2.1

*metastable*states of the system and refer to each of its elements as

*metastable*. We denote by \(\Gamma \) the quantity

## 3 Main Results

### 3.1 Mean Exit Time

In this section we will study the first hitting time of the configuration \(\mathbf {+1}\) when the system is prepared in \(\mathbf {-1}\), in the limit \(\beta \rightarrow \infty \). We will restrict our analysis to the cases given by the following condition.

### Condition 3.1

*N*be an integer such that \(N \ge 2\). We consider \(\Lambda = \{1, \dots , N\}\) and

*h*such that

By using the general theory developed in [25], we need first to solve two *model-dependent* problems: the calculation of the *minimax* between \(\mathbf {-1}\) and \(\mathbf {+1}\) (item 1 of Theorem 3.1) and the proof of a *recurrence* property in the energy landscape (item 3 of Theorem 3.1).

### Theorem 3.1

- 1.
\(\Phi (-\mathbf {1},\mathbf {+1})=\Gamma +H_{\Lambda ,h}(\mathbf {-1})\),

- 2.
\(V_{\mathbf {-1}}= \Gamma > 0\), and

- 3.
\(V_\sigma <\Gamma \) for any \(\sigma \in \Omega _\Lambda {\setminus } \{\mathbf {-1}, \mathbf {+1}\}\).

As a corollary we have that \(-\mathbf {1}\) is the only metastable state for this model.

### Corollary 3.1

Therefore, the asymptotic behaviour of the exit time for the system started at the metastable states is given by the following theorem.

### Theorem 3.2

- 1.for any \(\epsilon >0\)$$\begin{aligned} {\lim _{\beta \rightarrow \infty } \mathbb {P}\left( e^{\beta (\Gamma -\epsilon )}<\tau _{\mathbf {+1}}^{\mathbf {-1}}<e^{\beta (\Gamma +\epsilon )}\right) =1,} \end{aligned}$$
- 2.the limitholds.$$\begin{aligned} {\lim _{\beta \rightarrow \infty }\frac{1}{\beta }\log \left( \mathbb {E}\left( \tau _{\mathbf {+1}}^{\mathbf {-1}}\right) \right) =\Gamma } \end{aligned}$$

Once the model-dependent results in Theorem 3.1 have been proven, the proof of Theorem 3.2 easily follows from the general theory present in [25]: item 1 follows from Theorem 4.1 in [25] and item 2 from Theorem 4.9 in [25].

### 3.2 Nucleation of the Metastable Phase

We are going to show that for small enough external magnetic field, the size of the critical droplet is a macroscopic fraction of the system, while for *h* sufficiently large, the critical configuration will be a mesoscopic fraction of the system.

### Proposition 3.1

- 1.Case \(h < h_{L-1}^{(N)}\), we have$$\begin{aligned} H_{\Lambda ,h}(\mathscr {P}^{(L)}) > \max _{\begin{array}{c} 0 \le k \le N\\ k \ne L \end{array}} H_{\Lambda ,h}(\mathscr {P}^{(k)}). \end{aligned}$$
- 2.Case \(h_{k}^{(N)}< h < h_{k-1}^{(N)}\) for some \(k \in \{1,\dots , L-1\}\), we have$$\begin{aligned} H_{\Lambda ,h}(\mathscr {P}^{(k)}) > \max _{\begin{array}{c} 0 \le i \le N \\ i \ne \bar{k} \end{array}} H_{\Lambda ,h}(\mathscr {P}^{(i)}). \end{aligned}$$
- 3.Case \(h = h_{k}^{(N)}\) for some \(k \in \{1,\dots , L-1\}\), we have$$\begin{aligned} H_{\Lambda ,h}(\mathscr {P}^{(k)}) = H_{\Lambda ,h}(\mathscr {P}^{(k+1)}) > \max _{\begin{array}{c} 0 \le i \le N \\ i \ne k, i \ne k+1 \end{array}} H_{\Lambda ,h}(\mathscr {P}^{(i)}). \end{aligned}$$

