2.1. Configuration space

We consider three-dimensional lattices with well-defined distance between nearest neighbors (to be normalized to 1) that fulfill two conditions. Firstly, the nearest-neighbor edges of the lattice have to define a tessellation of $\mathbb{R}^3$ by regular tetrahedra and octahedra. Secondly, the lattice has to be translation-invariant in three linearly independent directions. We remark that regular tetrahedra and octahedra can be replaced by any rigid polyhedron (a polyhedron with all faces being triangles) that satisfies an analog of the rigidity estimates in Lemmas 2.1 and 2.2, and their volume has positive partial derivatives with respect to their edge lengths. We note that by Cauchy’s theorem, the volume is uniquely defined for rigid polyhedra when the edge lengths are given.

Examples of such lattices are the face-centered cubic lattice and the hexagonal close-packed lattice. For definitions see [Reference Lazzaroni, Palombaro and Schlömerkemper13]. Note that being translation-invariant does not mean that the lattice has to be a Bravais lattice, i.e. of the form $\mathbb{Z} n_1 + \mathbb{Z} n_2+\mathbb{Z} n_3$ for some vectors $n_i\in\mathbb{R}^3$. Bravais lattices are translation-invariant, but a union of Bravais lattices might still be translation-invariant but no longer a Bravais lattice, for which the hexagonal close-packed lattice serves as an example.

Let the set $I\subset\mathbb{R}^3$ denote one of the lattices that fulfill both criteria. We assume $0\in I$ and think of I as an index set which is going to be used to parametrize countable point configurations in $\mathbb{R}^3$. Let I have translational symmetry by the linearly independent vectors $t_1, t_2, t_3\in\mathbb{R}^3$ and define the set $T=\mathbb{Z} t_1+ \mathbb{Z} t_2+ \mathbb{Z} t_3$. Define the quotient space $I_n\,:\!=I / nT$ for $n\in\mathbb{N}$. We will think of $I_n$ as a specific set of representatives in the half-open parallelepiped $U_n$ spanned by $nt_1, nt_2, nt_3$, i.e. $U_n=n\{x t_1+ y t_2+ z t_3\mid x,y,z\in[0,1)\}$.

A parametrized point configuration in $\mathbb{R}^3$ is a map $\omega\colon I\rightarrow \mathbb{R}^2$, $x\mapsto \omega(x)$ that determines the point configuration $\{ \omega(x)\mid x\in I \}\subset \mathbb{R}^3$. For the set of all parametrized point configurations we introduce the character $\Omega=\{\omega\colon I\rightarrow \mathbb{R}^2\}$. Note that a single point configuration $\{ \omega(x)\mid x\in I \}$ can be parametrized by many different $\omega\in\Omega$.

Let $\alpha\in(0,1]$ be an arbitrary but fixed real to be fixed later. An n-periodic parametrized point configuration with edge length $l\in(1, 1+\alpha)$ is a parametrized configuration $\omega$ which satisfies the boundary conditions

(1)\begin{equation}\omega(x+nt_i)=\omega(x)+lnt_i \quad \text{for all $ x \in I $ and $ i\in\{1,2,3\}$.}\end{equation}

The set of n-periodic parametrized configurations with edge length l is denoted by $\Omega^{\text{per}}_{n, l}\subset \Omega$. From now on we will omit the word parametrized because in this section we are going to work solely with point configurations which are parametrized by I. An n-periodic configuration is uniquely determined by its values on $I_n$. Therefore we identify n-periodic configurations $\omega\in\Omega^{\text{per}}_{n,l}$ with functions $\omega\colon I_n\rightarrow \mathbb{R}^2$.

The bond set $E\subset I\times I$ contains index-pairs with Euclidean distance one; this is $E=\{(x,y)\in I\times I\mid |x-y|=1\}$. We set $E_n=E/ nT $; we can think of $E_n$ as a bond set $E_n\subset I_n\times I_n$. Let $\mathcal{T}$ denote the set of convex polyhedra, as in the definition of I, whose edges are in E and provide a tessellation of $\mathbb{R}^3$, which is the Delaunay pre-triangulation; see [Reference Lazzaroni, Palombaro and Schlömerkemper13]. Define $\mathcal{T}_n=\mathcal{T}/ nT$. Each $\triangle \in\mathcal{T}$ can be triangulated into tetrahedra (not necessarily uniquely); let us fix such a T-periodic triangulation of $\mathcal{T}$. The set of all (necessarily not all regular) tetrahedra created in this way forms a tessellation of $\mathbb{R}^3$ and is denoted by $\textrm{triang}(\mathcal{T})$. We define $\textrm{triang}(\mathcal{T}_n)\,:\!=\textrm{triang}(\mathcal{T})/ nT$.

2.2. Probability space

By definition of $\Omega$ and $\Omega^{\text{per}}_{n, l}$, we have $\Omega=(\mathbb{R}^3)^I$ and can identify $\Omega^{\text{per}}_{n, l}=(\mathbb{R}^3)^{I_n}$. Both sets are endowed with the corresponding product $\sigma$-algebras $\mathcal{F}=\bigotimes_{x\in I}\mathcal{B}(\mathbb{R}^3)$ and $\mathcal{F}_n=\bigotimes_{x\in I_n}\mathcal{B}(\mathbb{R}^3)$, where $\mathcal{B}(\mathbb{R}^3)$ denotes the Borel $\sigma$-algebra on each factor. The event of admissible n-periodic configurations $\Omega_{n, l}\subset \Omega^{\text{per}}_{n, l}$ is defined by the properties $(\Omega 1)$–$(\Omega 4)$.

• $(\Omega 1)$$ |\omega(x)-\omega(y)|\in (1, 1+\alpha)$ for all $(x,y)\in E$.

For $\omega\in\Omega$ we define the extension $\hat\omega\colon \mathbb{R}^3\to\mathbb{R}^3$ such that $\hat\omega(x)=\omega(x)$ if $x\in I$. On the closure of a tetrahedron $\triangle\in\textrm{triang}(\mathcal{T})$, the map $\hat\omega$ is defined to be the unique affine linear extension of the mapping defined on the corners of that tetrahedron.

• $(\Omega 2)$ The map $\hat\omega\colon \mathbb{R}^3\to\mathbb{R}^3$ is injective (and thus bijective).

• $(\Omega 3)$ The map $\hat\omega$ is almost everywhere orientation-preserving, that is, $\det(\nabla\hat\omega(x))>0$ for almost every $x\in\mathbb{R}^3$ with the Jacobian $\nabla\hat\omega\colon \mathbb{R}^3\to\mathbb{R}^{3\times3}$.

• $(\Omega 4)$ The image $\hat\omega(\triangle)$ of a polyhedron $\triangle\in\mathcal{T}$ is a convex polyhedron.

The conditions $(\Omega 3)$ and $(\Omega 4)$ follow from conditions $(\Omega 1)$ and $(\Omega 2)$ up to the sign of the determinant in $(\Omega 3)$, as was also noted in [Reference Heydenreich, Merkl and Rolles11, p. 4]. Since the proof is more analytic than stochastic, we also omit the proof and require them as technical conditions. Define the set of admissible n-periodic configurations with edge length l as

\begin{equation*}\Omega_{n, l}=\{\omega\in\Omega_{n, l}^{\text{per}}\mid \omega\ \text{satisfies (\Omega1)--(\Omega4)}\}.\end{equation*}

The set $\Omega_{n, l}$ is open and consists of non-empty subsets of $(\mathbb{R}^3)^{I_n}$. The scaled lattice $\omega_l(x)=lx$ for $x\in I$ and $1<l<1+\alpha$ is an element of $\Omega_{n, l}$. Any configuration $\omega\in \Omega_{n, l}$ is determined by a finite number of locations in $\mathbb{R}^3$. Each property $(\Omega 1)$–$(\Omega 4)$ is satisfied after sufficiently small perturbation of these locations; therefore any $\omega\in \Omega_{n, l}$ has a neighborhood that is fully contained in $\Omega_{n, l}$, hence the openness of $\Omega_{n, l}$.

Clearly $0<\delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l})<\infty$ with the Lebesgue measure $\lambda$ on $\mathbb{R}^3$ and the Dirac measure $\delta_0$ in $0\in\mathbb{R}^3$. The lower bound holds because $\Omega_{n, l}^0$ is non-empty and open in $(\mathbb{R}^3)^{I_n\setminus\{0\}}$ (similarly to the case of $\Omega_{n, l}$ above); the upper bound is a consequence of the parameter $\alpha$ in $(\Omega 1)$. Let the probability measure $\mathbb{P}_{n, l}$ be

\begin{equation*}\mathbb{P}_{n, l}(A)=\dfrac{\delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l}\cap A)}{\delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l})}\end{equation*}

for any Borel-measurable set $A\in \mathcal{F}_n$, so $\mathbb{P}_{n, l}$ is the uniform distribution on the set $\Omega_{n, l}$ with respect to the reference measure $\delta_0\otimes\lambda^{I_n \setminus\{0\}}$. The first factor in this product refers to the component $\omega(0)$ of $\omega\in\Omega$.

2.3. Result

We have the following finite-volume result.

