Phase Separation for the Long Range One-dimensional Ising Model

Abstract

We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long–range two–body interaction, $J(n)=n^{-2+\alpha }$ where $n\in \mathbb {N}$ denotes the distance of the two spins and $\alpha \in ]0,\alpha _{+}[$ with $\alpha _+=(\log 3)/(\log 2) -1$. We prove that given $m\in ]-1,+1[$, if the temperature is small enough, then typical configuration for the $\mu ^{+}$ Gibbs measure conditionally to have a empirical magnetization of the order m are made of a single interval that occupy almost a proportion $\frac{1}{2}(1-\frac{m}{m_\beta })$ of the volume with the minus phase inside and the rest of the volume is the plus phase, here $m_{\beta }>0$ is the spontaneous magnetization.

Appendix 1: Triangles, Contours and Polymers

In this section we regroup all the definitions and estimates that comes from [4] for the Peierls argument and from [5] for the cluster expansion.

1.1 Triangles Configurations

In this section we start recalling the content of [4]. For all $i^*\in \Lambda ^*$, we consider an interval $[i^*- \frac{1}{100}, i^* + \frac{1}{100} ] \subset {I{R}}$ and choose one point in each interval, say $r_{i^*}\in {I{R}}$ in such a way that for any four distinct points $r_j$, $j=1, \dots , 4$ $|r_1-r_2| \ne |r_3-r_4|$.

Given a spin configuration $\underline{\sigma }_\Lambda \in \mathcal{S}_\Lambda$, consider the set of its spin flip points $\mathcal{L}^*(\sigma _\Lambda )$ i.e. the set of $i^* \in \Lambda ^*$ such that $\sigma _{i^*-\frac{1}{2}}=-\sigma _{i^*+\frac{1}{2}}$ and the corresponding points $(r_{i^*}, i^* \in \mathcal{L}^*(\sigma _\Lambda ))\subset {I{R}}$.

We next embed ${I{R}}$ in ${I{R}}^2$ where the line containing the $r_{i^*}$ represents the state at $t=0$, and the orthogonal axis represents the evolving time of a process of growing “$\vee$-lines”: each point $r_{i^*}$ branches into two twin lines growing at velocity 1 in the positive half plane, in the directions respectively of angles $\pi /4$ and $3/4 \pi$, until one of the two meets another line coming from a different $r_{j^*}$. At the instant when two branches of different $\vee$-lines meet, they are frozen and stop their growth, at the same instant their twin lines disappear, while all the other $\vee$-line associated to the other points are undisturbed and keep growing. The collision of two lines is represented graphically in the (r, t) plane by a triangle whose basis is the interval between the two points $r_{i^*},r_{j^*}$, roots of the two lines that collided, and the third vertex is the point representing the collision in the plane (r, t).

This construction is a way to construct a pairing of spin flips with a criterion of minimal distance. Our choice of $r_{i^*}$ makes the definition of triangles non–ambiguous.

For any finite $\Lambda$ the process stops at a finite time $t\le |\Lambda |+1$ giving rise to a configuration of triangles. The triangles will be denoted by T and a family of triangles will be denoted by $\underline{T}$.

Definition 6.1

  Given a spin configuration $\underline{\sigma }_\Lambda \in \mathcal{S}_\Lambda$, we denote by $\underline{T}(\underline{\sigma }_\Lambda )$ the configuration of triangles obtained following the above mentioned procedure.

$x_-, x_+$ will denote respectively the left and right root of the associated $\vee$-lines. We will also write:

$$\begin{aligned} \Delta (T)=[x_-(T),x_+(T)] \cap {Z{Z}}\quad \hbox {the basis of the triangle}\,\, T; \end{aligned}$$

(6.1)

$$\begin{aligned} |T|=\#\{\Delta (T)\}\equiv |\Delta (T)| \quad \text{ the } \text{ mass } \text{ of } \text{ the } \text{ triangle }\,\, T; \end{aligned}$$

(6.2)

$$\begin{aligned} \mathrm{sf}(T)= \big \{ \inf ( \Delta (T))-1, \inf ( \Delta (T)), \sup ( \Delta (T)), \sup ( \Delta (T))+1\big \} \end{aligned}$$

(6.3)

where ${Z{Z}}$ is equipped with its natural order;

$$\begin{aligned} \mathrm{dist}(T,T')= \mathrm{dist} (\mathrm{sf}(T),\mathrm{sf}(T')). \end{aligned}$$