The first case of Proposition 3.1 describes the less interesting and, in a way, artificial, situation of very low external magnetic fields: in this regime the *bulk* term is negligible so that the energy of the droplet increases until the positive spins are the majority (i.e. \(k=L\), see Fig. 3). Therefore, the second case contains the most interesting situation, where there is an interplay between the bulk and the *surface* term. The following Corollary is a consequence of Proposition 3.1 when *N* is large enough and gives a characterisation of the critical size \(k_c\) of the critical droplet.

### Corollary 3.2

*N*is sufficiently large.

*critical configurations*\(\mathscr {P}_c\) is given by

*N*large enough. The following result shows the reason why configurations in \(\mathscr {P}_c\) are referred to as

*critical*configurations: they indeed trigger the transition towards the stable phase.

### Lemma 3.1

- 1.
any path \(\gamma \in \Omega (\mathbf {-1},\mathbf {+1})\) such that \(\Phi _\gamma - H_{\Lambda ,h}(-\mathbf {1})=\Gamma \) visits \(\mathscr {P}_c\), and

- 2.the limitholds.$$\begin{aligned} \lim _{\beta \rightarrow \infty } \mathbb {P}(\tau _{\mathscr {P}_c}^{-\mathbf {1}}<\tau _{+\mathbf {1}}^{-\mathbf {1}})=1 \end{aligned}$$

The proof of the previous Theorem is a straightforward consequence of Theorem 5.4 in [25].

### 3.3 Examples

Let us give two interesting examples of the general theory so far developed.

#### 3.3.1 Example 1: Exponentially Decaying Coupling

*J*and \(\lambda \) are positive real numbers with \(\lambda > 1\).

### Proposition 3.2

### Proof

*k*, as expected.

#### 3.3.2 Example 2: Polynomially Decaying Coupling

*J*and \(\alpha \) are positive real numbers with \(\alpha > 1\). As it is shown in Figs. 2 and 3, for the polynomially decaying coupling model, we have that, for

*h*small enough the critical droplet is essentially the half interval, while for large enough magnetic external magnetic field, the critical droplet is the configuration with \(k_c\) plus spins at the sides, with \(k_c\approx \left( \frac{J}{h(\alpha -1)}\right) ^{\frac{1}{\alpha -1}} \).

We can prove indeed the following proposition.

### Proposition 3.3

*N*is large enough.

### Proof

## 4 Proof Theorem 3.1

*k*spins with the value 1, in order to find such configurations with minimal energy, it is sufficient to minimize the first term of the right-hand side of Eq. (4.1).

### Proposition 4.1

*N*be a positive integer and \(k \in \{0, \dots , N\}\), if we restrict to all \(\sigma \in \mathscr {M}_k\), then

### Proof

As an immediate consequence of Proposition 4.1 the next results follows.

### Theorem 4.1

### 4.1 Proof of Theorem 3.1.1(minimax)

### Proof of Theorem 3.1.1

*k*such that \(0 \le k \le N-1\), and

*f*satisfies

### 4.2 Proof of Theorems 3.1.2 and 3.1.3

*spin-flipped*configuration \(\theta _{k}\sigma \) is defined as:

*k*from the configuration \(\sigma \) is given by

### Proposition 4.2

### Proof

The proof of the converse statement is straightforward. \(\square \)

As an immediate consequence of the result above, the next result follows.

### Corollary 4.1

Under Condition 3.1, for every configuration \(\sigma \) different from \(\mathbf {-1}\) and \(\mathbf {+1}\), there is a path \(\gamma = (\sigma ^{(1)}, \dots , \sigma ^{(n)})\), where \(\sigma ^{(1)} = \sigma \) and \(\sigma ^{(n)} \in \{\mathbf {-1},\mathbf {+1}\}\), such that \(H_{\Lambda ,h}(\sigma ^{(i+1)}) < H_{\Lambda ,h}(\sigma ^{(i)})\).