Theorem 2.1. For $\alpha$ sufficiently small we have

(2)\begin{equation}\lim_{l\downarrow 1}\sup_{n\in\mathbb{N}}\sup_{\triangle\in\textrm{triang}(\mathcal{T}_n)} \mathbb{E}_{\mathbb{P}_{n,l}}[ |\nabla\hat\omega(\triangle)-\textnormal{Id} |^2 ]=0\end{equation}

with the constant value of the Jacobian $\nabla\hat\omega(\triangle)$ on the tetrahedron $\triangle$ from the triangulation of $\mathcal{T}_n$ and some norm $| \cdot |$ on $\mathbb{R}^{3\times 3}$.

The choice of $\alpha$ has to be such that the volume of any tetrahedron and octahedron with side lengths in $[1, 1+\alpha]$ is uniquely minimized by the regular tetrahedron and octahedron with side length 1 respectively (see the proof of Theorem 2.1). The central argument is going to be the following rigidity theorem from [Reference Friesecke, James and Müller5, Theorem 3.1].

Theorem 2.2. (Friesecke, James and Müller) Let U be a bounded Lipschitz domain in $\mathbb{R}^d$, $ d\geq 2$. There exists a constant C(U) with the following property. For each $v\in W^{1,2}(U, \mathbb{R}^d)$ there is an associated rotation $R\in \textnormal{SO(\textit{d})}$ such that

\begin{equation*}\|\nabla v-R\|_{L^2(U)}\leq C(U)\|\textnormal{dist}(\nabla v, \textnormal{SO}(d))\|_{L^2(U)}.\end{equation*}

This is a generalization of Liouville’s theorem, which states that a map is necessarily a rotation whose Jacobian is a rotation in every point of its domain. We are going to set $v=\hat\omega|_{U_n}$ and $U=U_n$, which is a bounded Lipschitz domain. The function $\hat\omega|_{U_n}$ is linear on each triangle $\triangle\in\mathcal{T}_n$, and thus piecewise affine linear on $U_n$. As a consequence $\hat\omega|_{U_n}$ belongs to the class $W^{1,2}(U_n, \mathbb{R}^{3})$. The following remark, which also appears in [Reference Friesecke, James and Müller5] at the end of Section 3, is essential for achieving uniformity in (2) in the parameter n.

Remark 2.1. The constant C(U) in Theorem 2.2 is invariant under scaling: $C(\gamma U)=C(U)$ for all $\gamma>0$. Indeed, setting $v_\gamma(\gamma x)=\gamma v(x)$ for $x\in U$, we have $\nabla v_\gamma(\gamma x)=\nabla v(x)$ and hence

\begin{equation*} \|\nabla v_\gamma-R\|_{L^2(\gamma U)}=\gamma^{d/2}\|\nabla v-R\|_{L^2(U)} \end{equation*}

and

\begin{equation*} \|\textnormal{dist}(\nabla v_\gamma, \textnormal{SO}(d))\|_{L^2(\gamma U)}=\gamma^{d/2}\|\textnormal{dist}(\nabla v, \textnormal{SO}(d))\|_{L^2(U)}. \end{equation*}

This implies that for the domains $U_n$$(n\geq 1)$, the corresponding constant $C(U_n)$ can be chosen independently of n.

2.4. Proofs

We are going to show that the $L^2$-distance of the Jacobian $\nabla\hat\omega$ from the scaled identity matrix on $U_n$ can be controlled by the difference of the areas of $\hat\omega(U_n)$ and $U_n$. Because of the periodic boundary conditions, $\lambda(\hat\omega(U_n))$ does not depend on configurations $\omega$ with $(\Omega 2)$, so it provides a suitable uniform control on the set $\Omega_{n,l}$. Then we show that the expected square distance of $\nabla\hat\omega$ from the scaled identity matrix can be controlled by the expected square deviation of the polyhedra edge lengths from one, with ‘one’ being associated with the lattice constant of the unscaled lattice.

The following two lemmas from [Reference Lazzaroni, Palombaro and Schlömerkemper13] provide the desired rigidity estimate on tetrahedra and octahedra. They state that the distance from $\textrm{SO}(3)$ of a piecewise affine linear map defined on the polyhedron can be controlled by terms that measure how the map deforms the edge lengths of the polyhedron. We conjecture that any convex, rigid polyhedron satisfies such rigidity estimates via Dehn’s theorem and the inverse function theorem. However, in this paper, as our main concern is not rigidity theory, we will only consider tetrahedra and octahedra for which these estimates are already proven. Let $|M|=\sqrt{\textnormal{tr}(M^tM)}$ denote the Frobenius norm of a matrix $M\in \mathbb{R}^{3\times 3}$ and $|w|$ the Euclidean norm of $w\in\mathbb{R}^3$.

Lemma 2.1. (Lemma 3.2 of [Reference Lazzaroni, Palombaro and Schlömerkemper13]) There is a positive constant $C_1$ such that, for all linear maps $A\colon \mathbb{R}^3 \rightarrow \mathbb{R}^3$ with $\textnormal{det}(A)>0$ and

\begin{alignat*}{3} w_1&=(1, 0, 0),&\quad w_2&=\biggl(\dfrac{1}{2}, \dfrac{\sqrt3}{2}, 0\biggr),\\* w_3&=w_2-w_1,&\quad w_4&=\biggl(\dfrac{1}{2}, \dfrac{\sqrt3}{6}, \dfrac{\sqrt6}{3}\biggr),\\* w_5&=w_4-w_2,&\quad w_6&=w_4-w_1,\quad l\geq1, \end{alignat*}

the following inequality holds:

\begin{equation*} \textnormal{dist}^2 (A,\ \textnormal{SO}(3))\,:\!=\inf_{R\in \textnormal{SO}(3)} |A-R|^2\leq C_1 \sum_{i=1}^6 (|Aw_i|-1)^2.\end{equation*}

A similar theorem holds for octahedra. Let $\mathcal{O}$ denote an octahedron with vertices $P_i$, $i\in\{1,\dots,6\}$, and edges $P_iP_j$ for $i\not= j$ (mod 3).

Lemma 2.2. (Lemma 3.4 of [Reference Lazzaroni, Palombaro and Schlömerkemper13]) There is a constant $C_2>0$ such that

\begin{equation*} \textnormal{dist}^2(\nabla u,\ \textnormal{SO}(3))\leq C_2 \sum_{i\not= j \, \text{(mod\ 3)}} (|u(P_iP_j)|-1)^2 \quad \text{almost everywhere in $ \mathcal{O}$,}\end{equation*}

for every $u \in \mathcal{C}^0(\mathcal{O};\, \mathbb{R}^3)$ such that u is piecewise affine with respect to the triangulation determined by cutting $\mathcal{O}$ along the diagonal $P_1P_4$, $\det (\nabla u) > 0$ a.e. in $\mathcal{O}$, and $u(\mathcal{O})$ is convex.

Now we prove the estimate mentioned, which provides control over the $L^2$-distance of $\nabla\hat\omega$ from the scaled identity matrix in terms of the edge length deviations.

Lemma 2.3. For a polyhedron $\triangle\in\mathcal{T}$, let $\mathcal{E}(\triangle)$ denote the set of edges of $\triangle$. There is a constant $c>0$ such that, for all $n\geq 1$ and $1<l<1+\alpha$, the inequality

(3)\begin{equation}\| \nabla\hat\omega-l\ \textnormal{Id} \|^2_{L^2(U_n)}\leq c \sum_{\triangle\in\mathcal{T}_n} \sum_{\{x, y\}\in {\mathcal{E}(\triangle)}} (|\omega(x)-\omega(y)|-1)^2\end{equation}

holds for all $\omega\in\Omega_{n, l}$, and hence

(4)\begin{equation}\mathbb{E}_{\mathbb{P}_{n,l}}[ \| \nabla\hat\omega-l\ \textnormal{Id} \|^2_{L^2(U_n)} ]\leq c \sum_{\triangle\in\mathcal{T}_n} \sum_{\{x, y\}\in {\mathcal{E}(\triangle)}} \mathbb{E}_{\mathbb{P}_{n,l}}[ (|\omega(x)-\omega(y)|-1)^2 ] {,}\end{equation}

where the $L^2$-norm is defined with respect to the scalar product on $\mathbb{R}^{3\times 3}$ that induces the Frobenius norm, and $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^3$.

Note that the right-hand side of equation (3) is strictly positive because of the boundary conditions (1) and because $l>1$, whereas the left-hand side is zero for $\omega=\omega_l\in\Omega^{\text{per}}_{n,l}$. Since the measure $\mathbb{P}_{n,l}$ is supported on the set $\Omega_{n, l}$, (4) follows from (3). Also, note that c does not depend on n.