(6.4)

From our construction it follows that for all triangles $T_i\ne T_j$,

$$\begin{aligned} { \mathrm dist}(T_i,T_j)\ge \min (|T_i|, |T_j|). \end{aligned}$$

(6.5)

In particular if a triangle $\widetilde{T}$ is interior to a triangle T, i.e. is such that $\Delta (\widetilde{T})\subset \Delta (T)$, then

$$\begin{aligned} |\widetilde{T}|\le \frac{1}{3} |T|. \end{aligned}$$

(6.6)

We denote $\mathcal{T}_\Lambda$ the set of configurations of triangles $\underline{T} =(T_1,\dots , T_n)$ that satisfy (6.5) and such that $\Delta (T_i)\subset \Lambda$ for all $i\in \{1,\dots ,n\}$. Since here the spins at the boundary are specified, $\bar{\sigma }_{-L-1}=\bar{\sigma }_{L+1} =1$, where we recall that $\Lambda =[-L,+L]\cap {Z{Z}}$, the above construction defines a one to one map from $\mathcal{S}_\Lambda$ to $\mathcal{T}_\Lambda$. In particular, if $\mathbbm {1}$ denotes the spin configuration in $\Lambda$ constantly equals to $+1$, $\mathbbm {1}$ is mapped to the empty configuration of triangles.

We say that two collections of triangles $\underline{S}'\in \mathcal{T}_\Lambda$ and $\underline{S}\in \mathcal{T}_\Lambda$ are compatible and we denote it by $\underline{S}' \simeq \underline{S}$ iff $\underline{S}' \cup \underline{S} \in \mathcal{T}_\Lambda$ (i.e. there exists a configuration in $\mathcal{S}_\Lambda$ such that its corresponding collection of triangles is the collection made of all triangles in $\underline{S}'$ and $\underline{S}$.)

The basic estimates for a collection of triangles $\underline{T}\in \mathcal{T}_\Lambda$ is given in Lemma 2.1 and appendix A.1 of [4]: calling

$$\begin{aligned} H^{++}(\underline{T})=h^{++}(\sigma _\Lambda (\underline{T}) \end{aligned}$$

(6.7)

note that from (1.1), the configuration with no triangles has an energy 0 and is a ground state.

Lemma 6.2

For all $0<\alpha \le -1+(\log 3/\log 2)$, for J large enough, for all $\underline{T}\in \mathcal{T}_\Lambda$

$$\begin{aligned} H^{++}(\underline{T})\ge \frac{2 \zeta _\alpha }{\alpha (1-\alpha )} \sum _{T\in \underline{T}} |T|^{\alpha } \end{aligned}$$

(6.8)

where $\zeta _\alpha =1-2(2^\alpha -1)>0$.

The proof is given in Lemma 2.1 and appendix A.1 of [4] by exploiting the property (6.5). Note that in [4] formula (3.4), there is a misprint where $\le$ should be replaced by $\ge$. Note also the factor $2/\alpha (1-\alpha )$ that was present in appendix A in [4], see the proof of Lemma A.1 there, is missing in formulae (3.4) and (2.9) in [4]. Unfortunately this missing factor $\alpha (1-\alpha )$ propagate also in [5].

Let us now give the

Proof of Lemma 2.10

Let us make a partition of the interval $\Lambda$ in segments of size $\epsilon _s|\Lambda |/2$, there are less than $2|\Lambda |/(\epsilon _s |\Lambda |)$ such segments. Given a family of triangles $\underline{T}^E\in \mathcal{T}^E_\Lambda (\rho )$, see (2.12), let $n(\underline{T}^E)$ be the maximum number of segments contained in $\Delta (\underline{T}^E)=\cup _{\tilde{T} \in \underline{T}^E} \Delta (\tilde{T})$. We have $\rho |\Lambda |/(\epsilon _s|\Lambda |)\le n(\underline{T}^E)\le 2\rho |\Lambda |/(\epsilon _s|\Lambda |)$. Given an integer n and a family of n such segments, the number of families of external triangles that can contain this particular family of segments is less than $(\epsilon _s|\Lambda |)^n$. Then taking $\epsilon _s=|\Lambda |^{-\gamma }$ we get

$$\begin{aligned} \sharp [\mathcal{T}^E_\Lambda (\rho )]\le \sum _{n= \rho |\Lambda |/(\epsilon _s|\Lambda |)}^{2 \rho |\Lambda |/(\epsilon _s|\Lambda |)} {|\Lambda |^\gamma \atopwithdelims ()n} (\epsilon _s|\Lambda |)^n\le |\Lambda |^\gamma (|\Lambda |^{1-\gamma })^{2\rho |\Lambda |^\gamma } 2^{|\Lambda |^\gamma } \le e^{(2-\gamma ) |\Lambda |^\gamma \log |\Lambda |}\nonumber \\ \end{aligned}$$

