Etienne Sandier

Sylvia Serfaty

2D Coulomb Gases and the Renormalized Energy

Keywords: Coulomb gas, one-component plasma, random matrices, Ginibre ensemble, Fekete sets, Abrikosov lattice, triangular lattice, renormalized energy, large deviations, crystallization. MSC classification: 82B05, 82D10, 82D99, 15B52

We study the statistical mechanics of classical two-dimensional "Coulomb gases" with general potential and arbitrary β, the inverse of the temperature. Such ensembles also correspond to random matrix models in some particular cases. The formal limit case β = ∞ corresponds to "weighted Fekete sets" and also falls within our analysis.

It is known that in such a system points should be asymptotically distributed according to a macroscopic "equilibrium measure," and that a large deviations principle holds for this, as proven by Ben Arous and Zeitouni [BZ].

By a suitable splitting of the Hamiltonian, we connect the problem to the "renormalized energy" W , a Coulombian interaction for points in the plane introduced in [SS1], which is expected to be a good way of measuring the disorder of an infinite configuration of points in the plane. By so doing, we are able to examine the situation at the microscopic scale, and obtain several new results: a next order asymptotic expansion of the partition function, estimates on the probability of fluctuation from the equilibrium measure at microscale, and a large deviations type result, which states that configurations above a certain threshhold of W have exponentially small probability. When β → ∞, the estimate becomes sharp, showing that the system has to "crystallize" to a minimizer of W . In the case of weighted Fekete sets, this corresponds to saying that these sets should microscopically look almost everywhere like minimizers of W , which are conjectured to be "Abrikosov" triangular lattices.

Introduction

We are interested in studying the probability law where Z β n is the associated partition function, i.e. a normalizing factor such that P β n is a probability measure, and where (1.2)

w n (x 1 , . . . , x n ) = - i =j log |x i -x j | + n n i=1 V (x i ),
most often called the Hamiltonian. Here the x i 's belong to R 2 (identified with the complex plane C), β > 0 is a parameter corresponding to (the inverse of) the temperature and V is a potential satisfying some growth and regularity assumptions, which we will detail below.

The probability law P β n is the Gibbs measure of what is called either a classical "twodimensional Coulomb system" or "Coulomb gas" or "two-dimensional log gas", or "twodimensional one-component plasma", or also "Gaussian β-ensemble". It was first pointed out by Dyson [Dy] that Coulomb gases are naturally related to random matrices. This is somehow due to the fact that e i =j log |x i -x j | is the square of the Vandermonde determinant i<j |x i -x j | and thus the law P β n , in the particular case when V (x) = |x| 2 and β = 2 corresponds, as shown in [G] (see also [Me], Chap. 15), to the law of eigenvalues for the Ginibre ensemble, which is the set of matrices with independent standard (complex) Gaussian entries. For the general background and references to the literature, we refer to the book by Forrester [Fo] and references therein.

The Gibbs measure P β n can also be studied for x i 's belonging to the real line (the onedimensional case). In the context of statistical mechanics (general β), this corresponds to "log gases", and in the context of random matrices (β = 1, 2, 4), to Hermitian or symmetric random matrices (whose eigenvalues are always real). We examine that case in our companion paper [SS2], showing the present study can be extended to handle it. We also point out that studying P β n with the x i 's restricted to the unit circle and with β = 1, 2, 4 also has a random matrix interpretation: it corresponds to the so-called circular ensembles, e.g. in the β = 2 case, eigenvalues of the unitary matrices distributed according to the Haar measure. We also plan on examining this case in the future.

All the above models have been studied quite extensively in the literature, particularly from the random matrix point of view (although there one can say the planar or complex Ginibre case has attracted less attention than the real or Hermitian case), but also from the statistical mechanics point of view, more particularly in the physics literature (particularly relevant are [AJ, SM, JLM]).

The current research on the random matrix aspect in the complex case focuses on studying the more general case of random matrices with entries that are not necessarily Gaussian and showing the average behavior is the same as for the Ginibre ensemble, see e.g. [Ba, TV]. This is referred to as universality. We are instead limited to exact Vandermonde factors but we emphasize that our results are valid for all β, hence they are not limited to random matrices and thus for the proof we cannot rely on any explicit random matrix model. Our results also have some universality feature in the sense that they are valid for a large class of potentials V .

The function w n can also be studied for its own sake: it can seen as the interaction energy between similarly charged particles confined by the potential V . The case where V (x) is quadratic arises for instance as the interaction energy for superconducting vortices in the Ginzburg-Landau theory, in the regime where their number is fixed, bounded (see [SS2], Chapter 11). In the case where V is equal to zero on a compact set K and to +∞ elsewhere, which is not treated here, the minimizers of w n are known as Fekete points or Fekete sets, cf. the book of Saff and Totik [ST] for general reference. These are interesting in their own right -they arise mainly in polynomial interpolation -and the literature on the question of their distribution in various situations is vast. When instead V is a general smooth enough function (the situation we treat here), the minimizers of w n are called "weighted Fekete points" or weighted Fekete sets, and are also of interest, cf. again [ST].

We will pursue the analysis of these weighted Fekete sets, which can be seen as the formal limit β → +∞ of (1.1), in parallel with the analysis of (1.1) for general β, and obtain new results in both cases.

In the case of the Ginibre ensemble, i.e. when V (x) = |x| 2 and β = 1, it is known that the "spectral measure" ν n := 1 n n i=1 δ x i converges to the uniform measure on the unit disc. More precisely P β n , seen as a probability on the space of probability measures on C (the spectral measures) converges to a Dirac mass at µ 0 = 1 π 1 B 1 dx. This is the celebrated "circular law", attributed in this case to Ginibre, Mehta, an unpublished paper of Silverstein in 1984, and then Girko [Gi]. The large deviations from this law was established by Ben Arous and Zeitouni [BZ] (see Theorem 4 below): they showed that a large deviations principle holds with speed n -2 and rate function

(1.3) I(µ) = R 2 ×R 2 -log |x -y| dµ(x) dµ(y) + R 2 |x| 2 dµ(x)
whose unique minimizer among probabilities is of course the "circle law" distribution 1 π 1 B 1 dx. For the case of a general V and a general β, the same large deviations principle holds with the rate function, analogue of (1.3), being (1.4)

I(µ) = R 2 ×R 2 -log |x -y| dµ(x) dµ(y) + R 2 V (x) dµ(x).
This can be easily readapted from the proof of [BZ], otherwise it is proven in a much more general setting in [Be]. Again the spectral measure ν n = 1 n n i=1 δ x i converges to the minimizer among probability measures of I, called the equilibrium measure, which we will denote µ 0 . In the case of weighted Fekete sets, the analogue to the circular law has been known to be true for a much longer time: it was proved by Fekete, Polya and Szëgo that 1 n n i=1 δ x i converges to the same equilibrium measure minimizing I (then also referred to as the electrostatic interaction energy), whose description goes back to Gauss, and was carried out with modern mathematical rigor by Frostman [FR].

We are interested in examining the "next order" behavior, or that of fluctuations around the limiting distribution µ 0 . Let us mention that such questions have already been addressed, often with the point of view of deriving explicit scaling limits (e.g. [BSi, G]) or laws for certain statistics of fluctuations. One can see for example [AHM] where the authors essentially prove that the law of the linear statistics of the fluctuations is a Gaussian with specific variance and mean, or also [Rid] for related results. Our approach and results are quite different.

Recalling I and µ 0 are found through the large deviations at speed n -2 , we look into the speed n -1 and, while we do not prove a complete large deviations principle at this speed, we show there is still a sort of rate function for which a "threshhold phenomenon" holds. This analysis is based on an expansion, through a crucial but simple "splitting formula" (which we present in Section 1.1 below) of w n (x 1 , . . . , x n ), as equal to n 2 I(µ 0 ) -n 2 log n plus a term of order n, which tends as n → +∞ to the "renormalized energy" W , a Coulombian interaction of points in the plane with a uniform neutralizing background, that we introduced in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and whose definition we will recall below in Section 1.2. To be more precise the limit term is the average value of W on the set of blow-up limits of the configuration of points x 1 , . . . , x n at the scale 1/ √ n. It is this average that partially plays the role of a rate function at speed n -1 . For a precise statement, see Theorem 5.

Another way of saying this is in the language of Γ-convergence (for a definition we refer to [Br, DM], suffice it to know that this is the right notion of convergence to ensure that minimizers of w n converge (via their empirical measures) to minimizers of I, i.e. to µ 0 ): it is not very difficult to show (for a short proof, see [SS2] Prop. 11.1) that wn n 2 Γ-converges as n → ∞ to I, defined in (1.4). Here we examine the next order in the "Γ-expansion" of w n , i.e. we study the Γ-convergence of1 n (w n -n 2 I(µ 0 )), and show that the Γ-limit is (the average of) W . Consequently, after blow-ups at scale √ n, minimizers of w n (i.e. weighted Fekete sets) should minimize the (average of the) renormalized energy W . For a precise statement, see Theorem 2.

Before yet giving a precise definition, let us mention that we introduced the renormalized energy W in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] for the study of interaction of vortices in the context of the Ginzburg-Landau energy of superconductivity (for general reference on the topic, cf. [SS2]). Configurations that minimize the Ginzburg-Landau energy with applied magnetic field, exhibit in certain regimes "point vortices" that are densely packed (there are n 1 of them) and are expected to arrange themselves in perfect triangular lattices (i.e. with 60 • angles), named Abrikosov lattices after the physicist who predicted them [A]. The Abrikosov lattices are indeed observed in experiments on superconductors 1 . In [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] we made this partly rigorous by showing that minimizers of the Ginzburg-Landau energy have vortices that minimize the renormalized energy W after blow-up at the scale √ n. The conjecture made in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], also supported by some mathematical evidence (see Section 1.2), is then that the minimal value of W is achieved by the triangular lattice ; if proven true this would completely justify why vortices form these patterns. Combining this conjecture with the above conclusion that weighted Fekete sets should (after blow-up) minimize W , we thus obtain the conjecture that they also should locally form Abrikosov (triangular) lattices.

