Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $\Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.


Introduction
In this paper we are interested in the variational analysis of the N -clock model (also known as planar Potts model or Z N -model) in the d-dimensional setting. The N -clock model is a nearest neighbors ferromagnetic spin model on the cubic lattice in which the spin field is constrained to take values in a set of N equispaced points of the unit circle S 1 . It plays a fundamental role in understanding phase transition phenomena in the theory of classical ferromagnetic spin fields, as it is closely related to the XY (planar rotator) model, for which the spin field is allowed to attain all the values of S 1 . In fact, the N -clock model is considered as an approximation of the XY model, as for N large enough it predicts Berezinskii-Kosterlitz-Thouless transitions [23], i.e., phase transitions mediated by the formation and interaction of topological singularities, the so-called vortices [8,24,25].
With the aim of describing the relation between the N -clock model and the XY model, probabilistic methods have been used in [22,26], while a variational analysis at zero temperature has been only very recently carried out in [20,21]. There the authors study the effective behavior of (suitably rescaled versions of) the energy of the N -clock model on the 2-dimensional square lattice εZ 2 , examining the case when the number N = N ε of equi-spaced points on S 1 depends on ε and diverges as ε → 0. The coarse grained model, which describes the microscopic/mesoscopic geometry of the spin field, is strongly affected by the rate of divergence of N ε → +∞ as ε → 0.
In this paper we advance the variational analysis of the N -clock model by considering the model on a d-dimensional cubic lattice εZ d , with d ≥ 2, in the case where the number N is fixed and independent of ε. We shall first identify the limit of the N -clock model as ε → 0 keeping N fixed and, at a second stage, we will let N → +∞. In contrast to the energy of the XY model, the energy resulting from this two-step limit process is by nature unfit to describe the concentration of energy around vortex-like singularities, indicating that the dependence of N on ε seems inevitable with the intent to approximate the XY model at zero temperature. To the best of our knowledge, the explicit identification of the limit energies in the ε → 0 and N → ∞ regimes and in any dimension makes the result contained in this paper the first quantitative answer to the question whether the N -clock model approximates the XY model at zero temperature. We shall see that the result is rather analogous to the limiting energy of the N ε -clock model in a specific rate of divergence N ε → +∞, chosen among those examined in the two-dimensional setting in [21]. To present in detail the results in this paper, we first summarize the analysis of [21], starting with some notation.
Given N ∈ N, we consider the set of N equispaced points on the unit circle S N := {exp ι 2π N k : k = 0, . . . , N − 1} , where ι is the imaginary unit. Given an open set Ω ⊂ R 2 , the energy associated to an admissible spin field u : εZ 2 → S Nε is given by where the sum is taken over ordered pairs of nearest neighbors i, j , i.e., (i, j) ∈ Z 2 ×Z 2 such that |i − j| = 1 and εi, εj ∈ Ω. We recall that a wide range of phenomena has been observed in [20,21] when exploring the possible regimes of N ε . Here we outline the one pertaining to the discussion in the present paper, namely N ε 1 ε| log ε| . The relevant scaling of the energy in this regime is Nε 2πε E Nε ε , sequences of spin fields u ε with equibounded energy accumulate to vector fields in BV (Ω; S 1 ), and the scaled energy Nε 2πε E Nε ε approximates an anisotropic total variation for maps in BV (Ω; S 1 ).