(6.9)

if $|\Lambda |$ is large enough and how large depends only on $\gamma$. $\square$

A contour will be a family of triangles $\Gamma \equiv \{T: T\in \Gamma \}$ that satisfy the properties listed in the Definition 6.3.

Let us define

$$\begin{aligned} |\Gamma |\equiv \sum _{T\in \Gamma }|T|\quad \hbox {the mass of the contour} \end{aligned}$$

(6.10)

and for $\alpha >0$ we define

$$\begin{aligned} \Vert \Gamma \Vert _\alpha \equiv \sum _{T\in \Gamma }|T|^\alpha . \end{aligned}$$

(6.11)

Recalling (6.1) to (6.4), let us denote

$$\begin{aligned} \Delta (\Gamma )\equiv \bigcup _{{T\in \Gamma }}\Delta (T) \end{aligned}$$

(6.12)

$$\begin{aligned} x_{-}(\Gamma )\equiv \min _{T\in \Gamma }x_{-}(T) \end{aligned}$$

(6.13)

$$\begin{aligned} \mathrm{sf}(\Gamma )\equiv \bigcup _{T\in \Gamma }\mathrm{sf}(T) \end{aligned}$$

(6.14)

$$\begin{aligned} \mathrm{dist}(\Gamma ,\Gamma ')\equiv \inf _{\begin{array}{c} {T\in \Gamma }\\ {T'\in \Gamma '} \end{array}}\mathrm{dist}(T,T') \end{aligned}$$

(6.15)

$$\begin{aligned} \underline{T}(\Gamma )=\{T: T\in \Gamma \} \end{aligned}$$

(6.16)

Definition 6.3

Given a configuration of triangles $\underline{T}$ in $\mathcal{T}_\Lambda$, a configuration of contours $\Gamma =\Gamma (\underline{T})$ is the result of the implementation of an algorithm $\mathcal{R}$ on the family of triangles $\underline{T}$, denoted by $\underline{\Gamma }(\underline{T}) =\mathcal{R}(\underline{T})$. It is a partition of $\underline{T}$ whose atoms, called contours are determined by the following properties P.0, P.1, P.2 :

P.0 Let ${\mathcal{R}}(\underline{T}) \equiv ( \Gamma _1,..,\Gamma _n)$, $\Gamma _i =\{ T_{j,i}, 1\le j \le k_i\}$, then $\underline{T}=\{T_{j,i}, 1\le i \le n,\;1\le j \le k_i\}$.

P.1 Contours are well separated from each other. Any pair $\Gamma \ne \Gamma '$ in $\mathcal{R}(\underline{T})$ verifies one of the following two alternatives. (i): $\Delta (\Gamma )\cap \Delta (\Gamma ')=\emptyset$, or (ii): $\Delta (\Gamma )\cap \Delta (\Gamma ')\ne \emptyset$, then either $T(\Gamma )\subset \Delta (\Gamma ')$ or $T(\Gamma ')\subset \Delta (\Gamma )$; moreover, supposing for instance that the former case is verified, (in which case we call $\Gamma$ an inner contour, $\Gamma '$ is external w.r.t $\Gamma$), then for any triangle $T'_i\in \Gamma '$, either $T(\Gamma )\subset T'_i$ or $T(\Gamma )\cap T'_i=\emptyset$. Namely either $\Delta (\Gamma )\cap \Delta (\Gamma ')=\emptyset$ or

$$\begin{aligned} \sum _{T'\in \Gamma '}\mathbbm {1}_{\{T(\Gamma )\subset \Delta (T')\}}+\sum _{T\in \Gamma }\mathbbm {1}_{\{T(\Gamma ')\subset \Delta (T)\}}= 1. \end{aligned}$$

(6.17)

In both cases

$$\begin{aligned} \mathrm{dist}(\Gamma ,\Gamma ') > C \,\min \left\{ |\Gamma |^3,|\Gamma '|^3\right\} \end{aligned}$$

(6.18)

where C is a constant chosen such that, as in [4], we have

$$\begin{aligned} \sum _{M} \frac{4M}{[C M^3]}\le \frac{1}{2} \end{aligned}$$

(6.19)

and dist$(\Gamma ,\Gamma ')$ is defined in (6.15).