We can phrase an analogous question for Coulomb gases or random matrices. Here P β n induces a probability measure on the family of blow-ups of (x 1 , . . . , x n ) around a given origin point in E -the parameter of the family -at the scale √ n, a blow-up scale after which the resulting points are typically separated by order 1 distances. In the limit n → ∞ this yields a probability measure on the set of configurations of points in the plane and we may ask if, almost surely, the blow-up configurations minimize W . Our results indicate that this is not the case, however we are able to prove that there is a treshhold phenomenon, in the sense that except with exponentially small probability, the average of W is below a certain constant, itself converging to the minimum of W as β → ∞, which indicates crystallisation, i.e. if the above conjecture is true, we should see Abrikosov lattices as β → ∞.

To our knowledge, this is the first time Coulomb gases or Fekete sets are rigorously connected to triangular lattices, in agreement with predictions in the physics literature (see [AJ] and references therein).

A corollary of our way of expanding w n is that we obtain a next order estimate of the partition function Z β n , a result we can already state:

Theorem 1. Let V satisfy assumptions (1.11) -(1.13) below. There exist functions f 1 , f 2 depending only on V , such that for any β 0 > 0 and any β ≥ β 0 , and for n larger than some n 0 depending on β 0 , we have

(1.5) nβf 1 (β) ≤ log Z β n -- β 2 n 2 I(µ 0 ) + βn 4 log n ≤ nβf 2 (β), with f 1 , f 2 bounded in [β 0 , +∞) and such that (1.6) lim β→∞ f 1 (β) = lim β→∞ f 2 (β) = - α 2
where α is some constant related to W , and explicited in (1.36) below.

This improves on the known results, which only gave the expansion log Z β n ∼ β 2 n 2 I(µ 0 ). Let us recall that an exact value for Z β n is only known for the Ginibre ensemble case of β = 2 and [Me], Chap. 15). Known asymptotics allow to deduce (cf. [Fo] eq. (4.184))

V (x) = |x| 2 : it is Z 2 n = n -1 2 n(n+1) π n n k=1 k! (see
(1.7) log Z 2 n = - 3n 2 4 + n 2 log n + n(-1 + 1 2 log 2 + 3 2 log π) + O(log n) as n → ∞,
where we note the value of I(µ 0 ) is indeed 3 4 for this potential. On the other hand, no exact formula exists for general potentials 2 , nor for quadratic potentials if β = 2. This is in contrast with the one-dimensional situation for which, at least in the case of quadratic V , Z β n has an explicit expression for every β, given by the famous Selberg integral formulas (see e.g. [AGZ]).

In statistical mechanics language, the existence of an exact asymptotic expansion up to order n for log Z β n is essentially the existence of a thermodynamic limit. This is established in [LN] for a three-dimensional Coulomb system, and in a nonrigorous way in [SM] in two dimensions.

As suggested by the strong analogy between Coulomb gases and interacting vortices in the Ginzburg-Landau model, we will draw heavily on methods we introduced in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], such as the splitting, blow-up, the use of the ergodic theorem, and particular Ginzburg-Landau tools.

The rest of the introduction is organized as follows: first we give some more notation, give the assumptions we need to make on V and state the splitting formula, then we present the definition of the renormalized energy and the main results from [SS1] that we will use, and finally we state our main results and comment on them.

The equilibrium measure and the splitting formula

We need to introduce some notation, and for this we need to describe the equilibrium measure µ 0 minimizing (1.4) among probability measures.

This description, which is now classical in potential theory (see again [ST]) says that, provided lim |x|→+∞ V (x) -log |x| = +∞ and log V is lower semicontinuous, this equilibrium measure exists, is unique, and is characterized by the fact that there exists a constant c such that, quasi-everywhere, (1.8)

U µ 0 + V 2 ≥ c and U µ 0 + V 2 = c on Supp(µ 0 ),
where for any measure µ, U µ denotes the potential generated by µ, defined by (1.9)

U µ (x) = - R 2 log |x -y| dµ(y) = -2π∆ -1 µ.
2 an exception is the result of [DGIL] for a quadrupole potential

Here and in all the paper, we denote by ∆ -1 the operator of convolution by

1 2π log | • |. It is such that ∆ • ∆ -1 = I,
where ∆ is the usual Laplacian. We denote the support of µ 0 by E.

Another way to characterize U µ 0 is as the solution of the following obstacle problem3 : It is a superharmonic function bounded below by c -V /2 and harmonic outside the so-called coincidence set

(1.10) E = {U µ 0 = c -V /2}.
This implies in particular that

U µ 0 is C 1,1 if V is (see [C]).
It is now a good time to state the assumptions on V that we assume are satisfied in the sequel.

(1.11)

V is C 1,1 and lim |x|→+∞ V (x) -log |x| = +∞, (1.12) In a neighborhood of E, V is C 3 and there exists m, m > 0 s.t. m ≤ ∆V ≤ m, (1.13) V is such that ∂E is C 1 .
The assumption on the growth of V is what is needed to apply the results from [ST] and to guarantee that (1.4) has a minimizer. The other conditions are technical, they are meant to ensure that µ 0 and its support are regular enough, which we will need for example when making explicit constructions, and that µ 0 never degenerates. Indeed, assumptions (1.12)-(1.13), together with (1.8), (1.9) and the regularity of V , ensure that

(1.14) µ 0 = m 0 (x) dx, where m 0 (x) = ∆V (x) 4π 1 E (x).
hence for some m, m > 0 we have

(1.15) m ≤ m 0 ≤ m. Next, we set ζ = U µ 0 + V 2 -c
where c is the constant in (1.8). This function satisfies

(1.16)

   ∆ζ = 1 2 ∆V 1 R 2 \E ζ = 0 in E ζ > 0 in R 2 \ E
From our assumptions on V , in particular assumption (1.12), it follows that for some c > 0 and x in a neighbourhood of E,

(1.17) ζ(x) ≥ c dist(x, E) 2 .
This follows for instance by using (1.16), (1.12) to deduce that ∆ζ > η 1 outside E, and then writing ∆ζ in coordinates (r, σ) where r is the distance to E, to obtain a lower bound for ζ rr outside but close enough to E (see [KS], [Fre], [Fri], [C]). In fact [C] shows that this rate is optimal.

The function ζ arises in the splitting formula for w n which we now present. As mentioned above, expanding the probability density to the next order goes along with blowing-up the points by a factor √ n. We then denote the blown-up quantities by primes. For example x i = √ nx i , m 0 (x ) = m 0 (x), etc . . . . The splitting formula, proven in Section 2 is the observation that, for any x 1 , . . . , x n , (1.18)

w n (x 1 , . . . , x n ) = n 2 I(µ 0 ) - n 2 log n + 1 π W (-∇ ⊥ H , 1 R 2 ) + 2n n i=1 ζ(x i ),
where

∇ ⊥ := (-∂ x 2 , ∂ x 1 ) and (1.19) H = -2π∆ -1 n i=1 δ x i -m 0 ,
and where, in agreement with formula (1.27) below,

(1.20) W (-∇ ⊥ H , 1 R 2 ) := lim η→0 1 2 R 2 \∪ n i=1 B(x i ,η) |∇H | 2 + πn log η .
Letting, for a measure ν

(1.21) F n (ν) :=    1 n 1 π W (-∇ ⊥ H , 1 R 2 ) + 2n ζ dν if ν is of the form n i=1 δ x i +∞ otherwise,
the relation (1.18) can be rewritten

(1.22) w n (x 1 , . . . , x n ) = n 2 I(µ 0 ) - n 2 log n + nF n n i=1 δ x i .
This allows to separate orders as announced since we will see that F n ( n i=1 δ x i ) is typically of order 1.

We may next cancel out leading order terms and rewrite the probability law (1.1) as

(1.23) dP β n (x 1 , . . . , x n ) = 1 K β n e -n β 2 Fn( i δx i ) dx 1 . . . dx n where (1.24) K β n := Z β n e β 2 (n 2 I(µ 0 )-n 2 log n) .
As we will see below log K β n is of order nβ, which leads to Theorem 1.

The renormalized energy

We now define precisely the "renormalized energy" W , which is a way of computing the Coulomb interaction between an infinite number of point charges in the plane with a uniform neutralizing background of density m. We point out that, to our knowledge, each of the analogous Coulomb systems studied in the physics literature (e.g. [SM, AJ]) comprise a finite number of point charges, and hence implicitly extend only to a bounded domain on which there is charge neutrality. Here we do not assume any local charge neutrality. We denote by B(x, R) the ball centered at x with radius R and let B R = B(0, R).

Definition 1.1. Let m be a nonnegative number. For any continuous function χ and any vector-field j in R 2 such that

(1.25) curl j = 2π(ν -m), div j = 0
where ν has the form

(1.26) ν = p∈Λ δ p for some discrete set Λ ⊂ R 2 ,
we let

(1.27) W (j, χ) = lim η→0   1 2 R 2 \∪ p∈Λ B(p,η) χ|j| 2 + π log η p∈Λ χ(p)   .
Definition 1.2. Let m be a nonnegative number. Let j be a vector field in R 2 . We say j belongs to the admissible class A m4 if (1.25), (1.26) hold and

(1.28)

ν(B R ) |B R | is bounded by a constant independent of R > 1.
For any family of sets {U R } R>0 in R 2 we use the notation χ U R for positive cutoff functions satisfying, for some constant C independent of R,

(1.29) |∇χ U R | ≤ C, Supp(χ U R ) ⊂ U R , χ U R (x) = 1 if d(x, U R c ) ≥ 1.
Definition 1.3. The renormalized energy W is defined, for j ∈ A m , by

(1.30) W (j) = lim sup R→∞ W (j, χ B R ) |B R | , with {χ B R } R satisfying (1.29) for the family {B R } R>0 .
In theory, many different j's could correspond to a given ν (one can always add the gradient of a harmonic function). But as it turns out, they only differ by a constant: Lemma 1.4. Let m ≥ 0 and ν = p∈Λ δ p , where Λ ⊂ R 2 is discrete, and assume there exists j such that (1.31) curl j = 2π(ν -m), div j = 0, and W (j) < +∞.