We have now all the element for proving item 2 and 3 of Theorem 3.1.

### Proof of Theorem 3.1.2

### Proof of Theorem 3.1.3

- 1.Case \(\eta = \mathbf {+1}\). According to Corollary (4.1), there is a path \(\gamma = (\sigma ^{(1)}, \dots , \sigma ^{(n)})\) from \(\sigma ^{(1)}= \sigma \) to \(\sigma ^{(n)} \in \{\mathbf {-1},\mathbf {+1}\}\) along which the energy decreases.
- (a)If \(\sigma ^{(n)} = \mathbf {-1}\), then the path \(\gamma _0:\sigma \rightarrow \eta \) given by \(\gamma _{0} = (\sigma ^{(1)}, \dots , \sigma ^{(n-1)}, L^{(0)}, \dots , L^{(N)})\) satisfies$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma _{0}}H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\\le & {} \left( \max _{\zeta \in \gamma }H_{\Lambda ,h}(\zeta )\right) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\sigma )\right) \\< & {} \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\mathbf {-1}) \\= & {} V_{\mathbf {-1}}. \end{aligned}$$
- (b)Otherwise, if \(\sigma ^{(n)} = \mathbf {+1}\), then$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma }H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \\< & {} V_{\mathbf {-1}}. \end{aligned}$$

- (a)
- 2.Case \(\eta = \mathbf {-1}\). According to Corollary (4.1), there is a path \(\gamma = (\sigma ^{(1)}, \dots , \sigma ^{(n)})\) from \(\sigma ^{(1)}= \sigma \) to \(\sigma ^{(n)} \in \{\mathbf {-1},\mathbf {+1}\}\) along which the energy decreases.
- (a)If \(\sigma ^{(n)} = \mathbf {+1}\), then the path \(\gamma _0:\sigma \rightarrow \eta \) given by \(\gamma _{0} = (\sigma ^{(1)}, \dots , \sigma ^{(n-1)}, L^{(N)}, \dots , L^{(0)})\) satisfies$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma _{0}}H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\\le & {} \left( \max _{\zeta \in \gamma }H_{\Lambda ,h}(\zeta )\right) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\sigma )\right) \\< & {} \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\mathbf {-1}) \\= & {} V_{\mathbf {-1}}. \end{aligned}$$
- (b)Otherwise, if \(\sigma ^{(n)} = \mathbf {-1}\), then$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma }H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \\< & {} V_{\mathbf {-1}}. \end{aligned}$$