P.2 Independence. Let $\{\underline{T}^{(1)},\dots ,\underline{T}^{(k)}\}$, be $k> 1$ configurations of triangles; $\mathcal{R}(\underline{T}^{(i)})=\{\Gamma ^{(i)}_j, j=1,\dots ,n_i\}$ the contours of the configuration $\underline{T}^{(i)}$. Then, if any distinct pair $\Gamma ^{(i)}_j$ and $\Gamma ^{(i')}_{j'}$ satisfies P.1

$$\begin{aligned} \mathcal{R}\left( \underline{T}^{(1)},\underline{T}^{(2)},\dots ,\underline{T}^{(k)} \right) =\{\Gamma ^{(i)}_j, j=1,..,n_i; i=1,..,k\}. \end{aligned}$$

(6.20)

Definition 6.4

(compatibility between contours) We say that two contours $\Gamma , \Gamma '$ are compatible if (6.18) is verified. We denote

$$\begin{aligned} \Gamma \sim \Gamma ' \quad&\Longleftrightarrow \quad \Gamma ,\Gamma ' \hbox { are compatible}\nonumber \\ \Gamma \not \sim \Gamma ' \quad&\Longleftrightarrow \quad \Gamma ,\Gamma ' \text{ are } \text{ incompatible }. \end{aligned}$$

(6.21)

Therefore we have a bijection between spin configurations in $\mathcal{S}_\Lambda$ and triangles in $\mathcal{T}_\Lambda$ and another one between $\mathcal{T}_\Lambda$ and its image by $\mathcal{R}$, in particular there is a one–to–one correspondence between spin configurations and contour configurations. We denote by $\mathcal{G}_{\Lambda }$ the set of all possible configurations of compatible contours associated to $\mathcal{S}_{\Lambda }$.

Moreover denoting by $\underline{T}(\underline{\Gamma })$ the configuration of triangles that are in $\underline{\Gamma }$, we define

$$\begin{aligned} H^{++} (\underline{\Gamma })\equiv H^{++} (\underline{T}(\underline{\Gamma })) \end{aligned}$$

(6.22)

For $\underline{\Gamma }\in \mathcal{G}_{\Lambda }$ let $\underline{\sigma }_\Lambda (\underline{\Gamma })\,\,$ be the corresponding spin configuration.

Definition 6.5

Given a collection of contours $\underline{\Gamma }$, a contour $\Gamma \in \underline{\Gamma }$ is external with respect to $\underline{\Gamma }$ if each triangle of $\Delta (\Gamma )$ is not contained in some triangle that belongs to the others contours of $\underline{\Gamma }$.

$$\begin{aligned} \forall T \in \Gamma , \quad \Delta (T) \not \subset \bigcup _{\Gamma '\in \underline{\Gamma }: \Gamma '\ne \Gamma } \Delta (\Gamma ') \end{aligned}$$

(6.23)

A contour which is not external is called internal.

The following Proposition regroups all the basic estimates that we need. It is proved in [4].

Proposition 6.6

  If $0\le \alpha <\alpha _+=-1+(\log 3)/(\log 2)$, then for any contour $\Gamma$,

$$\begin{aligned} H^{++}(\Gamma ) \ge \frac{\zeta _\alpha }{\alpha (1-\alpha )} \Vert \Gamma \Vert _\alpha \end{aligned}$$

(6.24)

with $\zeta _\alpha$ as in Lemma 6.2. and $\Vert \Gamma \Vert _\alpha$ is defined by (6.11).