Then any other j satisfying (1.31) is such that j -j is constant.

If there exists j ν such that (1.31) holds and such that

(1.32) lim R→∞ - B R j ν = 0,
then any other j satisfying (1.31) is such that W (j) > W (j ν ).

Proof. Let j, j be as above, then j -j can be seen as as complex function of a complex variable and from (1.31) it is holomorphic. From the finiteness of W (j) and W (j ) we deduce easily that B R |j -j | 2 ≤ CR 2 and using Cauchy's estimate for the coefficients of a power series together with a mean value argument it follows that j -j is constant.

For the second statement, we deduce from the first statement that j = j ν + c for some constant vector c = 0, and then

W (j, χ B R ) = W (j ν , χ B R ) + c • j ν χ B R + |c| 2 2 χ B R ,
so that dividing by |B R |, passing to the limit as R → +∞ and in view of (1.32), we find

W (j) = W (j ν ) + 1 2 |c| 2 .
Note that given ν, the above lemma shows that either for all j's satisfying (1.31) the limit lim R→∞ -B R j ν exists, or it exists for none of them. Both cases may occur.

The following additional facts and remarks about W are from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]:

-In the definition (1.30), the balls {B R } R can be replaced by other families of (reasonable) shapes {U R } R , this yields a definition of a renormalized energy W U , where the letter U stands for the family {U R }.

-The value of W does not depend on {χ B R } R as long as it satisfies (1.29). The corresponding statement holds for W U under certain assumptions on the family {U R } R>0 .

-W is bounded below and admits a minimizer over A 1 .

-It is easy to check that if j belongs to A m , m > 0, then j = 1

√ m j(•/ √ m) belongs to A 1 and (1.33) W (j) = m W (j ) - π 2 log m .
Consequently if j is a minimizer of W over A m , then j minimizes W over A 1 . In particular

(1.34) min Am W = m min A 1 W - π 2 log m .
-The minimizers and the value of the minimum of W U are independent of U . However there are examples of admissible j's for which W U (j) depends on the family of shapes {U R } R>0 which is used.

-If j ∈ A m then in the neighborhood of p ∈ Λ we have curl j = 2π(δ p -m), div j = 0, thus we have near p the decomposition j(x) = ∇ ⊥ log |x -p| + f (x) where f is C 1 , and it easily follows that the limit (1.27) exists if χ is compactly supported. It also follows that j belongs to L p loc for any p < 2.

-Because the number of points is infinite, the interaction over large balls needs to be normalized by the volume, as in a thermodynamic limit. Thus W does not feel compact perturbations of the configuration of points. Even though the interactions are longrange, this is not difficult to justify rigorously.

-In the case m = 1 and when the set of points Λ is periodic with respect to some lattice Z u + Z v then it can be viewed as a set of n points a 1 , . . . , a n over the torus

T ( u, v) := R 2 /(Z u + Z v) with |T ( u, v) | = n.
In this case, the infimum of W (j) among j's which satisfy (1.31) is achieved by j {a i } = -∇ ⊥ h, where h is the periodic solution to -∆h = 2π( i δ a i -1), and

(1.35) W (j {a i } ) = π |T ( u, v) | i =j G(a i -a j ) + π lim x→0 (G(x) + log |x|)
where G is the Green function of the torus with respect to its volume form, i.e. the solution to

-∆G(x) = 2π δ 0 - 1 |T ( u, v) | in T ( u, v) .
An explicit expression for G can be found via Fourier series and this leads to an explicit expression for W of the form i =j E(a i -a j ) where E is an Eisenstein series (for more details see [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and also [BSe]). In this periodic setting, the expression of W is thus much simpler than (1.30) and reduces to the computation of a sum of explicit pairwise interaction.

-When the set of points Λ is itself exactly a lattice Z u + Z v then W can be expressed explicitly through the Epstein Zeta function of the lattice. Moreover, using results from number theory, it is proved in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], Theorem 2, that the unique minimizer of W over lattice configurations of fixed volume is the triangular lattice. This supports the conjecture that the Abrikosov triangular lattice is a global minimizer of W , with a slight abuse of language since W is not a function of the points, but of their associated current j {a i } .

This last fact allows us to think of W as a way of measuring the disorder and lack of homogeneity of a configuration of points in the plane (this point of view is pursued in [BSe] with explicit computations for random point processes). Another way to see it is to view W as measuring the distance between p∈Λ δ p and the constant m in H -1 . However this distance is infinite, both because the domain is infinite and because Dirac masses do not belong to H -1 , which is why the actual definition is more involved and requires this "renormalized" computation using η → 0 (hence the name, borrowed from [BBH]).

We may now define the constant α which appears in Theorem 1 and in Theorem 2 below:

(1.36)

α := 1 π E min A m 0 (x) W dx = 1 π min A 1 W - 1 2 E m 0 (x) log m 0 (x) dx,
where we have used (1.34) and the fact that, from (1.14), E m 0 = 1. Note that α only depends on V , via the integral term, and on the (so far) unknown constant min A 1 W .

Statement of main results

Our first result identifies the Γ-limit of {F n } n , defined in (1.21) or (1.22). This in particular allows a description of the weighted Fekete sets minimizing w n at the microscopic level. Below we abuse notation by writing ν n = n i=1 δ x i when it should be ν n = n i=1 δ x i,n . For such a ν, we let ν = n i=1 δ x i be the measure in blown-up coordinates and j ν = -∇ ⊥ H , where H is defined by (1.19) -equivalently j ν is the solution of div j ν = 0, curl j ν = 2π(ν -m 0 ) in R 2 which tends to 0 at infinity. (To avoid confusion, we emphasize here that ν lives at the original scale while j ν lives at the blown-up scale.) We also let (1.37)

P νn = - E δ (x,jν n ( √ n(x+•)) dx,
i.e. the push-forward of the normalized Lebesgue measure on E by x → (x, j νn ( √ nx + •)). It is a probability measure on E × L p loc (R 2 , R 2 ) (couples of (blow-up centers, blown-up current around this center)).

The limiting object as n → +∞ in the Γ-limit of w n was ν = lim n u n . In taking the Γ-limit of F n , the limiting object is more complex, it is the limit P of P νn , i.e. a Young measure akin to the Young measures on micropatterns introduced in [AM]. Note that in the rest of the paper, the probability P has nothing to do with P β n , and depends on the realizations of the configurations of points.

We will here and below use the notation

(1.38) D(x , R) = ν n B x, R √ n -nµ 0 B x, R √ n ,
where x = √ nx as usual.

Theorem 2 (Microscopic behavior of weighted Fekete sets). Let the potential V satisfy assumptions

(1.11)-(1.13). Fix 1 < p < 2 and let X = E × L p loc (R 2 , R 2 ). A. Lower bound. Let ν n = n i=1 δ x i be a sequence such that F n (ν n ) ≤ C.
Then P νn defined by (1.37) is a probability measure on X and 1. Any subsequence of {P νn } n has a convergent subsequence converging to a probability measure on X as n → ∞. We denote by P such a limit.

2. The first marginal of P is the normalized Lebesgue measure on E. P is invariant by (x, j) → (x, j(λ(x) + •)), for any λ(x) of class C 1 from E to R 2 (we will say T λ(x)invariant).

3. For P almost every (x, j) we have j ∈ A m 0 (x) .

4. Defining α as in (1.36), it holds that

(1.39) lim inf n→∞ F n (ν n ) -2 ζ dν n ≥ |E| π W (j) dP (x, j) ≥ α.
B. Upper bound construction. Conversely, assume P is a T λ(x) -invariant probability measure on X whose first marginal is 1

|E| dx |E and such that for P -almost every (x, j) we have j ∈ A m 0 (x) . Then there exists a sequence

{ν n = n i=1 δ x i } n of empirical measures on E and a sequence {j n } n in L p loc (R 2 , R 2 ) such that curl j n = 2π(ν n -m 0 )
and such that defining P n as in (1.37), with j n replacing j νn , we have P n → P as n → ∞ and

(1.40) lim sup n→∞ F n (ν n ) ≤ |E| π W K (j) dP (x, j),
where W K is the renormalized energy relative to the family of squares

{K R = [-R, R] 2 } R .
C. Consequences for minimizers. If (x 1 , . . . , x n ) minimizes w n for every n and ν n = n i=1 δ x i , then the limit P of P νn as defined in (1.37) satisfies the following.

1. For P -almost every (x, j), the current j minimizes W over A m 0 (x) .

2. We have

lim n→∞ F n (ν n ) = |E| π W (j) dP (x, j) = α, lim n→∞ n i=1 dist 2 (x i , E) = 0.
3. There exists C > 0 such that for every x ∈ R 2 , R > 1 and using the notation (1.38) we have

(1.41) D(x , R) 2 min 1, |D(x , R)| R 2 ≤ Cn.
Note that part B of the theorem is only a partial converse to part A because the constructed j n need not be divergence free, hence in general j n = j νn . Also the lower bound (1.39) uses W and the upper bound (1.40) uses W K . In fact (1.39) is true with W U for any family {U R } R satisfying (6.2) below, and in particular we could replace W by W K there. We recall though that both have the same minimizers and minimum value, as proved in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF].

This theorem is the analogue of the main result of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] but for w n rather than the Ginzburg-Landau energy. It is technically simpler to prove, except for the possibility of a nonconstant weight m 0 (x) which was absent from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]. It can be stated as the fact that |E| π W dP , which can be understood as the average of W with respect to all possible blow-up centers in E (chosen uniformly at random), is the Γ-limit of w n at next order. Its minimum over all admissible probabilities is α.