Proof. Let $\omega\in\Omega_{n, l}$ and $\mathcal{E}(\triangle)$ be the set of edges of a polyhedron $\triangle\in\mathcal{T}_n$. By Lemmas 2.1 and 2.2, we conclude that on every polyhedron $\triangle\in\mathcal{T}_n$ we have

\begin{equation*}\textrm{dist}^2(\nabla\hat\omega|_\triangle ,\ \textrm{SO}(3)) \leq \max \{C_1, C_2\} \sum_{\{x, y\}\in {\mathcal{E}(\triangle)}} (|\omega(x)-\omega(y)|-1)^2{,}\end{equation*}

where we used $(\Omega 1)$, $(\Omega 3)$, and $(\Omega 4)$ to apply Lemmas 2.1 and 2.2 and with the constants $C_1, C_2$ from Lemmas 2.1 and 2.2. Orthogonality of functions which are non-zero only on disjoint polyhedra gives

\begin{equation*}\| \textrm{dist}(\nabla\hat\omega, \textrm{SO}(3)) \|^2_{L^2(U_n)}\leq C \sum_{\triangle\in\mathcal{T}_n} \sum_{\{x, y\}\in {\mathcal{E}(\triangle)}}(|\omega(x)-\omega(y)|-1)^2{,}\end{equation*}

with constant $C=\max\{ C_1,\ C_2\} \max\{\sqrt 2 /12, \sqrt 2 /3\}$, where the second factor is the maximum of the volumes of a regular tetrahedron and octahedron. Applying Theorem 2.2 about geometric rigidity, we find an $R(\omega)\in\textrm{SO}(3)$ such that

\begin{equation*}\| \nabla\hat\omega-R(\omega) \|^2_{L^2(U_n)}\leq K \| \textrm{dist}( \nabla\hat\omega, \textrm{SO}(3)) \|^2_{L^2(U_n)},\end{equation*}

with a constant $K>0$ that does not depend on n by Remark 2.1. Due to the periodic boundary conditions (1), the function $\hat\omega-l\ \textrm{Id}$ is n-periodic in the directions $t_1, t_2, t_3$, that is,

\begin{equation*} \hat\omega(x+nt_i)-l (x+nt_i)=\hat\omega(x)-lx \quad \text{for all $ x\in \mathbb{R}^3$ and $ i \in \{1,2,3\}$.}\end{equation*}

By the fundamental theorem of calculus, the gradient of a periodic function is orthogonal to any constant function, and therefore

\begin{equation*}\| \nabla\hat\omega-l\ \textrm{Id} \|^2_{L^2(U_n)}+\| l\ \textrm{Id}-R(\omega) \|^2_{L^2(U_n)}=\| \nabla\hat\omega-R(\omega) \|^2_{L^2(U_n)}\end{equation*}

by Pythagoras. Since $\mathbb{P}_{n,l}$ is supported on the set $\Omega_{n,l}$, the lemma is established with $c=C K$.

With Lemma 2.3 we can now prove Theorem 2.1.

Proof of Theorem 2.1. A generalization of Heron’s formula for tetrahedra gives the volume $\lambda(\triangle)$ of the tetrahedron $\triangle$ with edge lengths u, v, w, U, V, W (opposite edges denoted by the same letter, lower-case and upper-case):

(5)\begin{equation}\lambda(\triangle)=\dfrac{\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}}{192\ u v w}{,}\end{equation}

with

\begin{alignat*}{3} X &= (w - U + v) (U + v + w){,} &\quad a &= \sqrt{x Y Z}{,}\\* x &= (U - v + w) (v - w + U){,} &\quad b &= \sqrt{y Z X}{,}\\ Y& = (u - V + w) (V + w + u){,} &\quad c &= \sqrt{z X Y}{,}\\ y &= (V - w + u) (w - u + V){,} &\quad d &=\sqrt{x y z}{,}\\y &= (V - w + u) (w - u + V){,}\\Z &= (v - W + u) (W + u + v){,}\\*z &= (W - u + v) (u - v + W).\end{alignat*}

By first-order Taylor approximation of (5) at the regular tetrahedron $\triangle_1$, denoting the edge lengths $a_i$, $i\in\{1,\dots,6\}$, we obtain

\begin{equation*}\lambda(\triangle)-\lambda({\triangle_1})=\dfrac{1}{12\sqrt{2}}\sum_{i=1}^6(a_i-1)+o\Biggl(\sum_{i=1}^6 |a_i-1|\Biggr)\quad \text{as $ a_i\to1$ for all \textit{i}.}\end{equation*}

For the octahedron, we obtain ${{1}/{(6\sqrt{2})}}$ for the volume derivative in one edge $b_1$ at $b_1=1$ and the remaining 11 edges fixed at $b_i=1$. This can be achieved by dividing the octahedron into four tetrahedra that all have a common edge d that is a diagonal of the octahedron adjacent to x. Using the formula (5) and some elementary geometry of a regular trapezoid to see that $d=\sqrt{x+1}$, we obtain with the regular octahedron $\octagon_1$ of edge length 1:

\begin{equation*}\lambda(\octagon)-\lambda({\octagon_1})=\dfrac{1}{6\sqrt{2}}\sum_{i=1}^{12}(b_i-1)+o\Biggl(\sum_{i=1}^{12} |b_i-1|\Biggr)\quad \text{as $ b_i\to1$ for all \textit{i}.}\end{equation*}

We only need that the partial derivatives of the volume at $\triangle_1$ and $\octagon_1$ are positive. By continuity, in a small neighborhood of the regular polyhedra, increasing one edge length increases the volume. Therefore we can choose $\alpha>0$ from the definition of permitted configurations so small that the polyhedra of the tessellation obtain minimal volume as the edge lengths go to 1. We choose $c_1>12\sqrt{2}$ and a corresponding $\alpha>0$ so small that the inequalities

(6)\begin{equation} \begin{aligned}\sum_{i=1}^6(a_i-1) \leq c_1 (\lambda(\triangle)-\lambda({\triangle_1})){,} \\*\sum_{i=1}^{12}(b_i-1) \leq c_1 (\lambda(\octagon)-\lambda({\octagon_1}))\end{aligned}\end{equation}

are satisfied whenever $1<a_i<1+\alpha$ and $1<b_i<1+\alpha$. Let us fix such $c_1>0$ and $\alpha>0$ and assume that $\Omega^{\text{per}}_{n,l}$ is defined by means of this $\alpha.$ Using (6) we can also estimate the squared edge length deviations:

(7)\begin{equation} \begin{aligned}\sum_{i=1}^6(a_i-1)^2 \leq c_1\ \alpha\ (\lambda(\triangle)-\lambda({\triangle_{1}})){,} \\*\sum_{i=1}^{12}(b_i-1)^2 \leq c_1\ \alpha\ (\lambda(\octagon)-\lambda({\octagon_1})){.}\end{aligned}\end{equation}

By equation (3) from Lemma 2.3 and (7), we get an upper bound on $\|\nabla\hat\omega-l\ \textnormal{Id}\|_{L^2(U_n)}^2$ in terms of the area differences. By summing up the contributions (7) of the polyhedra $\triangle\in\mathcal{T}_n$, we conclude for all $\omega\in\Omega_{n,l}$ that

(8)\begin{equation}\| \nabla\hat\omega-l\ \textnormal{Id} \|_{L^2(U_n)}^2 \leq c_1 \ \alpha\ c \sum_{\triangle\in\mathcal{T}_n}(\lambda(\hat\omega(\triangle))-\lambda({\triangle})).\end{equation}

As a consequence of $(\Omega 2)$ and the periodic boundary conditions (1), the right-hand side of (8) does not depend on $\omega\in\Omega_{n,l}$. Hence, with $\omega_l\in\Omega_{n,l}$ we can compute

(9)\begin{equation}\sum_{\triangle\in\mathcal{T}_n}(\lambda(\hat\omega(\triangle))-\lambda({\triangle}))=\sum_{\triangle\in\mathcal{T}_n}(\lambda(\hat\omega_l(\triangle))-\lambda({\triangle}))=|U_n|(l^3-1).\end{equation}

Combining the equations (8) and (9) gives

(10)\begin{equation}\| \nabla\hat\omega-l\ \textnormal{Id} \|_{L^2(U_n)}^2 \leq c_1 \ \alpha\ c\ |U_n|\ (l^3-1).\end{equation}

The reference measure $\delta_0\otimes\lambda^{I_n\setminus\{0\}}$ and the set of permitted configurations $\Omega_{n,l}$ are invariant under the translations

\begin{equation*}\psi_b\colon \Omega^{\text{per}}_{n,l} \to \Omega^{\text{per}}_{n,l} \quad (\omega(x))_{x\in I} \mapsto (\omega(x + b) - \omega(b))_{x\in I}\end{equation*}

for $b\in T$. As a consequence the matrix-valued random variables $\nabla(\hat\omega(\triangle))$ are identically distributed for $\triangle, \widetilde\triangle\in\textrm{triang}(\mathcal{T}_n)$ such that $\triangle = \widetilde\triangle$ (mod T). Thus, for any $\triangle\in \textrm{triang}(\mathcal{T}_1)$, the random variables $\nabla(\hat\omega(\triangle+t))_{t\in T}$ are identically distributed. Therefore

\begin{equation*}\mathbb{E}_{\mathbb{P}_{n,l}} [ \| \nabla\hat\omega-l\ \textnormal{Id} \|_{L^2(U_n)}^2 ] =\sum_{\triangle\in \textrm{triang}(\mathcal{T}_1)} |U_n(\triangle)| \ \mathbb{E}_{\mathbb{P}_{n,l}}[ |\nabla\hat\omega(\triangle)-l\ \textnormal{Id}|^2 ]{,}\end{equation*}

with the regions $U_n(\triangle)$ of $U_n$ taken up by T-translates of $\triangle$. As the proportions $|U_n(\triangle)|/|U_n|$ are independent of n for any $\triangle\in \textrm{triang}(\mathcal{T}_1)$, this equation together with (10) implies

\begin{equation*}\lim_{l\downarrow1}\sup_{n\in\mathbb{N}}\sup_{\triangle\in\textrm{triang}(\mathcal{T}_n)} \mathbb{E}_{\mathbb{P}_{n,l}}[ |\nabla\hat\omega(\triangle)-l\ \textnormal{Id}|^2 ]=0.\end{equation*}

By means of the triangle inequality, we see that, for all $\triangle\in\textrm{triang}(\mathcal{T}_n)$ and $\omega\in\Omega_{n,l}$,

\begin{equation*}|\nabla\hat\omega(\triangle)- \textnormal{Id}|^2\leq |\nabla\hat\omega(\triangle)-l\ \textnormal{Id}|^2+c_2^2(l-1)^2+2 c_2\ |l-1|\ |\nabla\hat\omega(\triangle)-l\ \textnormal{Id}|{,}\end{equation*}

with $c_2=|\text{Id}|>0$. For $\omega\in\Omega_{n,l}$, the term $|\nabla\hat\omega(\triangle)-l\ \textnormal{Id}|$ is uniformly bounded for $l\in (1, \alpha)$ and $n\in\mathbb{N}$, which proves the theorem.