In the next theorem we state the result in the regime N ε 1 ε| log ε| rigorously. We denote by | · | 1 the 1-norm on vectors, by | · | 2,1 the anisotropic norm on matrices given by the sum of the Euclidean norms of the columns, and by d S 1 the geodesic distance on S 1 . For the notation concerning functions of bounded variation we refer to Subsection 5. Theorem 1.1. [21] Let Ω ⊂ R 2 be a bounded, open set with Lipschitz boundary. Assume that N ε 1 ε| log ε| . Then the following results hold true: i) (Compactness) Let u ε : Ω ∩ εZ 2 → S Nε be such that Nε 2πε E Nε ε (u ε ) ≤ C. Then there exists a subsequence (not relabeled) and a function u ∈ BV (Ω; S 1 ) such that u ε → u in L 1 (Ω; R 2 ). ii) ( Γ-liminf inequality) Assume that u ε : Ω ∩ εZ 2 → S Nε and u ∈ BV (Ω; S 1 ) satisfy u ε → u in L 1 (Ω; R 2 ). Then iii) ( Γ-limsup inequality) Let u ∈ BV (Ω; S 1 ). Then there exists a sequence u ε : Ω ∩ εZ 2 → S Nε such that u ε → u in L 1 (Ω; R 2 ) and We are now in a position to present the two main results in this paper. We shall consider Ω ⊂ R d a bounded, open set with Lipschitz boundary and the energy defined for admissible spin fields on the d-dimensional cubic lattice u : Ω ∩ εZ d → S N by where the sum is taken over ordered pairs of nearest neighbors i, j , i.e., (i, j) ∈ Z d ×Z d such that |i−j| = 1 and εi, εj ∈ Ω (the factor 1 2 accounts for the fact that each pair is counted twice). We state the first result concerning the limit of E N ε as ε → 0. For N fixed, the physical system is expected to behave like a classical Ising-type system with N phases. (See also [16,1,3,15,2,19,11,13,18,14] for the analysis of spin systems in the surface scaling.) According to the results proven for the Ising system, we expect the limit energy to be finite on functions of bounded variation with values in the finite set S N . In the next theorem we identify precisely the surface energy concentrated on the interfaces between the phases of the spin field. We denote by θ N := 2π N the smallest angle between two different vectors in S N . Theorem 1.2 (Limit as ε → 0). Let Ω ⊂ R d be a bounded, open set with Lipschitz boundary. Let N ≥ 2 and θ N := 2π/N . Then the following results hold true: Then there exists a subsequence (not relabeled) and a function u ∈ BV (Ω; iii) ( Γ-limsup inequality) Let u ∈ BV (Ω; S N ). Then there exists a sequence u ε : To clarify the expression of the limit functional in Theorem 1.2, we sketch here the proof of the Γ-limsup inequality in a very simple setting. Assume that Ω is the unit cube Q = (−1/2, 1/2) d and u is the pure-jump function with constant value In this case, the jump set is given by J u = (−1/2, 1/2) d−1 ×{0}. Then u ε is constructed by rotating k + times of an angle θ N starting from u − up to u + on hyperplanes parallel to the jump set, cf. Figure 1. More precisely, for 0 ≤ k ≤ k + we define and we put u ε (εi) = (1, 0) if εi · e d < 0 and u ε (εi) = exp(ιk + θ N ) if εi · e d > k + ε, instead. Between two hyperplanes there are 1 ε d−1 interacting pairs of nearest neighbors. For two such points εi, εj we have by a simple geometric argument |u ε (εi) − u ε (εj)| = 2 sin( θ N 2 ). Summing over all interactions we conclude that Since k + θ N = d S 1 (u − , u + ), the previous expression reduces to the one in Theorem 1.2 and makes clear the role of 4 sin 2 ( θ N 2 )/θ 2 N : it is the correcting factor which allows us to pass from the Euclidean distance between vectors to their geodesic distance. The proof of the upper bound is based on the construction in a more general setting of a recovery sequence which mimics the one presented here in the introduction, cf. Proposition 3.4. The proof of the lower bound is based on Lemma 3.1, which shows that the behavior described above is always the most convenient from an energetical point of view. Figure 1. On the left: a recovery sequence in the case of a jump set aligned with the lattice. The spin makes a transition from u − to u + jumping with the smallest possible non-zero angle θ N . On the right: Euclidean distance between two vectors of length 1 with angle θ N between them.