The estimate (1.41) gives a control on the "discrepancy" D (between the effective number of points and the expected one) at the scale R/ √ n. Note that in a recent paper [AOC], the authors also study the fine behavior of weighted Fekete sets. Using completely different methods, based on Beurling-Landau densities and techniques going back to [La], they are able to show the very strong result that lim sup

R→∞ lim sup n→∞ D(x n , R) R 2 = 0,
as long as dist(x n , ∂E ) ≥ log 2 n. This shows that the density of points follows µ 0 at the microscopic scale 1/ √ n and thus the configurations are very rigid. This still leaves however some uncertainty about the patterns they should follow. On the contrary, our result is less precise about D(x, R) since we only recover the optimal estimate when R grows faster than n 1/4 , but it connects the pattern formed by the points to the Abrikosov triangular lattice via the minimization of W .

We now turn to Coulomb gases. It is straightforward from the form (1.23) and the estimate (provided by Theorem 1) log K β n = O(nβ) where K β n is defined in (1.24), to deduce that F n ≤ C except on a set of small probability. This fact allows to derive various consequences, the first being estimates on the probability of certain rare events. Theorem 3. Let V satisfy assumptions (1.11)-(1.13).

There exists a universal constant R 0 > 0 and c, C > 0 depending only on V such that: For any β 0 > 0, any n large enough depending on β 0 , and any β > β 0 , for any x 1 , . . . , x n ∈ R 2 , any R > R 0 , any x 0 = √ nx 0 ∈ R 2 and any η > 0, letting ν n = n i=1 δ x i , we have the following:

(1.42) log P β n D(x 0 , R) ≥ ηR 2 ≤ -cβ min(η 2 , η 3 )R 4 + Cβ(R 2 + n) + Cn. (1.43) log P β n ζ dν n ≥ η ≤ - 1 2 nβη + Cn(β + 1).
Moreover, for any smooth bounded

U = √ nU ⊂ R 2 , (1.44) log P β n U D(x , R) 2 R 2 min 1, |D(x , R)| R 2 dx ≥ η ≤ nβ(-cη + C|U | + C) + Cn.
Finally, if q ∈ [1, 2) there exists c, C > 0 depending on V and q such that ∀η ≥ 1, R > 0,

(1.45) log

P β n 1 + R 2 n 1 2 -1 q ν -nµ 0 W -1,q (B R/ √ n ) ≥ η √ n ≤ -cnβη 2 + Cn(β + 1),
where W -1,q (Ω) is the dual of the Sobolev space W 1,p 0 (Ω) with 1 p + 1 q = 1; in particular W -1,1 is the dual of Lipschitz functions.

These estimates can roughly be read in the following way: as soon as η is large enough, the events in parentheses have probability decaying like e -cn . More precisely, we bound the probability that a ball contains too many or too few points compared to the expected number nµ 0 (B), but whereas the circular law does it for a macroscopic ball, i.e. for R comparable to √ n, the estimate (1.42) is effective at microscopic scales, of the order of n 1/4 . This is sometimes called in this context "undercrowding" or "overcrowding" of points, see [JLM, NSV, K]. In view of similar results in [JLM] and the result of [AOC], we could expect this to hold as soon as R 1, but this seems out of reach by our method. This can also be compared with analogous estimates without error terms proven in the case of Hermitian matrices, cf. [ESY]. These results, in the Hermitian case, are proven in the general setting of Wigner matrices, i.e. Hermitian matrices with random i.i.d. entries, which do not need to be Gaussian. They only concern some fixed β however.

The estimate (1.44) gives a global version of this result: it expresses a control on the average microscopic "discrepancy" D. This control is in L 2 for large values of the discrepancy, and in L 3 for small values. The estimate (1.43) allows, in view of (1.17), to control (again with some threshhold to be beaten) the probability that some points may be far from the set E. Note that since ν n is a non-normalized empirical measure, (1.43) ensures for example that the probability that a single point lies at a distance η from E is exponentially small as soon as η is larger than some constant. All these estimates rely on controlling D and F n by W .

Finally, (1.45) tells us that fluctuations around the law nµ 0 can be globally controlled (take for example R = √ n) by O( √ n) (except with exponentially small probability). We believe this estimate to be optimal. Its proof uses in a crucial manner the result of [ST], which controls, via Lorentz spaces, the difference ν n -nµ 0 in terms of the renormalized energy W .

Our last result mostly expresses Theorem 2 in a "moderate" deviations language. Before stating it, let us recall for comparison the result of [BZ]:

Theorem 4 (Ben Arous -Zeitouni). Let β = 2 and V (x) = |x| 2 . We denote by P β n the image of the law (1.1) by the map (x 1 , . . . , x n ) → ν n , where ν n = 1 n n i=1 δ x i . Then for any subset A of the set of probability measures on R 2 (endowed with the topology of weak convergence), we have

-inf µ∈ Å I(µ) ≤ lim inf n→∞ 1 n 2 log P β n (A) ≤ lim sup n→∞ 1 n 2 log P β n (A) ≤ -inf µ∈ Ā I(µ),
where I = I -min I.

Before stating our theorem, we introduce the following notation, which allows to embed C n into the set of probabilities on X = E ×L p loc (R 2 , R 2 ): For any n and x = (x 1 , . . . , x n ) ∈ C n we let i(x) = P νn , where ν n = n i=1 δ x i and P νn is as in (1.37), so that i(x) is an element of P(X), the set of probability measures on

X = E × L p loc (R 2 , R 2 ). Theorem 5. For any n > 0 let A n ⊂ C n . Denote (1.46) A ∞ = n>0 m>n i(A m ).
Then for any η > 0 there is C η > 0 depending on V and η only such that α being as in (1.36),

(1.47) lim sup

n→∞ log P β n (A n ) n ≤ - β 2 |E| π inf P ∈A∞ W (j)dP (x, j) -α -η - C η β .
Conversely, let A ⊂ P(X) be a set of T λ(x) -invariant probability measures on X and let Å be the interior of A for the topology of weak convergence. Then for any η > 0, there exists a sequence of subsets A n ⊂ E n such that

(1.48) - β 2 |E| π inf P ∈ Å W K (j)dP (x, j) -α + η + C η β ≤ lim inf n→∞ log P β n (A n ) n ,
and such that for any sequence {ν n = n i=1 δ x i } n such that (x 1 , . . . , x n ) ∈ A n for every n there exists a sequence of currents j n ∈ L p loc (R 2 , R 2 ) such that curl j n = 2π(ν n -m 0 ) and such that -defining P n as in (1.37) with j n replacing j νn -we have

(1.49) lim n P n ∈ Å.
Note that if P n was P νn , then (1.49) would be equivalent to saying that ∩ n ∪ m>n i(A m ) ⊂ Å. The difference betwen P νn and P n is that the latter is generated by a current j n which is not necessarily divergence free.

Compared to Theorem 4 this result can be seen as a next order (n instead of n 2 ) deviations result, where the average of W over blow-up centers plays the role of a rate function, with a margin which becomes small as β → ∞. While Theorem 4 said that empirical measures at macroscopic scale converge to µ 0 , except for a set of exponentially decaying probability, Theorem 5 says that within the empirical measures which do converge to µ 0 , the ones with large average of W (computed after blow-up) also have exponentially decaying probability, but at the slower rate e -n instead of e -n 2 . More precisely, there is a threshhold C/β for some C > 0, such that configurations satisfying

|E| π W dP ≥ α + C β
have exponentially small probability, where we recall α is also the minimum possible value of |E| π W dP . Since we believe that W measures the disorder of a (limit) configuration of (blown up) points in the plane, this means that most configurations have a certain order. The threshhold, or gap, C/β tends to 0 as β tends to ∞, hence in this limit, configurations have to be closer and closer to the minimum of the average of W , or have more and more order.

Modulo the conjecture that the minimum of W is achieved by the perfect "Abrikosov" triangular lattice, this constitutes a crystallisation result. Note that to solve this conjecture, it would suffice to evaluate α, which in view of Theorem 1 is equivalent to being able to compute the asymptotics of Z β n as β → ∞. More generally, the following open questions naturally arise in view of our results, and are closely related to one another:

-Prove that min A 1 W is achieved by the triangular lattice.

-Find whether a large deviations statement is true at speed n, and if it is, find the rate function.

-Describe the limit of the probability measures P β n on P(X) defined as the images of P β n by the embedding i.

In [SS2] we will see that all the results we have obtained here are also true in the case of points on the real line, i.e. for 1D log gases or Hermitian random matrices. There the minimization of W is solved (the minimum is the perfect lattice Z) and the crystallisation result is complete.

The rest of the paper is organized as follows: Section 2 contains the proof of the "splitting formula". In Section 3, we present the "spreading result" from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and some first corollaries. In Section 4, we present an explicit construction which yields the lower bound on Z β n , whose proof is postponed to Section 7. In Section 5, we show how W controls the overcrowding/undercrowding of points, and prove Theorem 3. In Section 6 we present the ergodic averaging approach (the abstract result) and apply it to conclude the proofs of Theorem 2, 5 and 1.

Proof of the splitting formula

The connection between w n and W originates in the following computation Lemma 2.1. For any x 1 , . . . , x n and letting ν = i δ x i the following holds

(2.1) F n (ν) = 1 nπ W (-∇ ⊥ H , 1 R 2 ) + 2 n i=1 ζ(x i ) = 1 n w n (x 1 , . . . , x n ) -n 2 I(µ 0 ) + n 2 log n ,
where F n is defined in (1.21), W is defined in (1.27), and H is defined in (1.19).