3.1 Definitions

Let us cite some definitions from [Reference Dereudre, Drouilhet and Georgii4]. We equip the plane $\mathbb{R}^2$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^2)$ and we let $\lambda$ denote the Lebesgue measure on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. The characters $\Lambda$ and $\Delta$ will always denote measurable regions in $\mathbb{R}^2$ and the notation $\Delta\Subset\mathbb{R}^2$ means that in addition $\Delta$ is bounded. Consider the set $\mathcal{X}\subset 2^{(\mathbb{R}^2)}$ of locally finite point configurations in $\mathbb{R}^2$. That means $X\in \mathcal{X}$ is a subset $X\subset \mathbb{R}^2$, and for any $\Delta\Subset\mathbb{R}^2$, the intersection $X_\Delta\,:\!=\textrm{pr}_\Delta(X)\,:\!=X\cap \Delta$ has finite cardinality $|X_\Delta|<\infty$. The counting variables $N_\Delta(X)\,:\!=|X_\Delta|$ generate a $\sigma$-algebra $\mathcal{A}\,:\!=\sigma(N_\Delta \colon \Delta\Subset\mathbb{R}^2 )$ on $\mathcal{X}$. The union of $X, Y\in \mathcal{X}$ will be denoted by XY; this will be used when defining the configuration $X_\Lambda Y_{\Lambda^c}$ that agrees with X on $\Lambda$ and with Y on the complement of $\Lambda$. In a sequence of set operations, unions XY are evaluated first in order to reduce brackets. On the measurable space $(\mathcal{X}, \mathcal{A})$, we consider the Poisson point process $\Pi^z$ with intensity $z>0$. The measure $\Pi^z$ is uniquely characterized by the properties that, for all $\Delta\Subset\mathbb{R}^2$ under $\Pi^z$, (i) $N_\Delta$ is Poisson-distributed with parameter $z\lambda(\Delta)$, and (ii) conditional on $N_\Delta=n$, the n points in $\Delta$ are independently and uniformly distributed on $\Delta$ for each integer $n\geq 1$. Similarly, configurations $\mathcal{X}_\Lambda=\{X_\Lambda\colon X\in \mathcal{X}\}$ in the set $\Lambda$ carry the trace $\sigma$-algebra $\mathcal{A}_\Lambda'\,:\!=\mathcal{A}|_{\mathcal{X}_\Lambda}$ and the reference measure $\Pi^z_\Lambda$, which is the law of $X_\Lambda$ if X is distributed according to $\Pi^z$. We will also need the pullback of $\mathcal{A}_\Lambda'$ to $\mathcal{X}$ defined by $\mathcal{A}_\Lambda\,:\!=\textrm{pr}_\Lambda^{-1} \mathcal{A}_\Lambda'\subset \mathcal{A}$. Finally we define the shift group $\Theta=\{\theta_r\colon r\in \mathbb{R}^2\}$, where $\theta_r\colon \mathcal{X}\to\mathcal{X}$ is the translation by $-r\in\mathbb{R}^2$, consequently $N_\Delta(\theta_r X)=N_{\Delta+r}( X)$ for all $\Delta \Subset \mathbb{R}^2$.

We fix $\alpha>0$ small enough; the size of $\alpha$ will be specified later. We change the notation of [Reference Gaál7] from $\epsilon$ to $\alpha$ at this point to emphasize that $\alpha$ is fixed and not particularly small. Let

\begin{equation*}\Lambda^{1+\alpha}\,:\!=\{x\in\mathbb{R}^2\colon |x-y|<1+\alpha \text{ for some } y\in\Lambda \}\end{equation*}

be the $(1+\alpha)$-enlargement of $\Lambda$. For $X\in\mathcal{X}$ we define the Hamiltonian $H_{\Lambda,Y}$ in $\Lambda$ with boundary condition $Y\in\mathcal{X}$ by

\begin{equation*}H_{\Lambda,Y}(X)\,:\!=\begin{cases}0 & \textrm{if $ |x-y|>1$ whenever $ x \in X_{\Lambda}Y_{\Lambda^{1+\alpha}\setminus \Lambda}$ and $ y\in X_\Lambda Y_{\Lambda^c} $ for $ x\not=y$,}\\&\textrm{and for all $ x \in X_{\Lambda}Y_{\Lambda^{1+\alpha}\setminus \Lambda}\colon | X_\Lambda Y_{\Lambda^c} \cap A_{1, 1+\alpha}(x) |=6$,} \\\infty & \textrm{otherwise}.\end{cases}\end{equation*}

That is, $H_{\Lambda,Y}(X)\in\{0,\infty\}$ takes the value 0 if and only if every point of $X_{\Lambda^{1+\alpha}}$ has distance greater than one from points in $X_\Lambda Y_{\Lambda^c}$ and has exactly six $X_\Lambda Y_{\Lambda^c}$-neighbors in the annulus $A_{1, 1+\alpha}(x)=\{y\in\mathbb{R}^2\colon |y-x|\in(1,1+\alpha)\}$; otherwise H is defined to be infinity. Note that the only part of the boundary condition Y relevant for $H_{\Lambda,Y}(X)$ is in the region $\Lambda^{2(1+\alpha)}\setminus \Lambda$.

Definition 3.1. We define the partition function $Z^z_{\Lambda, Y}$ by

\begin{equation*}Z^z_{\Lambda, Y}\,:\!=\Pi^z_\Lambda\{ X_\Lambda\colon H_{\Lambda,Y}(X_\Lambda)=0\}=\int \,\text{e}^{-H_{\Lambda,Y}(X)} \Pi^z_\Lambda(\text{d} X).\end{equation*}

We call a boundary condition $Y\in\mathcal{X}$ admissible for the region $\Lambda \Subset \mathbb{R}^2$ if $0<Z^z_{\Lambda, Y}$. We write $\mathcal{X}_*^{\Lambda, z}$ for the set of all these Y.

The set of admissible boundary conditions $\mathcal{X}_*^{\Lambda, z}$ is never empty as the $l\in(1,1+\alpha)$ scaling of a triangular lattice with lattice constant one is always in $\mathcal{X}_*^{\Lambda, z}$. We note that $H_{\Lambda,Y}(\emptyset)=0$ for $Y_{\Lambda^{1+\alpha}}=\emptyset$ and also for specifically chosen $\Lambda$ and possibly non-empty Y. The partition function $Z^z_{\Lambda, Y}$ is zero if neither $Y_{\Lambda^{1+\alpha}\setminus \Lambda}=\emptyset$ nor the boundary condition $Y_{\Lambda^{1+\alpha}\setminus \Lambda}$ can be extended to a near-triangular lattice configuration in $\Lambda^{1+\alpha}$.

Definition 3.2. For $Y\in \mathcal{X}_*^{\Lambda, z}$, we define the Gibbs distribution in the region $\Lambda \Subset \mathbb{R}^2$ with boundary condition Y by the formula

\begin{equation*}\gamma^z_{\Lambda}(F|Y)=\int_{\mathcal{X}_\Lambda} \mathds{1}_F(X Y_{\Lambda^c}) \,\text{e}^{-H_{\Lambda,Y}(X)} \Pi^z_\Lambda(\text{d} X)/Z^z_{\Lambda, Y},\end{equation*}

where $F\in\mathcal{A}$. Note that $\gamma^z_{\Lambda}(\cdot|Y)$ is a measure on the whole space $(\mathcal{X}, \mathcal{A})$.

In the case of $Y_{\Lambda^\alpha\setminus \Lambda}\not=\emptyset$, the $\mathcal{X}_\Lambda$-marginal of the measure $\gamma^z_{\Lambda}(\cdot|Y)$ is uniform on the configurations in $\mathcal{X}_\Lambda$ that extended $Y_{\Lambda^\alpha\setminus \Lambda}$ to a near-triangular lattice configuration in $\Lambda^\alpha$. Otherwise if $Y_{\Lambda^\alpha\setminus \Lambda}=\emptyset$, then $\gamma^z_{\Lambda}(\cdot|Y)=\delta_{Y_{\Lambda^c}}$. Note that $(F,Y)\in(\mathcal{A}, \mathcal{X})\mapsto \gamma^z_{\Lambda}(F|Y)$ is a probability kernel from $(\mathcal{X}, \mathcal{A}_{\Lambda^c})$ to $(\mathcal{X}, \mathcal{A})$, but the distribution $\gamma^z_{\Lambda}(\cdot|Y)$ has $\delta_{Y_{\Lambda^c}}$ as its marginal on $(\mathcal{X}_{\Lambda^c}, \mathcal{A}^{\prime}_{\Lambda^c})$.