In Section 5 we also study the Γ-convergence of the functionals E N ε as ε → 0 under volume constraints on the phases of the spin fields or under Dirichlet boundary conditions.
We are now interested in the limit as N → +∞ of the energy defined by where θ N := 2π/N , i.e., the energy resulting from the limit process ε → 0 in Theorem 1.2. Up to the factor 4 , which is close to 1 for N large, the energy E N coincides (for d = 2) with the limiting energy of Theorem 1.1 restricted to Caccioppoli partitions taking values in S N . In the second result of this paper we show that the Γ-limit of E N as N → +∞ agrees with the limiting energy of Theorem 1.1. This is rigorously proved in the next theorem, which holds for any dimension d.
iii) ( Γ-limsup inequality) Let u ∈ BV (Ω; S 1 ). Then there exists a sequence u N : The proof of the upper bound in Theorem 1.3 is based on the following remark: a map u ∈ BV (Ω; S 1 ) can be approximated in energy by maps W 1,1 (Ω; S 1 ) which are smooth outside manifolds of codimension 2; such maps can be suitably sampled far from the singularities to define a u N ∈ BV (Ω; S N ); a crucial observation is that the precise definition of u N close to the singularities is not important, as the energy E N (u N ) does not concentrate close to manifolds of codimension 2. It is worth noticing that the latter feature is peculiar of this regime: in the other regimes studied in [20] where N = N ε depends on ε and N ε 1 ε| log ε| the behavior of the recovery sequence around the singularities becomes relevant and makes the generalization to the d-dimensional setting of the results in [20] more delicate and out of the scope of the present paper.

Notation and preliminary results
Let S d−1 = {x ∈ R d : |x| = 1} be the unit sphere. If u, v ∈ S 1 , their geodesic distance on S 1 is denoted by d S 1 (u, v). It is given by the angle in [0, π] between the vectors u and v, i.e., We denote the imaginary unit by ι. When it is convenient we will tacitly identify R 2 with the complex plane C. Given a vector a = ( We define the (2, 1)-norm of a matrix A = (a ij ) d i,j=1 ∈ R d×d as the sum of the Euclidean norms of its columns, i.e., Given a unit vector ν ∈ S d−1 , we denote by Q ν a cube with two sides orthogonal to ν, namely, we consider an orthonormal basis (ν, ν 2 , . . . , ν d ) of R d and we define For two sequences α ε and β ε of positive numbers, we write α ε β ε if lim ε→0 αε βε = 0.

BV-functions. In this section we recall basic facts about functions of bounded variation.
For more details we refer to the monograph [7].
is a function of bounded variation if its distributional derivative Du is given by a finite matrix-valued Radon measure on O. In that case, where |Du| denotes the total variation measure of Du. The total variation with respect to the anisotropic norm |·| 2,1 is denoted by We say that a sequence u n converges weakly * in BV (O; R n ) to u if u n → u in L 1 (O; R n ) and Du n * Du in the sense of measures.
We state some fine properties of BV -functions. To this end, we need some definitions. A function u ∈ L 1 (O; R n ) is said to have an approximate limit at x ∈ O whenever there exists z ∈ R n such that Next we introduce so-called approximate jump points. Given x ∈ O and ν ∈ S d−1 we set The triplet (a, b, ν) is determined uniquely up to the change to (b, a, −ν). We denote it by (u + (x), u − (x), ν u (x)) and we let J u be the set of approximate jump points of u. The triplet (u + , u − , ν u ) can be chosen as a Borel function on the Borel set J u . Denoting by ∇u the approximate gradient of u, we can decompose the measure Du as the sum where D (c) u is the so-called Cantor part and D (j We will need the slicing properties of BV -functions. Given a unit vector ξ ∈ S d−1 , we denote by Π ξ the hyperplane orthogonal to ξ. For every set E ⊂ R d and z ∈ Π ξ , the section of E corresponding to z is the set E ξ z := {t ∈ R : z + tξ ∈ E}. Accordingly, for any function Moreover it is possible to reconstruct the distributional gradient Du from the gradients of the slices Du ξ z through the formula Du for every Borel set B ⊂ R d . More precisely, the same decomposition holds true for each part of the decomposition of Du, namely

2.2.