Proof. Let ν = n i=1 δ x i . First we note that since ν and nµ 0 have same mass and compact support we have H

(x) = O(1/|x|) and ∇H(x) = O(1/|x| 2 ) as |x| → +∞.
We prove that, denoting the diagonal in

R 2 × R 2 , (2.2) (R 2 ×R 2 )\ -log |x -y| d(ν -nµ 0 )(x) d(ν -nµ 0 )(y) = 1 π W (-∇ ⊥ H, 1 R 2 ).
First, using Green's formula,

B R \∪ i B(x i ,η) |∇H| 2 = ∂B R H ∂H ∂ν + i ∂B(x i ,η) H ∂H ∂ν + 2π B R \∪ i B(x i ,η) H d(ν -nµ 0 ). Let H i (x) := H(x) + log |x -x i |.
We have

H i = -log * (ν i -nµ 0 ), with ν i = ν -δ x i , and near x i , H i is C 1 . Therefore, using (1.19
) and the boundedness of µ 0 in L ∞ , we have that, as η → 0

∂B(x i ,η) H ∂H ∂ν = -2π log η + 2πH i (x i ) + o(1),
while the integral on ∂B R tends to 0 as R → +∞ from the decay properties of H. We thus obtain, as η → 0 and R → +∞,

B R \∪ i B(x i ,η) |∇H| 2 = -2πn log η + 2π i H i (x i ) -2πn H dµ 0 + o(1),
and therefore, by definition of W ,

(2.3) W (-∇ ⊥ H, 1 R 2 ) = π i H i (x i ) -πn H dµ 0 .
Second we note that given x ∈ R 2 , either x / ∈ {x i } and

R 2 \{x} -log |x -y| d(ν -nµ 0 )(y) = H(x), or x = x i and R 2 \{x} -log |x -y| d(ν -nµ 0 )(y) = H i (x i ). It follows that c -log |x -y| d(ν -nµ 0 )(x) d(ν -nµ 0 )(y) = i H i (x i ) -n H(x) dµ 0 (x),
which together with (2.3) proves (2.2).

On the other hand, we may rewrite w n as

w n (x 1 , . . . , x n ) = c -log |x -y| dν(x) dν(y) + n V (x) dν(x)
and, splitting ν as nµ 0 + ν -nµ 0 and using the fact that µ 0 ( ) = 0, we obtain

w(x 1 , . . . , x n ) = n 2 I(µ 0 ) + 2n U µ 0 (x) d(ν -nµ 0 )(x) + n V (x) d(ν -nµ 0 )(x) + c -log |x -y| d(ν -nµ 0 )(x) d(ν -nµ 0 )(y).
Since U µ 0 + V 2 = c + ζ and since ν and nµ 0 have same mass n, we have

2n U µ 0 (x) d(ν -nµ 0 )(x) + n V (x) d(ν -nµ 0 )(x) = 2n ζ d(ν -µ 0 ) = 2n ζ dν,
using the fact that ζ = 0 on the support of µ 0 . Therefore, in view of (2.2) we have found

(2.4) w(x 1 , . . . , x n ) = n 2 I(µ 0 ) + 2n ζ dν + 1 π W (-∇ ⊥ H, 1 R 2 ).
But, changing variables,

1 2 R 2 \∪ n i=1 B(x i ,η) |∇H| 2 = 1 2 R 2 \∪ n i=1 B(x i , √ nη) |∇H | 2 ,
and by adding πn log η on both sides and letting η → 0 we deduce that W (-

∇ ⊥ H, 1 R 2 ) = W (-∇ ⊥ H , 1 R 2 ) -π 2 n log n.
Together with (2.4) this proves (2.1).

3 A first lower bound on F n and upper bound on Z β n

The crucial fact that we now wish to exploit is that, even though W (j, χ) does not have a sign, there are good lower bounds for F n . This follows from the analysis of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], more specifically from the following "mass spreading result", adapted from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], Proposition 4.9 and Remark 4.10 (with slightly different notation), which itself is based on the so-called "ball construction method", a crucial tool in the analysis of Ginzburg-Landau equations. This result, that we will use here as a black box, says that even though the energy density associated to W (j, χ) is not positive (or even bounded below), it can be replaced by an energy-density g which is uniformly bounded below, at the expense of a negligible error.

For any set Ω, Ω denotes its 1-tubular neighborhood, i.e. {x ∈ R 2 , dist(x, Ω) < 1}.

Proposition 3.1. Assume Ω ⊂ R 2 is open and (ν, j) are such that ν = 2π p∈Λ δ p for some finite subset Λ of Ω and curl j = 2π(ν -m), div j = 0 in Ω, where m ∈ L ∞ ( Ω). Then there exists a measure g supported on Ω and such that -Given any constant 0 < ρ, there exists a family of disjoint closed balls B ρ covering Supp(ν), with the sum of the radii of the balls in B ρ intersecting with any ball of radius 1 bounded by ρ, and such that

(3.1) g ≥ -C( m L ∞ + 1) + 1 4 |j| 2 1 Ω\Bρ in Ω,
where C is a universal constant.

-

(3.2) g = 1 2 |j| 2 outside ∪ p∈Λ B(p, C)
where C is universal.

-For any function χ compactly supported in Ω we have

(3.3) W (j, χ) -χ dg ≤ CN (log N + m L ∞ ) ∇χ ∞ ,
where N = #{p ∈ Λ : B(p, C) ∩ Supp(∇χ) = ∅} for some universal C.

-For any U ⊂ Ω,

(3.4) #(Λ ∩ U ) ≤ C 1 + m 2 L ∞ | U | + g( U ) .
Note that the result in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] is not stated for any ρ but a careful inspection of the proof there allows to show that it can be readapted to make ρ arbitrarily small. Definition 3.2. Assume ν = n i=1 δ x i . Letting ν = n i=1 δ x i be the measure in blown-up coordinates and j ν = -∇ ⊥ H , where H is defined by (1.19), we denote by g ν the result of applying the previous proposition to (ν , j ν ) in R 2 .

Even though we will not use the following result in the sequel, we state it to show how we can quickly derive a first upper bound on Z β n from what precedes.

Proposition 3.3. We have

(3.5) log K β n ≤ Cnβ + n(log |E| + o(1))
and

(3.6) log Z β n ≤ - β 2 n 2 I(µ 0 ) + βn 4 log n + Cnβ + n(log |E| + o(1))
where o(1) → 0 as n → ∞ uniformly with respect to β > β 0 , for any β 0 > 0, and C depends only on V .

The proof uses two lemmas.

Lemma 3.4. For any ν = n i=1 δ x i , we have

(3.7) F n (ν) = 1 nπ R 2 dg ν + 2 ζ dν,
where F n is as in (1.21).

Proof. This follows from (3.3) applied to χ B R , where

χ B R is as in (1.29). If R is large enough then #{p ∈ Supp(ν) : B(p, C) ∩ Supp(∇χ) = ∅} = 0 and therefore (3.3) reads W (j ν , χ B R ) = χ B R dg ν .
Letting R → +∞ yields W (j ν , 1 R 2 ) = dg ν and the result, in view of (1.21).

Lemma 3.5. Letting ν n stand for n i=1 δ x i we have, for any constant λ > 0 and uniformly w.r.t. β greater than any arbitrary positive constant β 0 ,

(3.8) lim n→∞ C n e -λβn ζ dνn dx 1 . . . dx n 1 n = |E|.
Proof. By separation of variables, we have

C n e -λβn n i=1 ζ(x i ) dx 1 . . . dx n = C e -λβnζ(x) dx n
On the other hand, we have ζ ≥ 0 and {ζ = 0} = E by (1.16), hence we have e -λβnζ(x) → 1 E pointwise, as βn → ∞. By Lebesgue's dominated convergence theorem, it follows that (3.8) holds uniformly w.r.t. β > β 0 , for any β 0 > 0.

Proof of Proposition 3.3. Let again ν n stand for n i=1 δ x i . From (3.2) we have g νn ≥ 0 outside ∪ i B(x i , C) and from (3.1) we have g νn ≥ -C (depending only on µ 0 L ∞ hence on V ) in ∪ i B(x i , C). Inserting into (3.7) we deduce that

F n (ν n ) ≥ -C + 2 ζ dν n ,
where C depends only on V . Inserting into (1.23) and integrating over C n , we find

1 ≤ 1 K β n e Cnβ C n
e -nβ ζ dνn dx 1 . . . dx n .

Inserting (3.8) and taking logarithms, it follows that

log K β n ≤ Cnβ + n(log |E| + o(1)).
The relation (3.6) follows using (1.24).

A construction and a lower bound for Z β n

In this section, we construct a set of explicit configurations whose W is not too large, and show that their probability is not too small, which will lead to a lower bound on Z β n . This is the longest part of our proof. The method is borrowed from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] but requires various adjustments that we shall detail in Section 7. We will need (1.13) in order to simplify the construction and estimates near the boundary.

We recall the following result ([SS1], Proposition 4.2). In this proposition, the notation W K is used for the renormalized energy relative to the family of squares

K R = [-R, R] 2 , i.e. defined by W K (j) = lim sup R→∞ W (j,χ K R ) |K R |
where χ K R satisfies (1.29) relative to the family of centered squares.

The following proves Theorem 2, part B and contains a bit more information useful for proving Theorem 5.

Proposition 4.1. Let P be a T λ(x) -invariant probability measure on X = E × L p loc (R 2 , R 2 ) with first marginal dx |E /|E| and such that for P almost every (x, j) we have j ∈ A m 0 (x) . Then, for any η > 0, there exists δ > 0 and for any n a subset A n ⊂ C n such that |A n | ≥ n!(πδ 2 /n) n and for every sequence {ν n = δ y 1 + • • • + δ yn } n with (y 1 , . . . , y n ) ∈ A n the following holds.

i) We have the upper bound

(4.1) lim sup n→∞ 1 n w n (y 1 , . . . , y n ) -n 2 I(µ 0 ) + n 2 log n ≤ |E| π W K (j) dP (x, j) + η.
ii) There exists {j n } n in L p loc (R 2 , R 2 ) such that curl j n = 2π(ν n -m 0 ) and such that the image P n of dx |E /|E| by the map x → (x, j n (

√ nx + •)) is such that (4.2) lim sup n→∞ dist(P n , P ) ≤ η,
where dist is a distance which metrizes the topology of weak convergence on P(X).