For any familly of compatible contours $(\Gamma _0,\underline{\Gamma })$,

$$\begin{aligned} H^{++}(\Gamma _0\cup \underline{\Gamma })-H^{++}(\underline{\Gamma })\ge \delta \Vert \Gamma _0\Vert _\alpha \end{aligned}$$

(6.25)

where if $0<\alpha <\alpha _+$, $\delta \equiv c_{10}(\alpha ) /C$ with C as in (6.18) and $c_{10}(\alpha )=2\pi ^2(3\alpha (1-\alpha ))^{-1}$

There exists a $b_0\equiv b_0(\alpha )$ such that for all $b\ge b_0$, for all integers $m\ge 1$,

$$\begin{aligned} \sum _{\begin{array}{c} {\Gamma : x_-(\Gamma )=0}\\ { |\Gamma |=m} \end{array}}e^{-b \Vert \Gamma \Vert _\alpha } \le 2 e^{-b m^\alpha }. \end{aligned}$$

(6.26)

Cluster expansion

Let us recall the result on the cluster expansion given in [5]. Section 4 of [5] gives the derivation of the partition function as that of a gas of polymers with an hard core condition and Proposition 5.4 there proves that the cluster expansion for the logarithm of the partition function is convergent is

$$\begin{aligned} \sum _{ R\ni 0}\xi ^{++}( R)<<1 \end{aligned}$$

(6.27)

where $\xi ^{++}( R)$ is the activity of the polymer R and the sum is over al the polymers containing the origin.

In [5] it is proved that (6.27) follows basically from (6.8), or modification of it as (6.24) by taking $\beta$ large enough. The consequence of the convergence of the cluster expansion is that we have

$$\begin{aligned} \log Z_{\Lambda }^{++}= \sum _{x\in \Lambda } \xi ^{++}(R_x^1) \left[ 1+{ \mathcal B}(x,++)\right] \end{aligned}$$

(6.28)

where:

• (1) $R_x^1$ is the triangle of size 1 located in x, i.e. is the simplest contour and also the simplest polymer made of a singleton and

  $$\begin{aligned} \xi ^{++}(R_x^1)= e^{-2\beta (J+\zeta (2-\alpha ))}. \end{aligned}$$

  (6.29)

  where $\zeta (2-\alpha )$ is the Riemann zeta function and J is defined in (1.2).

• (2) ${ \mathcal B}(x,++)$ is an absolutely convergent series

  $$\begin{aligned} { |\mathcal B}(x,++)| \le e^{-\frac{\beta }{32}(\frac{\zeta _\alpha }{\alpha (1-\alpha )}-3\delta )} \end{aligned}$$

  (6.30)

  where $\zeta _\alpha$ is the same as in lemma 6.2 and $\delta$ is the same as in (6.25).

Here we are considering instead of (6.28) the logarithm of a constrained partition function as

$$\begin{aligned} {\widetilde{Z}^{++}_\Lambda (\underline{T}^E,t)}= \sum _{\underline{\sigma }_\Lambda \in {\mathcal{S}}_{\underline{T}^E}} e^{-\beta \left[ h^{++}(\underline{\sigma }_\Lambda )-h^{++}(\overline{\sigma }(\underline{T}^E)\right] } e^{\beta t \sum _{i\in \Lambda }(\sigma _i-\overline{\sigma }_i(\underline{T}^E))} \end{aligned}$$

(6.31)

the partition function restricted to the set of configurations ${\mathcal{S}}_{\underline{T}^E}$, see (2.9), where an external field t is present. Here $\underline{T}^E\in \mathcal{T}^E_\Lambda (\rho )$, see (2.12) for some $0<\rho <1$.

In the case $t=0$, the ground state is $\bar{\sigma }(\underline{T}^E)$, see (2.8) and (2.11) and the energy fluctuations $\underline{T}$ are in

$$\begin{aligned} \mathcal{T}_\Lambda ^f(\underline{T}^E)=\left\{ \underline{\widetilde{T}} \in \mathcal{T}_\Lambda : \underline{\widetilde{T}} \sim \underline{T}^E, \underline{T}^E[\underline{\sigma }_\Lambda ( \underline{\widetilde{T}} \cup \underline{T}^E)]=\underline{T}^E \right\} \end{aligned}$$

(6.32)

that is the set of family the triangles $\underline{\widetilde{T}}$ that are compatible with $\underline{T}^E$, their presences does not modify the family of external triangles $\underline{T}^E$ in particular the triangles of $\underline{\widetilde{T}}$ that are external to $\underline{T}^E$ are all small.