Definition 3.3. (Infinite-volume Gibbs measure) A probability measure $\mathbb{P}$ on $(\mathcal{X},\mathcal{A})$ is an infinite-volume Gibbs measure for $z>0$ if $\mathbb{P}(\mathcal{X}_*^{\Lambda, z})=1$ and

\begin{equation*}\int f \text{d} \mathbb{P} = \int_{\mathcal{X}_*^{\Lambda, z}} \dfrac{1}{Z^z_{\Lambda, Y}}\int_{\mathcal{X}_\Lambda} f(XY_{\Lambda^c}) \,\text{e}^{-H_{\Lambda,Y}(X)} \Pi^z_\Lambda(\text{d} X) \mathbb{P}(\text{d} Y)\end{equation*}

for every $\Lambda\Subset \mathbb{R}^2$ and every measurable $f\colon \mathcal{X} \to [0,\infty)$. We denote the set of infinite-volume Gibbs measures by $\mathcal{G}^z$.

Note that the right-hand side of the defining equality is equal to $\mathbb{E}_\mathbb{P}[\gamma^z_\Lambda(f| \cdot)]$. Therefore a measure $\mathbb{P}$ is an infinite-volume Gibbs measure if and only if $\mathbb{P}\gamma^z_\Lambda=\mathbb{P}$ for every $\Lambda\Subset \mathbb{R}^2$, where the product is understood as taking the average with $\mathbb{P}$ in the second variable of $\gamma^z_\Lambda$. We can easily obtain a degenerate measure $\delta_\emptyset\in\mathcal{G}^z$, but we will consider more interesting Gibbs measures. In fact, as soon as $\mathbb{P}(\emptyset)=0$ for a measure $\mathbb{P}\in\mathcal{G}^z$, we have that $\mathbb{P}$ is supported on hard disk configurations with infinitely many disks.

The Hamiltonian H implements an example of a k-nearest-neighbor interaction, as explained in [Reference Dereudre, Drouilhet and Georgii4, Chapter 4.2.1]. Therefore, by [Reference Dereudre, Drouilhet and Georgii4, Lemma 5.1], the kernels $\gamma_\Lambda^z$, $\gamma_\Delta^z$ for $\Lambda\subset \Delta \Subset \mathbb{R}^2$ and $Y\in \mathcal{X}_*^{\Lambda, z}$ satisfy the consistency conditions $\gamma_\Lambda^z(\mathcal{X}_*^{\Lambda, z} | Y)=1$ and $\gamma_\Delta^z\gamma_\Lambda^z=\gamma_\Delta^z$, where the product is understood as the product of probability kernels.

3.3. Proofs

For a configuration $X\in \mathcal{X}$, we say that $H(X)=0$ if, for all $x,y\in X$, we have $|x-y|>1$ and $|X\cap A_{1, 1+\alpha}(x)|=6$. This is the same as having $H_{\Lambda, X}(X)=0$ for any $\Lambda\Subset\mathbb{R}^2$. For a configuration $\emptyset \not=X\in \mathcal{X}$ with $H(X)=0$, we can define a simplicial complex K(X) consisting of zero, one, and two cells defined as follows. The set of zero-cells $K_0(X)$ is $X \subset \mathbb{R}^2$. The set of one-cells $K_1(X)$ are edges between zero-cells of distance between 1 and $1+\alpha$, and the two-cells are triangles with sides in $K_1(X)$. We will see in the following lemma that, by definition of H and some geometric considerations, for $\alpha$ small enough, the graph defined by the one- and two-skeleton of this complex is locally – and therefore also globally – isomorphic to the triangular lattice $I=\mathbb{Z}+\tau \mathbb{Z}$ with $\tau=\text{e}^{{{{\text{i}}\pi}/{3}}}$ with edge set $E=\{\{i,j\}\subset I\colon |i-j|=1\}$. The set of triangles surrounded by three edges in E is denoted by $\mathcal{T}$; these are two-cells if we regard I as a simplicial complex.

The most important lemma linking the theorem above to [Reference Gaál7, Theorem 4.1] is as follows.

Later on, we will choose $\alpha$ sufficiently small that Lemma 3.1 and Theorem 3.1 both work for that $\alpha$. From the proof of the lemma it will be obvious that the choice of $\alpha$ does not need to be particularly small for it (and any smaller choice) to work.

Proof. For $i\in I$ we define its closest neighborhood $N(i)\subset I$ by $N(i)=\{j\in I\colon |i-j|\leq 1\}$. Let $X \in \mathcal{X}$ such that $H(X)=0$. A map $\omega\colon N(i)\to X$ is called a local isomorphism at i if, for all $j,k\in N(i)$, we have $|j-k|=1$ if and only if $|\omega(j)-\omega(k)|\in(1,1+\alpha)$. By taking $\alpha>0$ small enough, we can ensure that for all $i\in I$ and $x\in X$ there is a local isomorphism $\omega$ at i such that $\omega(i)=x$. To see this, observe that as $\alpha\to 0$, for every $y\in A_{1,1+\alpha}(x)$ there are exactly two points $y_1,y_2\in A_{1,1+\alpha}(x) \setminus\{y\}$ such that $|y_i-y|\to 1$; for other $z\in A_{1,1+\alpha}(x) \setminus\{y\}$, we have $\liminf_{\alpha\to 0} |z-y|\geq\sqrt 3$. Since we know that $|X\cap A_{1, 1+\alpha}(y)|=6$, a simple geometric consideration related to the kissing problem gives that $y_1, y_2\in A_{1, 1+\alpha}(y)$, since if $y_i \not \in A_{1, 1+\alpha}(y)$ for $i\in \{1,2\}$, for $\alpha$ small enough there was not enough space to place six points in $A_{1, 1+\alpha}(y)$ having distance larger than 1 from each other and from $y_i$. To be more precise, for all $i\in I$ and $x\in X$ there will be twelve such local isomorphisms taking rotations and reflection into account. We fix $\alpha$ sufficiently small that the local isomorphism property holds.

Let us construct a map $\omega\colon I\to X$ as follows. We fix an arbitrary $x_0\in X$ and define $\omega|_{N(0)}$ to be one of the six orientation-preserving local isomorphisms at 0 with $\omega(0)=x_0$. Fix a spanning tree T of I. For each $i\in I$, there is a unique path on nearest neighbors in T connecting 0 to i. Since there are local isomorphisms at each pair of points of I and X, we can uniquely extend $\omega$ to vertices of T by choosing the unique one of the six orientation-preserving local isomorphisms that is consistent with T. That is to say, if, for a neighbor i of j in T, we have already assigned a point $\omega(i)$, then we choose a local isomorphism at i with $i\mapsto \omega(i)$. Let us assign j to the point in X that is determined by this local isomorphism. Now there is only one local isomorphism at j which is consistent with the local isomorphism chosen at i in the sense that i has identical images under the two local isomorphisms. We use this local isomorphism to proceed with the construction and map all neighbors of j in T into X.

It remains to show that the map $\omega\colon I\to X$ is an isomorphism. To conclude that $\omega$ is an isomorphism onto its image, we fix a loop $\gamma$ starting and ending in $i\in I$ composed of a path in T and an edge between i and one of its neighbors in I to which it is not connected in T. We need to show that the map induced along $\gamma$ with the initial orientation-preserving local isomorphism $\omega|_{N(i)}$ at i maps to a loop in K(X) starting and ending in $\omega(i)$. To this end, we can show a seemingly more general but equivalent statement. Take any loop $\gamma=(i_0, i_1, i_2, \dots, i_n)$ at $0\in I$ (i.e. $i_0=i_n=0$) and $x\in X$, fix a local isomorphism at 0 with $0\mapsto x$, and show that the map induced along $\gamma$ maps $\gamma$ to a loop $\omega(\gamma)$ in X at x. Here $\omega$ is locally defined along the curve $\gamma$.

We can deform the loop $\gamma$ to the boundary of a two-cell that contains 0 by successively ‘removing’ two-cells that intersect $\gamma$ and are inside it. By removing a two-cell, we mean one of the following. Two subsequent edges $(i_{k-1}, i_{k})$, $(i_{k}, i_{k+1})$ of $\gamma$, can be exchanged for the unique edge $(i_{k-1},i_{k+1})$ if $|i_{k-1}-i_{k+1}|=1$, or we can exchange one edge $(i_k, i_{k+1})$ of $\gamma$ for two edges $(i_k, j)$ and $(j,i_{k+1})$ in I. For every such transformation of $\gamma$, we obtain a modified $\gamma'$ and a map $\omega'$ that is uniquely determined by the local isomorphism at $i_{k}$ and is the unique extension of the local isomorphism at 0 along $\gamma'$. Note that $\omega=\omega'$ on the domain that they are both defined and $\omega(\gamma)$ is closed if and only if $\omega'(\gamma')$ is. After removing finitely many two-cells, we arrive at $\gamma'=(0, i, j, 0)$ being the boundary of a two-cell that contains the origin. Since $\omega'|_{\gamma'}$ should be the unique extension of the local isomorphism at 0 along $\gamma$, we see that $\omega'(\gamma')$ is closed and therefore so is $\omega(\gamma)$.