Applying the above proposition with η = 1/k we get a subset A n,k in which we choose any n-tuple (y i,k ) 1≤i≤n . This yields in turn a family {P n,k } of probability measures on X. A standard diagonal extraction argument then yields Corollary 4.2 (Theorem 2, Part B). Under the same assumptions as Proposition 4.1, there exists a sequence

{ν n = δ x 1 + • • • + δ xn } n and a sequence {j n } n in L p loc (R 2 , R 2 ) such that curl j n = 2π(ν n -m 0 ) and (4.3) lim sup n→∞ 1 n w n (x 1 , . . . , x n ) -n 2 I(µ 0 ) + n 2 log n ≤ |E| π W K (j) dP (x, j).
Moreover, if we assume in addition that P is T λ(x) -invariant then, denoting P n the image of dx |E /|E| by the map x → (x, j n ( √ nx + •)), we have P n → P as n → +∞.

Another consequence of Proposition 4.1 is, recalling (1.36) and (1.24):

Corollary 4.3 (Lower bound part of Theorem 1). For any η > 0 there exists C η > 0 such that for any β > 0 we have

(4.4) lim inf n→+∞ log K β n n ≥ - β 2 (α + η) -C η .
Proof of the corollary. We use Proposition 4.1. Choose j 0 ∈ A 1 to be a minimizer for min A 1 W , and let P be the image of the normalized Lebesgue measure on E by the map x → (x, σ m 0 (x) j 0 ), where (4.5) σ m j(y) := √ m j( √ my).

Then by construction P -almost every (x, j) satisfies j ∈ A m 0 (x) and the first marginal of P is dx |E /|E|. Given η > 0, applying Proposition 4.1 and using the notation there we have

|A n | ≥ n!(δ 2 /n) n and from (1.23) we have (4.6) 1 ≥ An 1 K β n e -n β 2 Fn(νn) dy 1 . . . dy n ,
where ν n = n i=1 δ y i . From (1.22) and (4.1), when (y 1 , . . . , y n ) ∈ A n we have

F n (ν n ) ≤ η + |E| π W K (j) dP (x, j) = η + 1 π E W K (σ m 0 (x) j 0 ) dx.
From (1.33), which is true for W K as well as W , and since E m 0 = 1, we obtain

1 π E W K (σ m 0 (x) j 0 ) dx = 1 π W K (j 0 ) - 1 2 E m 0 (x) log m 0 (x) dx = α,
by definition (1.36) and using the fact that W and W K have the same minimum and minimizers (see the remark after (1.34)). We deduce

F n (ν n ) ≤ η + α.
Together with (4.6) we find 1 ≥ |An|

K β n e -n β 2 (η+α) .
Taking logarithms, we find

log K β n ≥ log n! + n log δ 2 -n log n - 1 2 nβ(η + α).
From Stirling's formula, log n! ≥ n log n -Cn and we deduce (4.4), with

C η = -log δ 2 + C.
Note that the dependence on η comes from δ.

5 Consequences for fluctuations: proof of Theorem 3

In this section, we prove Theorem 3. The first step is to find, via the method first introduced in [SS3] and tools from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]ST] how F n and W control the discrepancy between ν n and the measure nµ 0 .

Proposition 5.1. Let ν = n i=1 δ x i , and g ν be as in Definition 3.2. There exists a universal constant R 0 > 0 such that for any R > R 0 , and any x 0 = √ nx 0 ∈ R 2 , we have

(5.1)

B 2R (x 0 ) dg ν ≥ cD(x 0 , R) 2 min 1, |D(x 0 , R)| R 2 -CR 2 ,
where c > 0 and C depend only on V , and where D was defined in (1.38).5

Proof. From Proposition 3.1 we have, defining j ν as in Definition 3.2,

(5.2)

g ν ≥ -C + 1 4 |j ν | 2 1 R 2 \Bρ ,
where B ρ is a set of disjoint closed balls covering Supp(ν ), and the sum of the radii of the balls in B ρ intersecting any given ball of radius 1 is bounded by ρ and we may take ρ < 1 8 . We distinguish two cases. Either D(x 0 , R) ≥ 0 or D(x 0 , R) < 0. Let us start with the first case. We must introduce a modified distance function to x 0 , as follows. For any x, f (x) is the infimum over the set of curves γ joining a point in B R (x 0 ) to x of the length of γ \ B ρ . This is also the distance to x 0 for the degenerate metric which is Euclidean outside B R (x 0 ) ∪ B ρ and vanishes on B R (x 0 ) ∪ B ρ . We claim the following:

Lemma 5.2. If |x -x 0 | ≥ R + 2 then |x -x 0 | -R 4 ≤ f (x) ≤ |x -x 0 | -R.
Proof. The upper bound is obvious so we turn to the lower bound. Let γ(t) be a continuous curve joining x = γ(0) to B R (x 0 ). Let us build by induction a sequence x 0 = x, . . . , x K with x k+1 defined as follows: let t k+1 be the smallest t > t k such that γ(t) / ∈ B(x k , 1) ∩ γ or γ(t) ∈ B R (x 0 ). This procedure terminates after a finite number of steps at x K ∈ ∂B R (x 0 ). By triangle inequality we have

(5.3) |x -x 0 | ≤ K-1 k=0 |x k+1 -x k | + |x K -x 0 | ≤ K + R.
On the other hand, by property of B ρ , for any 0

≤ k ≤ K -1 we have (γ[t k , t k+1 ] \ B ρ ) ≥ |x k+1 -x k | -2ρ.
Summing this over k and using (5.3), we find

(γ \ B ρ ) ≥ K -1 -2Kρ ≥ |x -x 0 | -R -1 -2ρ |x -x 0 | -R .
Taking the infimum over all curves γ we deduce that

f (x) ≥ |x -x 0 | -R (1 -2ρ) -1.
Since by assumption |x -

x 0 | -R ≥ 2, we obtain f (x) ≥ (|x -x 0 | -R) (1 -2ρ -1
2 ) and the result follows since ρ < 1/8. We proceed with the proof of Proposition 5.1. Since f is Lipschitz with constant 1, almost every t is a regular value of f . For such a t the curve γ t := {f = t} is Lipschitz and does not intersect B ρ , since ∇f = 0 there. Moreover, restating Lemma 5.2 we have (5.5)

(5.4) f (x) < t =⇒ x ∈ B R+4t (x 0 ) ∪ B R+2 (x 0 ), thus γ t ⊂ B 2R (x 0 ) if R + 4t < 2R, i
B 2R (x 0 ) dg ν ≥ -CR 2 + 1 4 R/4 t=0 γt |j ν | 2 dt.
We proceed by estimating the innermost integral on the right-hand side. If t > 1/2 then B R+2 ⊂ B R+4t and using (5.4) we find, using Definition 3.2 and writing D instead of D(x 0 , R) in the course of this proof,

γt j ν • τ = {f <t} curl j ν ≥ 2πν B R √ n (x 0 ) -2πnµ 0 B R+4t √ n (x 0 ) ≥ 2πD -C (R + 4t) 2 -R 2 ,
where we have used (1.15). The right-hand side is bounded below by πD if R+4t < √ R 2 + cD and c is small enough. Thus, if

(5.6) 1/2 < T := R 4 1 + c D R 2 -1 ,
it follows using Cauchy-Schwarz's inequality that for every t ∈ (1/2, T ) we have

(5.7)

γt |j ν | 2 ≥ π 2 D 2 |γ t | .
Inserting into (5.5) while reducing integration to 1/2 < t < min(T, R/4) we obtain

(5.8)

B 2R (x 0 ) dg ν ≥ -CR 2 + π 2 D 2 min(T,R/4) t=1/2 1 |γ t | dt.
The evaluation of the last integral is complicated by the fact that γ t is the level set for f rather than the usual circle, however the result will be comparable to the one we would get if we had |γ t | = 2π(R + t), this is proven as follows: From Lemma 5.2 we have for every

t ∈ [1/2, min(T, R/4)] that γ t ⊂ {x : 0 < |x -x 0 | -R < min(4T, R)}.
From the coarea formula and the fact that |∇f | ≤ 1 it follows that min(T,R/4)

1 2 |γ t | dt ≤ |{x : R < |x -x 0 | < R + min(4T, R)}| ≤ CR min(4T, R).
Then, using the convexity of x → 1/x and Jensen's inequality in (5.8) we obtain for some c > 0 that min(T,R/4)

1 2 1 |γ t | dt ≥ c min(T, R/4) -1 2 2 R min(4T, R) .
Inserting into (5.8) we obtain assuming (5.6) and

(5.9) 1 < min(T, R/4) that (5.10)

B 2R (x 0 ) dg ν ≥ -CR 2 + c D 2 R min(T, R/4) - 1 2 .
One may check that (5.6), (5.9) are satisfied if R > 4 and D > C 0 R for a large enough C 0 > 0. Then it is not difficult to deduce (5.1) from (5.10) by distinguishing the cases T < R/4 and T ≥ R/4, i.e. D < C 1 R 2 and D ≥ C 1 R 2 for a well chosen C 1 . Finally, if D < C 0 R then (5.1) is trivially satisfied, if C is chosen large enough.

Let us turn to the case D(x 0 , R) ≤ 0, which implies

|D(x 0 , R)| ≤ nµ 0 (B R/ √ n (x 0 )) ≤ πR 2 m.
In this case we define f (x) to be the distance of x to the complement of B R (x 0 ) with respect to the metric which is Euclidean on B R (x 0 ) \ B ρ . As in Lemma 5.2 and assuming |x -

x 0 | < R -2 we have R -|x -x 0 | 4 ≤ f (x) ≤ R -|x -x 0 |.
and as above, if R > 2 and for almost every 1/2 < t < R/4 the curve γ t = {f = t} is a Lipschitz curve which does not intersect B ρ and {f < t} ⊂ B R (x 0 ) \ B R-4t (x 0 ). It follows that, writing as before D for D(x 0 , R)

(5.11)

γt j ν • τ = {f <t} curl j ν = B R (x 0 ) curl j ν - B(x 0 ,R)\{f >t} curl j ν ≤ 2πD + 2πnµ 0 B R √ n \ B R-4t √ n ≤ 2πD + C R 2 -(R -4t) 2 .
The proof then proceeds as in the first case by using Cauchy-Schwarz's inequality and integrating with respect to t ∈ [1/2, min(T, R/4)], where

T = R 4 1 -1 + c D R 2 ,
which ensures that the right-hand side in (5.11) is bounded above by πD. Note that D is nonpositive, but bounded below by -CR 2 hence if c > 0 is small enough the quantity inside the square root above is positive.