- (a)
- 3.Case \(\eta \notin \{\mathbf {-1}, \mathbf {+1}\}\). Let \(\gamma _{1} = (\sigma ^{(1)}, \dots , \sigma ^{(n)})\) and \(\gamma _{2} = (\eta ^{(1)}, \dots , \eta ^{(m)})\) be paths from \(\sigma ^{(1)} = \sigma \) to \(\sigma ^{(n)} \in \{\mathbf {-1},\mathbf {+1}\}\) and from \(\eta ^{(1)} = \eta \) to \(\eta ^{(m)} \in \{\mathbf {-1},\mathbf {+1}\}\), respectively, along which the energy decreases.
- (a)If \(\sigma ^{(n)} = \eta ^{(m)}\), define the path \(\gamma : \sigma \rightarrow \eta \) given by \(\gamma _{0} = (\sigma ^{(1)}, \dots , \sigma ^{(n-1)}, \eta ^{(m)},\dots , \eta ^{(1)})\) in order to obtain$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma _{0}}H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\= & {} \left( \max _{\zeta \in \gamma _{1}}H_{\Lambda ,h}(\zeta )\right) \vee \left( \max _{\zeta \in \gamma _{2}}H_{\Lambda ,h}(\zeta )\right) - H_{\Lambda ,h}(\sigma ) \\= & {} H_{\Lambda ,h}(\sigma ) \vee H_{\Lambda ,h}(\eta ) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \\< & {} V_{\mathbf {-1}}. \end{aligned}$$
- (b)If \(\sigma ^{(n)} = \mathbf {-1}\) and \(\eta ^{(m)} = \mathbf {+1}\), let us define the path \(\gamma _{0}: \sigma \rightarrow \eta \) given byit satisfies$$\begin{aligned} \gamma _{0} = (\sigma ^{(1)}, \dots , \sigma ^{(n-1)}, L^{(0)}, \dots , L^{(N)}, \eta ^{(m-1)}, \dots , \eta ^{(1)}) \end{aligned}$$(4.29)$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma _{0}}H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\= & {} \left( \max _{\zeta \in \gamma _{1}}H_{\Lambda ,h}(\zeta )\right) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) \\&\vee \left( \max _{\zeta \in \gamma _{2}}H_{\Lambda ,h}(\zeta )\right) - H_{\Lambda ,h}(\sigma ) \\= & {} H_{\Lambda ,h}(\sigma ) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) \vee H_{\Lambda ,h}(\eta ) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\sigma )\right) \\< & {} \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\mathbf {-1}) \\= & {} V_{\mathbf {-1}}. \end{aligned}$$
- (c)If \(\sigma ^{(n)} = \mathbf {+1}\) and \(\eta ^{(m)} = \mathbf {-1}\), let us define the path \(\gamma _{0}: \sigma \rightarrow \eta \) given byit satisfies$$\begin{aligned} \gamma _{0} = (\sigma ^{(1)}, \dots , \sigma ^{(n-1)}, L^{(N)}, \dots , L^{(0)}, \eta ^{(m-1)}, \dots , \eta ^{(1)}) \end{aligned}$$(4.30)$$\begin{aligned} \Phi (\sigma ,\eta ) - H_{\Lambda ,h}(\sigma )\le & {} \max _{\zeta \in \gamma _{0}}H_{\Lambda ,h}(\zeta ) - H_{\Lambda ,h}(\sigma ) \\= & {} \left( \max _{\zeta \in \gamma _{1}}H_{\Lambda ,h}(\zeta )\right) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) \vee \left( \max _{\zeta \in \gamma _{2}}H_{\Lambda ,h}(\zeta )\right) \\&- H_{\Lambda ,h}(\sigma ) \\= & {} H_{\Lambda ,h}(\sigma ) \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)})\right) \vee H_{\Lambda ,h}(\eta ) - H_{\Lambda ,h}(\sigma ) \\= & {} 0 \vee \left( \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\sigma )\right) \\< & {} \max _{0 \le k \le N}H_{\Lambda ,h}(L^{(k)}) - H_{\Lambda ,h}(\mathbf {-1}) \\= & {} V_{\mathbf {-1}}. \end{aligned}$$

- (a)

## 5 Proofs of the Critical Droplets Results

### Proof of Proposition 3.1

*f*decreases for all

*i*greater than

*L*, and since \(\Delta ^2 f<0\), we conclude that

*f*attains a unique strict global maximum at

*L*. In the second case, we have \(\Delta f(k-1) = 2(h_{k-1}^{(N)} - h) > 0\) and \(\Delta f(k) = 2(h_{k}^{(N)} - h) < 0\), so,

*f*attains a unique strict global maximum at

*k*. Finally, in the third case, we have \(\Delta f(k) = 0\), that is, \(f(k) = f(k+1)\). Using the fact that \(\Delta f(k+1)< 0 < \Delta f(k-1)\), we conclude that the global maximum of

*f*can we only be reached at

*k*and \(k+1\). \(\square \)

### Proof of Corollary 3.2

*N*sufficiently large such that \(\left\lfloor \frac{N}{2} \right\rfloor > k_{c}\) and

*N*large enough, \(k_{c}\) satisfies

## Notes

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