Let us define, for $\underline{T} \in \mathcal{T}_\Lambda ^f(\underline{T}^E)$

$$\begin{aligned} H^{++}_{\underline{T}^E} (\underline{T})=h^{++} (\underline{\sigma }(\underline{T} \cup \underline{T}^E))-h^{++}(\overline{\sigma }(\underline{T}^E)) \end{aligned}$$

(6.33)

that will be simply denoted $H^{++} (\underline{T})$ if no confusion could arise. One can check that if $\underline{\sigma }_\Lambda \in S_{\underline{T}^E}$ and $\underline{T}=\underline{T}(\underline{\sigma }_\Lambda )$ is the associated family of triangles in $\mathcal{T}_\Lambda ^f(\underline{T}^E)$ then

$$\begin{aligned} h^{++}(\underline{\sigma }_\Lambda )-h^{++}(\overline{\sigma }(\underline{T}^E)-t \sum _{i\in \Lambda }(\sigma _i-\overline{\sigma }_i(\underline{T}^E)) \ge H^{++}(\underline{T}) -|t|\sum _{T \in \underline{T}} | T|. \end{aligned}$$

(6.34)

Therefore, in presence of a magnetic field t as in (6.31), an analogous of the condition (6.8) is: For all $\underline{T}\in \mathcal{T}^f_\Lambda (\underline{T}^E)$

$$\begin{aligned} H^{++}(\underline{T})-|t|\sum _{T\in \underline{T}}| T|\ge \frac{ \zeta _\alpha }{\alpha (1-\alpha )} \sum _{T\in \underline{T}} |T|^{\alpha }, \end{aligned}$$

(6.35)

This condition is satisfied if for all $\underline{T}\in \mathcal{T}^f_\Lambda (\underline{T}^E)$

$$\begin{aligned} \frac{2 \zeta _\alpha }{\alpha (1-\alpha )} |T|^{\alpha }-|t||T|\ge \frac{ \zeta _\alpha }{\alpha (1-\alpha )} |T|^{\alpha }, \end{aligned}$$

(6.36)

i.e. if

$$\begin{aligned} |t|\le \frac{ \zeta _\alpha }{\alpha (1-\alpha )} (\frac{1}{3} |\underline{T}^E|)^{\alpha -1}\le \inf _{T\in \mathcal{T}^f_\Lambda (\underline{T}^E)} \frac{ \zeta _\alpha }{\alpha (1-\alpha )} |T|^{\alpha -1}. \end{aligned}$$

(6.37)

where we have used (3.29).

We regroup in the following Lemma the contribution of the dominant term of the constrained free energy obtained by cluster expansion. The proof is the same as the one of (6.28) given in [5]. On the other hand since all polymers that can occur in the expansion of $\log \widetilde{Z}^{++}_\Lambda (\underline{T}^E,t )$ occur also in the expansion of (6.28), the error terms satisfy the same bound.

Lemma 6.7

There exists a $\beta _1=\beta _1(\alpha )$ such that if $\beta \ge \beta _1(\alpha )$, for all $\rho \in [0,1]\cap Q_\Lambda$, for $\epsilon _ s=|\Lambda |^{-\gamma }$ with $0<\gamma <2/3$ and for $\Lambda$ so large to have $\rho >\epsilon _s$ then, for $t\le \zeta _\alpha 3^{1-\alpha } /(4\alpha (1-\alpha ) (\rho |\Lambda |)^{1-\alpha })\equiv t^\star _{\Lambda ,1}(\rho )$ where $\zeta _\alpha =1-2(2^\alpha -1)>0$ we have, for all $\underline{T}^E\in \mathcal{T}^E_\Lambda (\rho )$,

$$\begin{aligned} \log \widetilde{Z}^{++}_\Lambda (\underline{T}^E,t )=\sum _{x\in \Lambda } \mathbbm {1}_{\{\mathrm{dist}(x,sf(\underline{T}^E))\ge 1\}} \xi ^{\overline{\sigma }(\underline{T}^E)}(x) e^{-2\beta t \overline{\sigma }_x(\underline{T}^E) } (1+\mathcal{B}(x, \underline{T}^E, t))\nonumber \\ \end{aligned}$$

(6.38)

where $\xi ^{\overline{\sigma }(\underline{T}^E)}=e^{-\beta [h^{++}(T^{\{x\}}\overline{\sigma }(\underline{T}^E))-h^{++} (\overline{\sigma }(\underline{T}^E))]}$ and $T^{\{x\}}\overline{\sigma }(\underline{T}^E)$ is the configuration equals to $\overline{\sigma }(\underline{T}^E)$ everywhere but at x where the spin $\overline{\sigma }_x(\underline{T}^E)$ is reversed and $\mathcal{B}(x, \underline{T}^E, t)$ can be written as an explicit absolutely convergent series and satisfies

$$\begin{aligned} |\mathcal{B}(x, \underline{T}^E, t)| \le e^{-\frac{\beta }{64}(\frac{\zeta _\alpha }{\alpha (1-\alpha )}-3\delta )} \end{aligned}$$

(6.39)

where as in Proposition 6.6,

$$\begin{aligned} \delta =2\pi ^2(3\alpha (1-\alpha ))^{-1} C^{-1} \end{aligned}$$

(6.40)

and C is the constant that appears in the definition of contours (6.18).