We now proceed to the proof of Theorem 3, starting with (1.42). If R > R 0 and |D(x 0 , R)| ≥ ηR 2 then from Proposition 5.1 and using the fact -from Proposition 3.1 -that g ν is positive outside ∪ n i=1 B(x i , C) and that g ν ≥ -C everywhere, we deduce from (3.7) and (5.1) that (5.12)

F n (ν) ≥ 1 n -CR 2 + c min(η 2 , η 3 )R 4 + 2 ζ dν.
Inserting into (1.23) we find

P β n D(x 0 , R) ≥ ηR 2 ≤ 1 K β n exp CβR 2 -cβ min(η 2 , η 3 )R 4 e -nβ ζ dν dx 1 . . . dx n .
Then, using the lower bound (4.4) and Lemma 3.5 we deduce that if β > β 0 > 0 and n is large enough depending on β 0 then log

P β n D(x 0 , R) ≥ ηR 2 ≤ -cβ min(η 2 , η 3 )R 4 + CβR 2 + Cnβ + Cn,
where c, C > 0 depend only on V . Thus (1.42) is established.

We next prove (1.44). By Fubini's theorem, and using again the facts that g ν is positive outside ∪ n i=1 B(x i , C) and ≥ -C everywhere we have

R 2 dg ν ≥ U - B(x ,2R) dg ν dx -C|U | -Cn.
Combining with Proposition 5.1 it follows that

R 2 dg ν ≥ -C(|U | + n) + 1 R 2 U -CR 2 + cD(x , R) 2 min 1, |D(x , R)| R 2 dx .
i.e., changing the constants if necessary, (5.13)

R 2 dg ν ≥ -C(|U | + n) + c R 2 U D(x , R) 2 min 1, |D(x , R)| R 2 dx.
It follows, using as above (1.23), (3.7), (4.4) and Lemma 3.5, and since

|U | = n|U |, that log P β n U D(x , R) 2 R 2 min 1, |D(x , R)| R 2 dx ≥ η ≤ -cnβη + Cnβ (|U | + 1) + Cn,
where c, C > 0 depend only on V , where β > β 0 > 0 and where n > n 0 (β 0 ).

We next turn to (1.43). Arguing as above, from (3.7) we have F n (ν) ≥ -C + 2 ζ dν. Splitting 2 ζ dν as ζ dν + ζ dν, inserting into (1.23) and using (4.4) we are led to

P β n ζ dν ≥ η ≤ e -1 2 nβη+Cn(β+1) e -nβ ζ dν dx 1 . . . , dx n ,
where C depends only on V . Then, using Lemma 3.5 we deduce (1.43).

There remains to prove (1.45). Let ν = n i=1 δ x i and χ be a nonnegative function such that χ = 1 on

U := E ∪ ∪ n i=1 B(x i , 1 2 ) ∪ B R and χ ∞ , ∇χ ∞ ≤ 1, compactly supported on U = {x : d(x, U ) ≤ 1}. We have | U | ≤ C(n + R 2 )
, where C depends only on E, i.e. on V . Then Corollary 1.2 in [ST] asserts that, for any q < 2, (5.14)

√ χj ν L q ≤ C q (n + R 2 ) 1 q -1 2 (W (j ν , χ) + n) 1 2 .
where we use that by construction, #{p ∈ Λ|B(p, 1 2 ) ∩ {0 ≤ χ < 1} = ∅} = 0. This same fact implies that ν = 0 in the support of 1 -χ, hence

W (j ν , 1 -χ) = 1 2 (1 -χ)|j ν | 2 ≥ 0. In particular W (j ν , χ) ≤ W (j ν , χ) + W (j ν , 1 -χ) = W (j ν , 1 R 2 ).
It then follows from (5.14) and the fact that

F n (ν) = 1 πn W (j ν , 1 R 2 ) + 2 ζ dν (cf. (1.21)), that (5.15) B R |j ν | q ≤ C q (n + R 2 ) 1-q 2 n q 2 F n (ν) -2 ζ dν + 1 q 2 .
But, reasoning as above after (5.12), the probability that F n (ν) -2 ζ dν ≥ η is bounded above for any β > β 0 > 0 an n large enough depending on β 0 by exp(-1 2 nβη + Cn(β + 1)), where C depends on V only. In view of (5.15),

P β n B R |j ν | q ≥ C q n(η + 1) q 2 1 + R 2 n 1-q 2 ≤ exp(- 1 2 nβη + Cn(β + 1)).

Rescaling we have

Ω |∇H| q = n q 2 -1 Ω |j ν | q , where H = -2π∆ -1 (ν -nµ 0 ), while ν - nµ 0 W -1,q (Ω) ≤ C ∇H L q (Ω) . Therefore P β n 1 + R 2 n 1 2 -1 q ν -nµ 0 W -1,q (B R/ √ n ) ≥ C q n 1 2 (1 + η) 1 2 ≤ exp(- 1 2 nβη + Cn(β + 1)).
After a slight rewriting, this concludes the proof of Theorem 3.

6 Lower bounds via the ergodic theorem and conclusions

Abstract result via the ergodic theorem

In this section, we present the ergodic framework introduced in [SS1] for obtaining "lower bounds for 2-scale energies". We cannot directly use the result there because it is written for a uniform "macroscopic environment", which would correspond to the case where m 0 (x) is constant on its support (as in the circular law). To account for the possibility of varying environment or weight at the macroscopic, we can however adapt Theorem 3 of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and easily prove the following variant: Let X denote a Polish metric space, when we speak of measurable functions on X we will always mean Borel-measurable. We assume there is a d-parameter group of transformations θ λ acting continuously on X. More precisely we require that -For all u ∈ X and λ, µ ∈ R d , θ λ (θ µ u) = θ λ+µ u, θ 0 u = u.

-The map (λ, u) → θ λ u is continuous with respect to each variable (hence measurable with respect to both).

Typically we think of X as a space of functions defined on R d and θ as the action of translations, i.e. θ λ u(x) = u(x + λ). Then we consider the following d-parameter group of transformations T ε λ acting continuously on R d × X by T ε λ (x, u) = (x + ελ, θ λ u). We also define T λ (x, u) = (x, θ λ u).

For a probability measure P on R d × X we say that P is translation-invariant if it is invariant under the action T , and we say it is T λ(x) -invariant if for every function λ(x) of class C 1 , it is invariant under the mapping (x, u) → (x, θ λ(x) u). Note that T λ(x) -invariant implies translation-invariant.

Let G denote a bounded compact set in R d such that (6.1)

|G| > 0, lim ε→0 |(G + εx) G| |G| = 0,
for every x ∈ R 2 . We let {f ε } ε and f be measurable nonnegative functions on G × X, and assume that for any family {(x ε , u ε )} ε such that

∀R > 0, lim sup ε→0 B R f ε (T ε λ (x ε , u ε )) dλ < +∞ the following holds. 1. (Coercivity) {(x ε , u ε )} ε admits a convergent subsequence (note that {x ε } ε subsequen- tially converges since G is compact). 2. (Γ-liminf) If {(x ε , u ε )} ε converges to (x, u) then lim inf ε→0 f ε (x ε , u ε ) ≥ f (x, u).

Then we consider an increasing family of bounded open sets

{U R } R>0 such that (6.2) (i) {U R } R>0 is a Vitali family, (ii) lim R→+∞ |(λ + U R ) U R | |U R | = 0
for any λ ∈ R d , where Vitali means (see [Ri]) that the intersection of the closures is

{0}, that R → |U R | is left continuous, and that |U R -U R | ≤ C|U R |.
We have Theorem 6. Let G, X, {θ λ } λ , and {f ε } ε , f be as above. Let

F ε (u) = - G f ε (x, θ x ε u) dx.
Assume that {F ε (u ε )} ε is bounded. Let P ε be the probability on G × X which is the image of the normalized Lebesgue measure on G under the map x → (x, θ x ε u ε ). Then P ε converges to a Borel probability measure P on G×X whose first marginal is the normalized Lebesgue measure on G, which is T λ(x) -invariant, such that P -a.e. (x, u) is of the form lim ε→0 (x ε , θ xε ε u ε ) and such that

(6.3) lim inf ε→0 F ε (u ε ) ≥ f (x, u) dP (x, u).
Moreover,

(6.4) f (x, u) dP (x, u) = E P lim R→+∞ - U R f (x, θ λ u) dλ ,
where E P denotes the expectation under the probability P .

Proof. We only sketch the proof as it is very similar to [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]. The following points have to be checked:

1. P ε is tight hence has a limit P . This follows from the coercivity property of f ε as in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF].

2. P is T λ(x) -invariant. Let Φ be bounded and continuous, and let P λ be the push-forward of P by (x, u) → (x, θ λ(x) u). Then from the definition of P λ , P , P ε , we have

Φ(x, u) dP λ (x, u) = Φ(x, θ λ(x) u) dP (x, u) = lim ε→0 Φ(x, θ λ(x) u) dP ε (x, u) = lim ε→0 - G Φ(x, θ x ε +λ(x) u ε ) dx = lim ε→0 - (I+ελ)(G) Φ((I + ελ) -1 (y), θ y ε u ε ) | det(I + εDλ((I + ελ) -1 (y))| dy,
where the last equality follows by the change of variables y = (I + ελ)(x). Using the boundedness of Φ, the C 1 character of λ, the compactness of G and (6.1), we may replace (I + ελ)(G) by G and the denominator by 1 in the last integral and we find, using the definition of P ε

(6.5) Φ(x, u) dP λ (x, u) = lim ε→0 Φ((I + ελ) -1 (x), u) dP ε (x, u).
Since {P ε } ε is tight, for any δ > 0 there exists K δ such that P ε (K δ c ) < δ for every ε. Then by uniform continuity of Φ on K δ the map (x, u) → Φ((I + ελ) -1 (x), u) converges uniformly on K δ to (x, u) → Φ(x, u) and thus

lim ε→0 K δ Φ((I + ελ) -1 (x), u) dP ε (x, u) = lim ε→0 K δ Φ(x, u) dP ε (x, u).
Since this is true for any δ > 0, and using the boundedness of Φ we get

lim ε→0 Φ((I + ελ) -1 (x), u) dP ε (x, u) = lim ε→0 Φ(x, u) dP ε (x, u) = Φ(x, u) dP (x, u),
by definition of P . Thus in view of (6.5) we have P λ = P and P is thus T λ(x) -invariant.

lim inf

ε→0 F ε (u ε ) ≥ f dP . This follows from Lemma 2.2 of [SS1], since F ε (u ε ) = f ε dP ε .
To conclude, as in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], the fact that P is T λ(x) -invariant (which implies T λ -invariant) and Wiener's multiparametric ergodic theorem (see e.g. [Be]) imply that

f (x, u) dP (x, u) = E P lim R→+∞ - U R f (T λ (x, u)) dλ = E P lim R→+∞ - U R f (x, θ λ u) dλ .