Known results for general models with finite phases. We recall here some results that were proved for more general energies defined for functions taking values in a given finite set. In [12], Braides together with the first and third author consider energies E ε defined for spin variables u : εL → S, where S is a finite set and L is a so-called thin stochastic lattice. In general, these points sets are located in a fixed neighborhood of a lower-dimensional subspace such that there is a minimal distance between points and there are no arbitrarily large holes in the neighborhood of the subspace. The energies in [12] can be of the form where the energy density f : R d ×S 2 → [0, +∞) has to satisfy certain growth and decay conditions. We do not state them explicitly here, but we mention that they cover in particular the case when L = Z d is a periodic lattice that is completely contained in the subspace R d and With c = N 4π and S = S N we recover the energy N 2πε E N ε , so that all results of [12] can be applied. In particular, we can use an integral representation result and the characterization of the corresponding integrand through an asymptotic cell formula. Indeed, by [12,Theorem 5.8] we know that in the case of spatially homogeneous interactions the Γ-limit as ε → 0 of N 2πε E N ε exists, is finite only on BV (Ω; S N ), and for u ∈ BV (Ω; S N ) it is of the form where the integrand is given by an asymptotic minimization problem with a suitable boundary conditions. More precisely, denoting by u s,r ν : R d → R (ν ∈ S d−1 and s, r ∈ S N ) the function then in the case of just nearest neighbor interactions the function ϕ(s, r, ν) is given by [12, Remarks 5.9 & 4.2(i)] for the fact that the width of the discrete boundary layer can be taken as 2ε. In the above formula, Q ν denotes a unit cube centered at the origin with two faces orthogonal to ν as in (2.2). The energy E N ε (u, Q ν ) denotes the energy restricted to the set Q ν . More in general, for any non-empty set A ⊂ R d and u : εZ d → S N let us introduce for later purposes the localized functional

Continuum limit for fixed N as lattice spacing vanishes
In this section we identify the variational limit of the N -clock model as ε → 0 for the scaled energy N 2πε E N ε . We start with the following auxiliary result that will be crucial to establish the lower bound.
Next we establish a lower-semicontinuity result which helps to prove the lower bound. otherwise.
By with respect to the strong topology of L 1 (I; R 2 ). In particular, it is lower semicontinuous. We next fix an open set A ⊂ Ω and v n , v ∈ L 1 (A; R 2 ) such that v n → v strongly in L 1 (A; R 2 ). We want to prove that Without loss of generality, we assume that the right-hand side in (3.2) is finite and that the lim inf is actually a limit. Since |Dv n |(A) ≤ E(v n ; A) we obtain v ∈ BV (A; S 1 ) and v n * v weakly* in BV (A; R 2 ). Note further that Let us fix a direction ξ ∈ S 1 , which plays the role of one of the coordinate directions e . In the following we use the notation and the properties of slicing recalled in Subsection 2.1. We start by extracting a subsequence of n (possibly depending on ξ and which we do not relabel) such that the liminf is actually a limit. Moreover, since v n → v strongly in L 1 (A; R 2 ), by Fubini's Theorem we extract a further subsequence (possibly depending on ξ and which we do not relabel) such that We observe now that the coarea formula (cf. [7, formula (272) Hence, by the equality above and by Fatou's Lemma, we deduce that

(3.4)
From the one-dimensional lower semicontinuity result we infer that lim inf for H d−1 -a.e. z ∈ Π ξ . Integrating the inequality above with respect to z ∈ Π ξ , again by the coarea formula, and by (3.4) we obtain that We conclude the proof of (3.2) by evaluating the last inequality for ξ = e 1 , . . . , e d , by (3.3), and employing the superadditivity of the lim inf.