Remark

For future reference we notice that: if $x\in \Lambda$

$$\begin{aligned} \xi ^{\overline{\sigma }(\underline{T}^E)}(x) =\xi ^{++}(\beta ) \times {\left\{ \begin{array}{ll}e^{2\beta \sum _{y\in {Z{Z}}\setminus \Delta (\underline{T}^E)} J(x-y)}, &{} \mathrm{if } \ x\in \Delta (\underline{T}^E)\setminus \mathrm{sf}(\underline{T}^E);\\ e^{2\beta \sum _{y\in \Delta (\underline{T}^E)\setminus \{x\} } J(x-y)}, &{} \mathrm{if } \ x\notin \Delta (\underline{T}^E) \cup \mathrm{sf}(\underline{T}^E).\\ \end{array}\right. } \end{aligned}$$

(6.41)

Let us now prove Lemmata 2.13 and 3.2

For the proof of (2.24), given $\underline{t}=(t_i, i\in \Lambda )\in {I{R}}^\Lambda$, let us define for $\underline{T}^E \in \mathcal{T}^E_\Lambda (\rho )$,

$$\begin{aligned} {\widetilde{Z}^{++}_\Lambda (\underline{T}^E,\underline{t})}= \sum _{\underline{\sigma }_\Lambda \in {\mathcal{S}}_{\underline{T}^E}}e^{-\beta \left[ h^{++}(\underline{\sigma }_\Lambda )-h^{++}(\overline{\sigma }(\underline{T}^E)\right] } e^{\beta \sum _{i\in \Lambda }t_i(\sigma _i-\overline{\sigma }_i(\underline{T}^E))}. \end{aligned}$$

(6.42)

By elementary computations, one can check that

$$\begin{aligned} \mu _{\Lambda }^{++}[\sigma _i|\mathcal{S}_{\underline{T}^E}]=\overline{\sigma }_i(\underline{T}^E) + \frac{1}{\beta } \frac{\partial }{\partial t_i}[\ln {\widetilde{Z}^{++}_\Lambda (\underline{T}^E,\underline{t})}]\bigg |_{\underline{t}=0}. \end{aligned}$$

(6.43)

Then it is immediate to check that

$$\begin{aligned} \frac{1}{|\Lambda |} \sum _{i\in \Lambda } \overline{\sigma }_i(\underline{T}^E)=(1-2\rho ) \end{aligned}$$

(6.44)

since $\overline{\sigma }(\underline{T}^E)$ is made of $\rho |\Lambda |$ sites in $\Lambda$ with $-1$ and what remains is $+1$. Now recalling (6.38), the dominant terms that comes from the second term in (6.43) gives a contribution to the mean

$$\begin{aligned} \frac{1}{|\Lambda |} \sum _{i\in \Lambda } (-2\overline{\sigma }_x(\underline{T}^E)) \mathbbm {1}_{\{\mathrm{dist}(x,sf(\underline{T}^E))\ge 1\}} \xi ^{\overline{\sigma }(\underline{T}^E)}(x). \end{aligned}$$

(6.45)

Recalling (1.5) and writing $\xi ^{\overline{\sigma }(\underline{T}^E)}(x) =\left[ \xi ^{\overline{\sigma }(\underline{T}^E)}(x) -\xi ^{++}(\beta )\right] + \xi ^{++}(\beta )$ in (6.45), to the term with $\xi ^{++}(\beta )$ corresponds

$$\begin{aligned} \frac{1}{|\Lambda |} \sum _{i\in \Lambda } (-2\overline{\sigma }_x(\underline{T}^E))\xi ^{++}(\beta )= -2\xi ^{++}(\beta )(1-2\rho ). \end{aligned}$$

(6.46)

Recalling (1.4), we get (2.24) since, by similar arguments as in the end of Sect. 4

$$\begin{aligned} \frac{2}{|\Lambda |} \sum _{i\in \Lambda } \left| \xi ^{\overline{\sigma }(\underline{T}^E)}(x) -\xi ^{++}(\beta )\right| \le \frac{10\xi ^{++}(\beta ) }{\alpha (1-\alpha )} |\Lambda |^{\alpha -1}. \end{aligned}$$

(6.47)

from which we get (2.24). Using (2.22) we get (2.23).