We showed that for neighbors i, j in I, $\omega(i)$, $\omega(j)$ are also neighbors in X. To obtain the converse statement and the injectivity of $\omega$, we repeat the above procedure for the same map $\omega$ but with exchanged roles of I and X. This concludes the proof that $\omega$ is an isomorphism onto its image.

It remains to show that $\omega$ is surjective. Now take a curve $\hat \gamma$ in K(X) from $x_0$ to some $y\in K(X)$. Note that K(X) is a connected graph, as for small enough $\alpha$ and $x\not=y$ we can always find a neighbor z of x which is closer to y than x. The curve $\hat\gamma$ corresponds to a curve $\gamma$ in I from 0 to some $i\in I$. Applying the above procedure to the concatenation of the path from 0 to i in T and the reverse of $\gamma$, we see that $\omega(i)=y$.

This lemma can also be proved with the formalism of Čech cohomology using the de Rham isomorphism and can be generalized to configurations with point defects (missing points). The usefulness of the Čech cohomology and de Rham’s theorem was pointed out to us by Franz Merkl. We decided to give another proof using less formalism.

To construct $\mathbb{P}_\rho$, we use measures on periodic configurations. For $l>1$ and $n\in\mathbb{N}$, let us define measures $P_{n,l}$ on n-periodic configurations as in [Reference Gaál7]. A periodic, enumerated configuration $\omega\in\Omega^{\text{per}}_{n, l}$ is a map $I\to\mathbb{R}^2$ such that Theorem 3.2 holds true for this choice of $\alpha$. Thus

(11)\begin{equation}\omega(i+nj)=\omega(i)+lnj \quad \text{for all $ i, j \in I$.}\end{equation}

It suffices to define an n-periodic, enumerated configuration on a set of $n^2$ representatives $I_n\subset I$ as equation (11) uniquely defines the configuration on the complement $(I_n)^c$. The event of admissible, n-periodic, enumerated configurations $\Omega_{n, l}\subset \Omega^{\text{per}}_{n, l}$ is defined by the properties $(\Omega 1)$–$(\Omega 3)$.

• $(\Omega 1)$$ |\omega(i)-\omega(j)|\in (1, 1+\alpha)$ for all $\{i,j\}\in E$.

For $\omega\in\Omega$ we define the extension $\hat\omega\colon \mathbb{R}^2\to\mathbb{R}^2$ such that $\hat\omega(i)=\omega(i)$ if $i\in I$, and on the closure of any triangle $\triangle\in\mathcal{T}$, the map $\hat\omega$ is defined to be the unique affine linear extension of the mapping defined on the corners of $\triangle$.

• $(\Omega 2)$ The map $\hat\omega\colon \mathbb{R}^2\to\mathbb{R}^2$ is injective.

• $(\Omega 3)$ The map $\hat\omega$ is orientation-preserving, that is, $\det(\nabla\hat\omega(x))>0$ for all $\triangle\in\mathcal{T}$ and $x\in\triangle$ with the Jacobian $\nabla\hat\omega\colon \cup\mathcal{T}\to\mathbb{R}^{2\times2}$.

Define the set of admissible, n-periodic, enumerated configurations as

\begin{equation*}\Omega_{n, l}=\{\omega\in\Omega_{n, l}^{\text{per}}\mid \omega\ \text{satisfies}\ (\Omega1)\textrm{--}(\Omega3)\}.\end{equation*}

Let the probability measure $\mathbb{P}_{n, l}$ be

\begin{equation*}\mathbb{P}_{n, l}(A)=\dfrac{\delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l}\cap A)}{\delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l})}\end{equation*}

for any Borel-measurable set $A\in \mathcal{F}_n=\bigotimes_{i\in I_n}\mathcal{B}(\mathbb{R}^2)$; thus $\mathbb{P}_{n, l}$ is the uniform distribution on the set $\Omega_{n, l}$ with respect to the reference measure $\delta_0\otimes\lambda^{I_n \setminus\{0\}}$. The first factor in this product refers to the component $\omega(0)$. The parameter l in the definition of $\Omega_{n,l}$ and $\mathbb{P}_{n,l}$ controls the density of periodic configurations such that $\rho={{2}/{(l^2\sqrt{3})}}$. We quote Theorem 4.1 from [Reference Gaál7], which will be the major ingredient of the proof of Theorem 3.1.

Theorem 3.2. For any $\alpha>0$ small enough we have

\begin{equation*}\lim_{l\downarrow 1}\sup_{n\in\mathbb{N}}\sup_{\triangle\in\mathcal{T}} \mathbb{E}_{\mathbb{P}_{n,l}}[ |\nabla\hat\omega(\triangle)-\textnormal{Id} |^2 ]=0 {,}\end{equation*}

with the constant value of the Jacobian $\nabla\hat\omega(\triangle)$ on the set $\triangle\in\mathcal{T}$.

We note that the theorem holds for any $\alpha\in(0,\sqrt 3 -1)$, but we omit the proof, which is just a more careful consideration of arguments in the proof of [Reference Gaál7, Theorem 4.1], and will refer to small enough $\alpha$. The main observation needed for this explicit range of $\alpha$ where the theorem holds is that the area of triangles with side lengths in the range $[1,\sqrt 3)$ is uniquely minimized by the regular triangle with side length 1. This observation is then utilized as in the similar proof of Theorem 2.1 in the three-dimensional case. We note that Theorem 3.2 might work with $\alpha \geq \sqrt 3 -1$, but looking for the optimal upper bound is not the concern of this paper.

In the following we construct $\mathbb{P}_\rho$ as a limit of translation-invariant versions of $\mathbb{P}_{n,l}$ and show that this measure is a Gibbs measure in $\mathcal{G}^z$ for any $z>0$. We follow ideas from [Reference Dereudre, Drouilhet and Georgii4] to construct a limiting measure. Fix $l>1$ and define the measures $\mathbb{G}_n$ on $(\mathcal{X}, \mathcal{A})$ by specifying its marginal $(\mathbb{G}_n)_{\Lambda_n}$ on $(\mathcal{X}_{\Lambda_n},\mathcal{A}_{\Lambda_n}')$:

\begin{equation*}(\mathbb{G}_n)_{\Lambda_n}=\biggl(\dfrac{1}{\lambda( \Lambda_n)}\int_{ \Lambda_n} \textrm{Im}[\mathbb{P}_{n,l}]\circ \theta_r\, \text{d} r\biggr)_{\Lambda_n},\end{equation*}

with the image measure $\textrm{Im}[\mathbb{P}_{n,l}]$ of $\mathbb{P}_{n,l}$ under the map $\textrm{Im}\colon \omega\mapsto \{\omega(x) \colon x\in I\}$ and the domain $\Lambda_n=l\{x+y\tau \colon x,y\in[-n/2, n/2)\}$. The averaging over $r\in \Lambda_n$ is necessary to obtain a translation-invariant measure on the torus, since $\omega(0)=0$ holds $\mathbb{P}_{n,l}$-a.s. The measure $\mathbb{G}_n$ is then defined by having independent and identically distributed projections on the sets $\{ \Lambda_n+inl\}_{i\in I}$, which form a tiling of $\mathbb{R}^2$. In order to have translation-invariant probability measures on $(\mathcal{X}, \mathcal{A})$, we consider the averaged measures

\begin{equation*}\hat{ \mathbb{G}}_n=\dfrac{1}{\lambda( \Lambda_n)} \int_{ \Lambda_n} \mathbb{G}_n \circ \theta_r\ \text{d} r.\end{equation*}

By definition and the periodicity of $\mathbb{G}_n$, $\hat{ \mathbb{G}}_n$ are translation-invariant. We will show that the sequence $(\hat{ \mathbb{G}}_n)_{n\in\mathbb{N}}$ is tight in the topology of local convergence on translation-invariant probability measures on $\mathcal{X}$ generated by $\mathbb{P}\to \int f \,\text{d} \mathbb{P}$ for functions f that are $\mathcal{A}_\Lambda$-measurable for some $\Lambda\Subset \mathbb{R}^2$. We call such functions local, and denote the set of local functions by $\mathcal{L}$.

The only difference to the definitions after [Reference Dereudre, Drouilhet and Georgii4, Lemma 5.1] are in the nature of the measures $(\mathbb{G}_n)_{\Lambda_n}$. In our case $(\mathbb{G}_n)_{\Lambda_n}$ are measures that inherit geometric constraints from the structure of $\mathbb{P}_{n,l}$ that are defined on tori of different sizes. In [Reference Dereudre, Drouilhet and Georgii4], in contrast, the authors use measures $\mathbb{G}^{z}_{\Lambda_n, \bar\omega}$ that have fixed boundary condition $\bar\omega$ on the complement of $\Lambda_n$.