Now we can prove the lower bound for the Γ-limit of the functionals N 2πε E N ε . Proposition 3.3. Let u ε : εZ d → S N and u ∈ BV (Ω; S N ) be such that u ε → u in L 1 (Ω; R 2 ). Then Proof. To simplify the notation we denote θ N by θ. Let A ⊂⊂ Ω be an open set. By (2.1) it holds that |u ε (εi) − u ε (εj)| = 2 sin 1 2 d S 1 (u ε (εi), u ε (εj) . Since u ε takes values in S N , the geodesic distance d S 1 (u ε (εi), u ε (εj)) is an integer multiple of θ, i.e., there exists a k ∈ N (depending on i, j, and ε) such that d S 1 (u ε (εi), u ε (εj)) = kθ. Note that kθ ≤ π. Hence from Lemma 3.1 we infer that Since u ε is piecewise constant on cubes of the form Q = (−ε/2, ε/2) d + z with z ∈ Z d , we obtain that for ε small enough where we also used that N = 2π/θ and that the discrete energy counts each interaction twice. Note that by Lemma 3.2 the functional is L 1 (A; R 2 )-lower semicontinuous on BV (A; S N ), as it is the restriction of a lower semicontinuous functional to a closed subset of BV (A; S N ). Thus letting ε → 0 we deduce that The claim now follows from the arbitrariness of A ⊂⊂ Ω.
We next prove that the corresponding upper bound for the Γ-limit.
Proposition 3.4. Let u ∈ BV (Ω; S N ). Then there exists a sequence u ε : εZ d → S N such that u ε → u in L 1 (Ω; R 2 ) and Proof. To simplify the notation we denote θ N by θ. Due to the discussion in Section 2.2, the Γ-limit of N 2πε E N ε has the form (2.3). To prove the upper bound it suffices to define a suitable candidate for the minimum problem (2.4) whose energy can be bounded in the limit as ε → 0 by 4 sin 2 ( θ 2 )θ −2 d S 1 (s, r)|ν| 1 . Write s = exp(ιk s θ) and r = exp(ιk r θ) with 0 ≤ k s , k r ≤ N − 1. We will treat the case when k r = 0, i.e. r = (1, 0), and 0 < k s θ ≤ π. The construction we provide can then be composed with a rotation in the co-domain to cover the general case. The idea is to define a candidate whose angular variable jumps by θ along the discretization of k s parallel hyperplanes orthogonal to ν, where all hyperplanes are O(ε)-close to the hyperplane Π ν := {x ∈ R d : x · ν = 0}. The correction in order to satisfy the boundary condition will be of lower order. In formulas, let u ε : εZ d → S N be defined by where x denotes the integer part of x. Hence for all εi ∈ εZ d ∩ Q ν such that εi · ν ≤ 0 we have u ε (εi) = r, while for all εi ∈ εZ d with εi · ν ≥ k s ε we have u ε (εi) = s, so that for non-vanishing interactions at least one point belongs to the set H ks ε := {x ∈ Q ν : x · ν ∈ (0, εk s )} .
Therefore, given t > 1, for ε = ε(t) small enough the intersection point is contained in tQ ν ∩ H ν . Since by definition the mapping I k,ε (εi, εj) → εj − (εj · e )e is injective, we obtain that where Π x =0 denotes the projection onto the subspace {x = 0}. In particular, it holds that By elementary geometric considerations we can bound the cardinality via a (d − 1)-dimensional volume as Since t > 1 was arbitrary we deduce that lim sup We claim that the right hand side term equals |ν |, which then concludes the proof summing over . This is a consequence of the coarea formula in the form [7, Theorem 2.93] taking f to be the projection Π x =0 and E = Q ν ∩H ν and using the fact that the (d−1)-dimensional coarea factor of the projection Π x =0 on the tangent space H ν is given by |ν | (cf. [7, formula (3.110)]).