Let us now complete the proof the Lemma 3.2. It is enough to have an estimate uniform in the magnetic field r which satisfies (3.20) or (3.22) for the two points truncated correlation function (3.25). For notational simplicity, we consider only the case of (3.23), the case of (3.21) can be proved mutatis mutandis. Here we have to start with a $T_0$ with $|T_0|=\rho |\Lambda |$. Considering ${\widetilde{Z}^{++}_\Lambda (T_0,\underline{t})}$ as in (6.42) we take first $\underline{t}=r+\underline{t}(i,j)$ where $t_k(i,j)=0$, if $k\ne i, j$ while $t_i(i,j)=t_i$, $t_j(i,j)=t_j$ and then we get easily

$$\begin{aligned} \mu ^{+}_\Lambda (r)[\sigma _i,\sigma _j| \mathcal{S}_{T_0}]= \frac{1}{\beta ^2} \frac{\partial }{\partial t_i \partial t_j} \log {\widetilde{Z}^{++}_\Lambda (T_0,r+t(i,j))} \bigg |_{t(i,j)=0} \end{aligned}$$

(6.48)

Note first that if the presence ot $T_0$ impose that $\sigma _i$ or $\sigma _j$ are fixed, that is when i or j are in $\mathrm{sf}(T_0)$, see (6.3), then the corresponding truncated correlation is zero. On the other hand, assuming that $|i-j|\ge 2$, depending if $|i-j|$ is smaller or larger than C, the dominant terms in (6.48) coming from the cluster expansion of $\log {\widetilde{Z}^{++}_\Lambda (T_0,r+t(i,j))}$ are different. In the first case it comes from a single polymer made of a single contour with two unit triangles located at site i and j, say $T_i,T_j$ that should be compatible with the presence of $T_0$. In the second case it comes from a single polymer made of two contours each one made of an unit triangle located at site i and j, that should be compatible with the presence of $T_0$ .

It follows from [5], appendix 2, and proposition 5.4 there, that the corresponding activity is

$$\begin{aligned} \xi ^{\overline{\sigma }(T_0)}(i)\xi ^{\overline{\sigma }(T_0)}(j)e^{-\beta (r+t_i) 2\overline{\sigma }_i(T_0)} e^{-\beta (r+t_j) 2\overline{\sigma }_j(T_0)} \left[ e^{-\beta \left[ H^{++} _{T_0}(T_i,T_j)-H^{++}_{T_0}(T_i)-H^{++}_{T_0}(T_j)\right] } -1\right] \nonumber \\ \end{aligned}$$

(6.49)

where we have used (6.33). It can be checked by simple algebra that

$$\begin{aligned} H^{++} _{T_0}(T_i,T_j)-H^{++}_{T_0}(T_i)-H^{++}_{T_0}(T_j)=-\, \frac{2{\overline{\sigma }}_i(T_0){\overline{\sigma }}_j(T_0) }{|i-j|^{2-\alpha }}. \end{aligned}$$

(6.50)

Therefore, taking into account the error terms that come from the cluster expansion, we get

$$\begin{aligned}&\mu ^{+}_\Lambda (r)[\sigma _i,\sigma _j| \mathcal{S}_{T_0}]\nonumber \\&\quad =\xi ^{\overline{\sigma }(T_0)}(i)\xi ^{\overline{\sigma }(T_0)}(j)e^{-\beta r 2\overline{\sigma }_i(T_0)} e^{-\beta r 2\overline{\sigma }_j(T_0)} 4\overline{\sigma }_i(T_0)\overline{\sigma }_j(T_0)\nonumber \\&\qquad \times \left[ e^{\frac{2{\overline{\sigma }}_i(T_0){\overline{\sigma }}_j(T_0) }{|i-j|^{2-\alpha }}}-1\right] \left( 1 \pm e^{-\frac{\beta }{64}(\frac{\zeta _\alpha }{\alpha (1-\alpha )}-3\delta )}\right) \end{aligned}$$

(6.51)

inserting it in (3.24), it implies (3.23).