For a shift-invariant probability measure $\mathbb{P}$ on $(\mathcal{X}, \mathcal{A})$ and $\Lambda\Subset \mathbb{R}^2$, define the measure $\mathbb{P}_{\Lambda}\,:\!=\mathbb{P}\circ \textrm{pr}_{\Lambda}^{-1}$ and the relative entropy with respect to $\Pi_{\Lambda}^z$ as

\begin{equation*}I(\mathbb{P}_{\Lambda} \mid \Pi_{\Lambda}^z)\,:\!=\begin{cases}\displaystyle\int f \ln f \,\text{d} \Pi_{\Lambda}^z &\text{if $ \mathbb{P}_{\Lambda} \ll \Pi_{\Lambda}^z $ with density \textit{f},} \\\infty &\text{otherwise}.\end{cases}\end{equation*}

The specific entropy of $\mathbb{P}$ with respect to $\Pi^z$ is then defined by

\begin{equation*}I(\mathbb{P})\,:\!= \lim_{n\to \infty} \dfrac{1}{\lambda(\Delta_n)} I\big(\mathbb{P}_{\Delta_n} \mid \Pi_{\Delta_n}^z\big),\end{equation*}

where $\Delta_n\Subset\mathbb{R}^2$ is a cofinite increasing sequence of sets. We refer to [Reference Georgii8] and [Reference Georgii and Zessin9] for existence and properties of the specific entropy. We will set $z=1$ and compute entropies relative to $\Pi_{\Delta_n}^1$. By [Reference Georgii and Zessin9, Proposition 2.6], the sublevel sets of I are sequentially compact in the topology of local convergence. Therefore we only need to show that the specific entropies of the measures $\{\hat{ \mathbb{G}}_n\}_{n\in\mathbb{N}}$ are bounded by some constant. We start with a proposition that also appeared in [Reference Gaál6, Lemma 5.2] and provides a lower bound on the partition sum.

Proposition 3.1. For all $\alpha\in (0,1]$ and $l\in(1,1+\alpha)$, there is an $r=r(\alpha, l)\in(0,1/2)$ such that for $n\in\mathbb{N}$ we have

\begin{equation*}\delta_0\otimes\lambda^{I_n\setminus\{0\}}(\Omega_{n,l})\geq (\pi r^2)^{|I_n|-1}.\end{equation*}

Proof. For $r>0$, we define, like in (3.2) in [Reference Heydenreich, Merkl and Rolles11], the set of configurations which are close to the scaled, enumerated, standard configuration $\omega_l(i)=li$ for $i\in I$:

\begin{equation*}S_{n,l,r}=\{\omega\in \Omega_{n,l}^{\text{per}}\mid |\omega(i)-\omega_l(i)|<r \text{ for all } i\in I \}.\end{equation*}

For sufficiently small $r>0$, depending on $\alpha$ and l, we conclude, like in the proof of [Reference Heydenreich, Merkl and Rolles11, Lemma 3.1], that $S_{n,l,r}\subset\Omega_{n,l}$. To prove this inclusion, we have to show the properties $({\Omega 1})$–$({\Omega 3})$ for all $\omega\in S_{n,l,r}$. For $(i,j)\in E$ and $\omega\in S_{n,l,r}$, let us compute

\begin{align*}|\,|\omega(i)-\omega(j)|-l|&= |\,|\omega(i)-\omega(j)|- |\omega_l(i)-\omega_l(j)|\,|\\[4pt] &\leq |\omega(i)-\omega_l(i)| + |\omega(j)-\omega_l(j)|\\[4pt] &<2r.\end{align*}

If we choose $2r<\max\{l-1, 1+\alpha-l\}<1$, then $\omega$ satisfies $({\Omega 1})$. Condition $({\Omega 2})$ is a consequence of the inequality $\langle v, \nabla\hat\omega(x)v\rangle>0$ for all $v\in\mathbb{R}\setminus\{0\}$, and for all $x\in\mathbb{R}^2$ where $\hat\omega$ is differentiable. This inequality holds for small enough r since $\nabla\hat\omega$ is close to the identity uniformly on $\mathbb{R}^2$. Hence $\hat\omega$ is a bijection onto its image. Here we applied a theorem from analysis which states that a $\mathcal{C}^1$-map f from an open convex domain $U\subset\mathbb{R}^n$ into $\mathbb{R}^n$ with $\langle v, \nabla f(x)v\rangle>0$ for all $v\in\mathbb{R}^n\setminus\{0\}$ and $x\in U$ is a diffeomorphism onto its image. However, $\nabla\hat\omega(x)$ is only piecewise differentiable, but on the straight line L connecting $x,y\in \mathbb{R}^2$ with $x\not=y$ there are only finitely many points $z\in \mathbb{R}^2\cap L$ where the curve $(\hat\omega(ty+(1-t)x))_{t\in (0,1)}$ is not differentiable. Assume that $\langle v, \nabla\hat\omega(x)v\rangle>0$ holds whenever $\hat\omega$ is differentiable in x. The curve is piecewise linear, and on each of these pieces, the derivative of the curve forms an acute angle with $y-x$, so the curve cannot be closed. Thus the condition $({\Omega 2})$ is satisfied in the case of a sufficiently small r. Furthermore, condition $({\Omega 3})$ is satisfied by $\omega_l$, and therefore also by $\omega$ if r is sufficiently small. Hence $S_{n,l,r}\subset\Omega_{n,l}$ for some $r\in(0,1/2)$, and we conclude that

\begin{equation*}\delta_0\otimes\lambda^{I_n\setminus\{0\}}(\Omega_{n,l})\geq\delta_0\otimes\lambda^{I_n\setminus\{0\}}(S_{n,l,r})=(\pi r^2)^{|I_n|-1}{,}\end{equation*}

where the last equality is obtained by integrating over each $\omega(i)$ with $i\not= 0$ successively along a fixed spanning tree of $I_n$ which gives a factor $\pi r^2$, and considering that $\omega_l(0)=0$ and that the measure $\delta_0\otimes\lambda^{I_n\setminus\{0\}}$ fixes $\omega(0)=0$.

Proposition 3.2. The set $\{I(\hat{ \mathbb{G}}_n)\colon n\in\mathbb{N} \}$ is bounded, thus the set $\{\hat{ \mathbb{G}}_n\colon n\in\mathbb{N} \}$ is sequentially compact in the topology of local convergence. Therefore there is a sequence $n_k\to\infty$ and a shift-invariant measure $\mathbb{P}_\rho$ on $(\mathcal{X}, \mathcal{A})$ such that

\begin{equation*} \lim_{k\to\infty}\int f \,\text{d} \hat{ \mathbb{G}}_{n_k}=\int f \,\text{d} \mathbb{P}_\rho\quad \text{for any $f\in \mathcal{L}$.} \end{equation*}

Proof. As also noted in the proof of [Reference Dereudre, Drouilhet and Georgii4, Proposition 5.3], the definition of $\hat{ \mathbb{G}}_n$ implies that

\begin{equation*}I^z(\hat{ \mathbb{G}}_n)=\dfrac{1}{\lambda( \Lambda_n)}I \big((\mathbb{G}_n)_{ \Lambda_n} \mid \Pi_{\Lambda_n}^1\big).\end{equation*}

The relative entropy $I((\mathbb{G}_n)_{ \Lambda_n} \mid \Pi_{\Lambda_n}^1)$ can be explicitly computed as follows. The measure $(\mathbb{G}_n)_{ \Lambda_n}$ is supported on configurations that have $n^2$ points in $ \Lambda_n$, and if $\Lambda_n$ is folded into a torus, then each point x has exactly six neighbors in the annulus $A_{1,1+\alpha}(x)$ around it and no points closer than distance one. These configurations $\mathcal{X}_{n,l}$ are images of enumerated configurations $\mathcal{X}_{n,l}=(\textrm{Im}\ \Omega_{n,l})_{ \Lambda_n}$. By Lemma 3.1, $(\mathbb{G}_n)_{\Lambda_n}$ is the uniform distribution on these configurations with respect to $\Pi_{\Lambda_n}^1$. The density of $(\mathbb{G}_n)_{\Lambda_n}$ with respect to $\Pi_{\Lambda_n}^1$ is given by $f=\mathds{1}_{\mathcal{X}_{n,l}}/ \Pi_{\Lambda_n}^1(\mathcal{X}_{n,l})$. To find the constant $\Pi_{\Lambda_n}^1(\mathcal{X}_{n,l})$ more explicitly, consider the expectation

\begin{equation*}\Pi^1_{\Lambda_n}[g]=\text{e}^{-\lambda(\Lambda_n)} \sum_{k=0}^\infty \int_{\Lambda_n^k}\dfrac{1}{k!}g(\{x_1, \dots, x_k\}) \ \lambda^k|_{\Lambda_n^k}(\text{d} x_1, \dots, \text{d} x_k){.}\end{equation*}

Consequently we have

\begin{equation*}\Pi_{\Lambda_n}^1(\mathcal{X}_{n,l})=\dfrac{\text{e}^{-\lambda(\Lambda_n)}}{n^2}\lambda(\Lambda_n) \ \delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l}).\end{equation*}

This follows since a factor ${{\text{e}^{-\lambda(\Lambda_n)}}/{(n^2)!}}$ comes from the density of $ \Pi_{\Lambda_n}^1$ conditioned on $n^2$ points with respect to

\begin{equation*}\lambda^{(n^2)}\big|_{\Lambda_n^{(n^2)}}(\text{d} x_1, \ldots, \text{d} x_n^2).\end{equation*}