Limit of the continuum functional for large N
In this section we study the Γ-convergence of the limit functionals E N defined on L 1 (Ω; R 2 ) by as N → +∞, where we write θ N to stress the dependence on N of the minimal angle between vectors in S N . We show that the Γ-limit of E N coincides with the functional derived in [21] in the regime N = N ε 1 ε| log ε| and d = 2. More precisely, we define the functional for u ∈ L 1 (Ω; R 2 ). We first state and proof the lower bound together with a compactness result. Then up to subsequences u N → u ∈ BV (Ω; S 1 ) strongly in L 1 (Ω; R 2 ). Moreover, for any sequence u N ∈ BV (Ω; S N ) and u ∈ BV (Ω; Note that θ N = 2π/N implies θ N → 0 as N → +∞. Hence Thus the compactness statement follows from the inclusion S N ⊂ S 1 and standard compactness results in BV (Ω; R 2 ). In order to prove the lower bound, note that for all u ∈ BV (Ω; S 1 ), cf. (4.1)-(4.2). The functional E is L 1 (Ω; R 2 )-lower semicontinuous by Lemma 3.2. Hence, the claim follows from (4.3).
We now establish the upper bound via several approximations combined with a relaxation result for integral functionals defined on W 1,1 (Ω; S 1 ).
Thanks to this property and to a diagonal argument, it is enough to prove the upper bound (4.4) assuming u ∈ W 1,1 (Ω; S 1 ).
where P (x, τ ) = (x, −τ ). Since Γ is bi-Lipschitz, we have thatũ ∈ W 1,1 (Ω; S 1 ) and by a change of variables we can bound the L 1 -norm of its gradient via where the constant C Γ depends only on the bi-Lipschitz properties of Γ and the dimension. With an abuse of notation we will denote the extended functionũ ∈ W 1,1 (Ω; S 1 ) again by u.
Hence, by a diagonal argument it is enough to prove the upper bound (4.4) assuming u ∈ R ∞ 1 (Ω; S 1 ). Let Ω be such that Ω ⊂⊂ Ω ⊂⊂Ω. For λ small enough we have Ω λ ⊂⊂ Ω ⊂⊂Ω. We now define the piecewise constant function u λ : Ω λ → S 1 as follows. Let z ∈ Z d be such that I λ (λz) ⊂ Ω λ . If I λ (λz) ∩ Σ = Ø, the map u is C ∞ in the interior of I λ (λz) and thus it admits a lifting ϕ z (unique up to a multiple integer of 2π), which is C ∞ in the interior of I λ (λz), namely u = exp(ιϕ z ) in I λ (λz). We consider the average and we set u λ (x) := exp(ιϕ z ) for x ∈ I λ (λz). If, instead, I λ (λz) ∩ Σ = Ø we put u λ (x) := e 1 for x ∈ I λ (λz) (the precise value e 1 being not relevant). We remark that u λ → u strongly in L 1 (Ω; R 2 ). Indeed, let B be a ball such that B ⊂⊂ Ω \ Σ. Since B is simply connected and u ∈ C ∞ (B; S 1 ), there exists a lifting ϕ ∈ C ∞ (B; R), namely, u = exp(ιϕ) in B. If I λ (λz) ∩ B = Ø, then I λ (λz) ∩ Σ = Ø for λ small enough. In particular, we can consider the lifting ϕ z of u in I λ (λz) used in the definition of u λ . By uniqueness of the liftings up to integer multiples of 2π, there exists a k z ∈ Z such that ϕ z = ϕ + 2πk z . This entails Given x ∈ B, we consider a family of cubes I λ (λz λ ) x. By Lebesgue's differentiation theorem for L d -a.e. x ∈ B. Then u λ → u a.e. in Ω and by dominated convergence we obtain u λ → u in L 1 (Ω; R 2 ). Let us prove that lim sup λ→0 