Then, conditioned on the position of $x_1$, the volume of the permitted configurations by their shift invariance on the torus is $(n^2-1)!\ \delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l})$; furthermore, the first point can be distributed uniformly in $\Lambda_n$. The relative entropy is $I((\mathbb{G}_n)_{ \Lambda_n} \mid \Pi_{\Lambda_n}^1)=-\ln(\Pi_{\Lambda_n}^1(\mathcal{X}_{n,l}))$ and the specific entropy can be bounded using Proposition 3.1 and $\lambda( \Lambda_n)=n^2 l^2 \sqrt{3}/2$ for sufficiently large n, we obtain

\begin{align*} \hspace*{90pt} I((\mathbb{G}_n)_{ \Lambda_n}) &=-\dfrac{\ln(\Pi_{\Lambda_n}^1(\mathcal{X}_{n,l}))}{\lambda( \Lambda_n)}\\[3pt] &= 1+\dfrac{n^2}{\lambda( \Lambda_n)}-\dfrac{\ln ( \lambda( \Lambda_n))}{\lambda( \Lambda_n)}-\dfrac{\ln( \delta_0\otimes\lambda^{I_n \setminus\{0\}} (\Omega_{n, l}))}{\lambda( \Lambda_n)}\\[3pt] &\leq1+\dfrac{n^2}{\lambda( \Lambda_n)}-\dfrac{\ln ( \lambda( \Lambda_n))}{\lambda( \Lambda_n)}-\dfrac{|I_n-1| \ln(\pi r^2)}{\lambda( \Lambda_n)}\\[3pt] &\leq1+\dfrac{2-2\ln (\pi r^2)}{l^2 \sqrt{3}}.\hspace*{150pt}\Box\end{align*}

The next proposition shows that $\mathbb{P}_\rho$ is an infinite-volume Gibbs measure. Note that $\hat{ \mathbb{G}}_n$ and $\Lambda_n$ depend on $l>1$, which we fixed previously.

Proof. Fix $\Lambda \Subset \mathbb{R}^2$, $z>0$ and $\rho<2/\sqrt{3}$ sufficiently large that $2/(\sqrt{3}(1+\alpha)^2)<\rho$, where $\alpha$ is such that Lemma 3.1 holds with that $\alpha$. Let $l>1$ such that $\rho=2/(l^2\sqrt{3})$. For $X\in \mathcal{X}$, let $\widetilde X_n$ be the periodic extension of $X_{\Lambda_n}$ to $\mathcal{X}$, i.e. $\widetilde X_n=\cup_{i\in I} X_{\Lambda_n}+lni$. Let $\kappa>0$ be so large that $\Lambda^{\kappa}\setminus \Lambda$ contains a connected ring of triangles from $K_2(\widetilde X_n)$ for $\mathbb{G}_n$-almost all X for all $n\in\mathbb{N}$. Consequently, for all $n\in\mathbb{N}$ sufficiently large that $\Lambda^{\kappa}\subset\Lambda_n$, the number of points in $\Lambda$ conditioned on $X_{\Lambda^c}$, is $\mathbb{G}_n$-a.s. determined by the configuration in $\Lambda^{\kappa}\setminus \Lambda$. The measure $(\mathbb{G}_n)_{\Lambda_n}$ is the uniform distribution of enumerable permitted configurations with $n^2$ points on the torus. By Lemma 3.1, the conditional distribution of $X_\Lambda$ given $X_{\Lambda^c}$ under $\mathbb{G}_n$ is therefore the uniform distribution on configurations $X_\Lambda$ such that $H_{\Lambda, X_{\Lambda^c}}(X_\Lambda)=0$. Uniform distribution makes sense, as the number of points in $\Lambda$ is almost surely constant with respect to the conditioned measure. Therefore the factorized version of the conditional distribution of $\mathbb{G}_n$ given $\mathcal{A}_{\Lambda^c}$ is given by $\gamma_\Lambda(\cdot \mid \cdot)$, that is,

(12)\begin{equation}\mathbb{G}_n(F)=\int_{\mathcal{X}}\gamma_{\Lambda}(F\mid Y) \mathbb{G}_n(\text{d} Y)\end{equation}

for any $F\in \mathcal{A}$ and $n\in\mathbb{N}$ large enough for $\Lambda^{\kappa}\subset\Lambda_n$. Since z is fixed, we can omit it as a superscript in $\gamma^z$.

The rest of the proof is as the proof of [Reference Dereudre, Drouilhet and Georgii4, Proposition 5.5]. Define

\begin{equation*}\Lambda^\circ_n\,:\!=\{r\in\mathbb{R}^2 \colon \Lambda^{\kappa}+r\subset \Lambda_n \}\end{equation*}

and the (subprobability) measures

\begin{equation*}\bar{ \mathbb{G}}_n\,:\!= \dfrac{1}{|\Lambda_n|}\int_{\Lambda^\circ_n} \mathbb{G}_n\circ \theta_r^{-1} \, \text{d} r.\end{equation*}

Then

\begin{equation*}\int f \,\text{d} \hat{\mathbb{G}}_n - \int f \,\text{d} \bar{\mathbb{G}}_n\to 0\end{equation*}

by the same argument as in [Reference Georgii and Zessin9, Lemma 5.7], so $\mathbb{P}_\rho$ can also be seen as an accumulation point of the sequence $(\bar{\mathbb{G}}_n)$. Let $F \in \cup_{\Delta\Subset \mathbb{R}^2}\mathcal{A}_{\Delta}$ be a local set; using (12), we obtain for $r\in\Lambda^\circ_n$

\begin{equation*}\mathbb{G}_n\circ \theta_r^{-1}(F)=\int_{\mathcal{X}} \gamma_{\Lambda}(F\mid Y)\mathbb{G}_n\circ \theta_r^{-1}(\text{d} Y).\end{equation*}

Therefore averaging over $r\in\Lambda^\circ_n$ gives

(13)\begin{equation}\bar{\mathbb{G}}_n(F)=\int_{\mathcal{X}} \gamma_{\Lambda}(F\mid Y)\bar{\mathbb{G}}_n(\text{d} Y).\end{equation}

Since the integrand on the right is a local function of Y, we can set $n=n_k$ and let $k\to \infty$, which gives (13) for $\mathbb{P}_\rho$ instead of $\bar{\mathbb{G}}_n$. Since local sets generate the $\sigma$-algebra $\mathcal{A}$, (13) holds for $\mathbb{P}_\rho$ and $F\in\mathcal{A}$, which by monotone convergence shows that $\mathbb{P}_\rho$ is an infinite-volume Gibbs measure.

Proof of Theorem 3.1. In Propositions 3.3 and 3.2 we showed the existence of a translation-invariant measure $\mathbb{P}_\rho\in\cap_{z>0}\mathcal{G}^z$ which is the local limit of the measures $(\mathbb{G}_{n_k})_{k\geq 1}$, so $\mathbb{P}_\rho$ satisfies property (ii). Property (i) holds as it can be expressed by a local function and $\mathbb{E}_{\mathbb{G}_{n_k}}[|X\cap B|]=\rho \lambda(B)$ for any $k\geq 1$ by the periodic boundary conditions. Similarly, property (iii) can be expressed by local functions depending on $\{x_0, x_1, \ldots, x_6\}\cap \Lambda_n$, where $x_0$ is the closest random point to the origin and $x_i$ is the ith closest point to $x_0$. For n large enough we have

\begin{equation*} \mathbb{G}_{n_k}(|\{x_0, x_1, \ldots, x_6\}\cap \Lambda_n|=7)=1 \end{equation*}

for any $k\geq 1$ and therefore

\begin{equation*} \mathbb{P}_\rho(|\{x_0, x_1, \ldots, x_6\}\cap \Lambda_n|=7)=1. \end{equation*}

By Theorem 3.2 we have

(14)\begin{equation}\lim_{\rho\uparrow 2/\sqrt{3}}\sup_{k\geq 1}\mathbb{E}_{\mathbb{G}_{n_k}}\Biggl[ \sum_{i=1}^6|\nabla\hat\omega(\triangle_i)-\textnormal{Id} |^2\Biggr]=0,\end{equation}

where $\{\triangle_i\}_{1\leq i\leq 6}$ are the random six triangles in $\mathcal{T}$ such that one of their vertices is mapped to $x_0$ under $\omega$. Let $f\colon \mathbb{C}^6\to \mathbb{R}$ be continuous, bounded, and permutation-invariant. We use the natural identification of topological spaces $\mathbb{C}\hat = \mathbb{R}^2$. Let $y_i=x_i-x_0$. By continuity of f, there is a constant $c>0$ such that

(15)\begin{equation}|f(y_1, \dots, y_6)-f(\text{e}^{{\text{i}}\pi/3}, \text{e}^{{\text{i}} 2\pi/3}, \dots, \text{e}^{{\text{i}} 2\pi}) |\leq c \sum_{i=1}^6|\nabla\hat\omega(\triangle_i)-\textnormal{Id} |^2\end{equation}

$\mathbb{G}_{n_k}$-a.s. for any $k\geq 1$. Combining equations (14) and (15), we obtain that

\begin{equation*}\lim_{\rho\uparrow 2/\sqrt{3}} \mathbb{E}_{\mathbb{P}_\rho} \big[|f(y_1, \dots, y_6)-f\big(\text{e}^{{\text{i}}\pi/3}, \text{e}^{{\text{i}} 2\pi/3}, \dots, \text{e}^{{\text{i}} 2\pi}\big) |\big]=0 {,}\end{equation*}

which concludes the proof of property (iii).