For i ∈ {1, . . . , d} we define the families of indices Let z ∈ G i (λ). As in the definition of u λ , we let ϕ z and ϕ z+ei be the liftings of u in I λ (λz) and I λ (λ(z +e i )), respectively. Moreover, since u is C ∞ in the interior of the rectangle I λ (λz)∪I λ (λ(z + e i )), it admits a C ∞ lifting ϕ such that u = exp(ιϕ) in I λ (λz) ∪ I λ (λ(z + e i )). By uniqueness of the liftings up to integer multiples of 2π, there exist k z , k z+ei ∈ Z such that ϕ z = ϕ + 2πk z in I λ (λz) and ϕ z+ei = ϕ + 2πk z+ei in I λ (λ(z + e i )). Note that Now we are in a position to estimate d S 1 u λ (λ(z + e i )), u λ (λz) = d S 1 exp(ιϕ z+ei ), exp(ιϕ z ) (4.10) Using the fact that Ω λ ⊂⊂ Ω , for λ small enough we obtain for λ small enough. Using the rough estimate d S 1 u λ (λ(z + e i )), u λ (λz) ≤ π we deduce that d i=1 z∈Bi(λ) λ d−1 d S 1 u λ (λ(z + e i )), u λ (λz) ≤ C Σ,d λ , (4.11) the constant C Σ,d being larger than the previous one. From (4.10) and (4.11) it follows that ≤ Ω ∇u 2,1 dx + C Σ,d λ and hence, letting λ → 0 and Ω Ω, (4.9). Thanks to this step, it suffices to prove the upper bound assuming that the S 1 -valued map is constant on each of the cubes I λ (λz) ⊂ Ω λ .
Step 5. (Construction of u N ). Let u λ : Ω λ → S 1 be a map that is constant on each of the cubes I λ (λz). We consider the discretization map P N : S 1 → S N defined as follows: given a ∈ S 1 , we let ϕ a ∈ [0, 2π) be the unique angle such that a = exp(ιϕ) and we set P N (a) := exp ιθ N ϕ a /θ N .

Then, by the triangle inequality
Letting N → +∞ and by (4.3) we conclude the proof.

Constrained problems
In this final section we apply the results for the discrete-to-continuum limit to some constrained minimization problem. Again here we can use the more abstract results of [12]. We consider the case of discrete Dirichlet boundary conditions and discrete phase constraints. We start with the latter. Note that in both cases we do not state separately the convergence of minimizers which is a standard consequence of the general theory of Γ-convergence.
Volume constraints in the N -clock model: Let V ∈ (0, 1) N be such that Then by [12, Theorem 6.2] we have the following Γ-convergence result.
Corollary 5.1. Let N ∈ N and for 1 ≤ k ≤ N let V k,ε ∈ (0, 1) satisfy (5.1). Then as ε → 0 the sequence of functionals E N ε,V Γ-converge with respect to the strong L 1 (Ω; R 2 ) to the functional E N,V : L 1 (Ω; R 2 ) → [0, +∞] defined by Dirichlet Boundary conditions: In order to define discrete Dirichlet boundary conditions and to derive a convergence result, we need to assume some-well preparedness of the boundary condition.
For the sake of simplicity we assume that u 0 ∈ BV loc (R d , S N ) is a polyhedral partition such that We define the set of configurations satisfying a discrete Dirichlet boundary condition u = u 0 by PC ε,u0 = u : εZ d ∩ Ω → S N : u(εi) = u 0 (εi) if dist(εi, ∂Ω) ≤ 2ε .
Since the Γ-limit result for the sequence E N ε remains unchanged for any set Ω ⊃⊃ Ω we can apply [12, Theorem 4.1 & Remark 4.2 (i)] to obtain the following corollary.