Proof of Theorem 2, part A

The proof follows essentially [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], Proposition 4.1 and below. Let {ν n } n and P νn be as in the statement of Theorem 2. We need to prove that any subsequence of {P νn } n has a convergent subsequence and that the limit P is a T λ(x) -invariant probability measure such that P -almost every (x, j) is such that j ∈ A m 0 (x) and (1.39) holds. Note that the fact that the first marginal of P is dx |E /|E| follows from the fact that, by definition, this is true of P νn . We thus take a subsequence of {P νn } (which we don't relabel). We may assume that it has a subsequence, denoted νn , which satisfies F n (ν n ) ≤ C, otherwise there is nothing to prove. This implies that νn is of the form n i=1 δ x i,n . We let n denote the current and ḡn the measures associated to νn as in Definition 3.2. As usual,

ν n = n i=1 δ √ nx i,n . Another useful consequence of F n (ν n ) ≤ C is that, using (2.1), we have w n (x i,n ) = n 2 I(µ 0 ) + O(n log n),
which in turn implies (see [ST]) that (6.6) 1 n νn → µ 0 .

Step 1: We set up the framework of Section 6.1

We will use integers n instead of ε to label sequences, and the correspondence will be ε

= 1/ √ n. We let G = E and X = M + × L p loc (R 2 , R 2 ) × M
, where p ∈ (1, 2), where M + denotes the set of positive Radon measures on R 2 and M the set of those which are bounded below by the constant -C V := -C( m 0 ∞ + 1) of Proposition 3.1, both equipped with the topology of weak convergence.

For λ ∈ R 2 and abusing notation we let θ λ denote both the translation x → x + λ and the action

θ λ (ν, j, g) = (θ λ #ν, j • θ λ , θ λ #g) .
Accordingly the action T n is defined for λ ∈ R 2 by

T n λ (x, ν, j, g) = x + λ √ n , θ λ #ν, j • θ λ , θ λ #g .
Then we let χ be a smooth cut-off function with integral 1 and support in B(0, 1) and define (6.7)

f n (x, ν, j, g) =    1 π R 2 χ(y) dg(y) if (ν, j, g) = θ √ nx (ν n , n , ḡn ), +∞ otherwise.
Finally we let, in agreement with Section 6.1, (6.8)

F n (ν, j, g) = - E f n x, θ x √ n (ν, j, g) dx.
We have the following relation between F n and F n , as n → +∞: (6.9)

F n (ν, j, g) is ≤ 1 |E| F n (ν n ) -2 ζ dν n + o(1) if (ν, j, g) = (ν n , n , ḡn ) = +∞ otherwise.
Indeed it is obvious from (6.7) that if (ν, j, g) = (ν n , n , ḡn ) then F n (ν, j, g) = +∞. On the other hand, if (ν, j, g) = (ν n , n , ḡn ), then from the definition of the image measure θ λ #ḡ n ,

F n (ν, j, g) = 1 π - E χ(y -x √ n) dḡ n (y) dx = 1 π|E | χ * 1 E dḡ n .
Since χ * 1 E is bounded above by 1 and is equal to 1 on

U := {x : dist(x , R 2 \ E ) ≥ 1} we deduce that (6.10) πF n (ν, j, g) ≤ ḡ+ n (R 2 ) -ḡ- n (U ) |E | = ḡn (R 2 ) + ḡ- n (U c ) n|E| = π |E| F n (ν n ) -2 ζ dν n + ḡ- n (U c ) n|E| .
Then we note that from (3.2) in Proposition 3.1 the measure ḡn is supported in the union of balls B(x , C) for x ∈ Supp(ν n ), and bounded above by a constant. Thus ḡn (U c ) is bounded by a constant times the number of balls intersecting U c , hence by C ν n {x : dist(x , U c ) ≤ C}. From (6.6) this is equal to Cnµ 0 {x : dist(x, ∂E) ≤ C/ √ n} + o(n), which from the assumption (1.13) on E is o(n). Plugging this into (6.10) proves (6.9).

Step 2: We check the hypotheses in Section 6.1

We must now check the Gamma-liminf and coercivity properties of {f n } n . Lemma 6.1. Assume that {(x n , ν n , j n , g n )} n converges to (x, ν, j, g). Then

lim inf n f n (x n , ν n , j n , g n ) ≥ f (x, ν, j, g) := 1 π χ dg.
Proof. We may assume that the left-hand side is finite, in which case

f n (x n , ν n , j n , g n ) = 1 π
χ dg n for every large enough n, from which the result follows by passing to the limit.

Lemma 6.2. Assume that for any R > 0 we have

(6.11) lim sup n→+∞ B R f n x n + λ √ n , θ λ (ν n , j n , g n ) dλ < +∞.
Then a subsequence of {(x n , ν n , j n , g n )} n converges to some (x, ν, j, g) ∈ E × X.

Proof. Assume (6.11). Then the integrand there is bounded for a.e. λ and from the definition (6.7) we deduce that θ λ (ν n , j n , g n ) = θ √ nxn+λ (ν n , n , ḡn ) and then that (ν n , j n , g n ) = θ √ nxn (ν n , n , ḡn ). Thus (6.11) gives, in view of (6.7), that for every R > 0 there exists C R > 0 such that for any n

B R χ(y - √ nx n -λ) dḡ n (y) dλ = χ * 1 B R ( √ nxn) dḡ n < C R .
This and the fact that ḡn is bounded below implies that ḡn (B R ( √ nx n )) is bounded independently of n and then, using (3.4), that the same is true of ν n (B R ( √ nx n )). In other words {ν n = θ √ nxn ν n } n is a locally bounded sequence of (positive) measures hence converges weakly after taking a subsequence, and the same is true of {g n = θ √ nxn ḡn } n . On the other hand {x n } n is a sequence in the compact set E hence converges modulo a subsequence.

It remains to study the convergence of

{j n = n •θ √ nxn+λ } n . From (3.3) in Proposition 3.1 and the local boundedness of {ν n } n we get that W ( n , χ * 1 B R ( √ nxn) ) = W (j n , χ * 1 B R
) is bounded independently of n for any R > 0 and then, using Corollary 1.2 in [ST], that {j n } n is locally bounded in L p loc (R 2 , R 2 ), for any 1 ≤ p < 2 hence a subsequence locally weakly converges in L p loc (R 2 , R 2 ). Strong local convergence follows as in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] using elliptic regularity with curl n = ν n -µ 0 and div n = 0.

Step 3: Conclusion

From the previous steps, we may apply Theorem 6 in this setting and we deduce in view of (6.9) that, letting Q n denote the push-forward of the normalized Lebesgue measure on E by the map x → (x, θ √ nx (ν n , n , ḡn )), and Q = lim n Q n , (6.12) lim inf

n 1 |E| F n (ν n ) -2 ζ dν n ≥ lim inf n F n (ν n , n , ḡn ) ≥ χ dg dQ(x, ν, j, g) = lim R→+∞ - B R χ(y -λ) dg(y) dλ dQ(x, ν, j, g) = lim R→+∞ 1 πR 2 χ * 1 B R dg dQ(x, ν, j, g).
Now we use the fact that for Q-almost every (x, ν, j, g):

i) There exists a sequence {x n } n in E such that (x, ν, j, g) = lim n (x n , θ √ nxn (ν n , n , ḡn )).
ii) As a consequence of the above 1

πR 2 χ * 1 B R dg converges to a finite limit as R → +∞.
The first point implies, since curl n = ν n -m 0 and div n = 0, that curl j = ν -m 0 (x) and div j = 0. The second point implies in particular using (3.4) that ν(B R ) ≤ CR 2 , proving that (j, ν) ∈ A m 0 (x) . Moreover the second point implies that for any C > 0 we have g(

B R+C \ B R-C ) = o(R 2 ) as R → +∞, and thus from point i) above lim R→+∞ lim n→+∞ 1 R 2 ḡn (B R+C ( √ nx n )) \ B r-C ( √ nx n )) = 0. Using (3.4) we deduce that lim R→+∞ lim n→+∞ 1 R 2 ν n (B R+C ( √ nx n )) \ B r-C ( √ nx n )) = 0 and then from (3.3), lim R→+∞ lim n→+∞ 1 R 2 W ( n , χ * 1 B R ( √ nxn) ) -χ * 1 B R ( √ nxn) dḡ n = 0.
Thus, using Lemma 4.8 in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] in the limit with respect to n we deduce

lim R→+∞ 1 R 2 W (j, χ * 1 B R ) -χ * 1 B R dg = 0.
Together with (6.12) this yields, by definition of W , (6.13) lim inf

n 1 |E| F n (ν n ) -2 ζ dν n ≥ W (j) dQ(x, ν, j, g)
and, we recall, Q-a.e. (j, ν) ∈ A m 0 (x) . Now we let P n (resp. P ) be the marginal of Q n (resp. Q) with respect to the variables (x, j). Then the first marginal of P is the normalized Lebesgue measure on E and P -a.e. we have j ∈ A m 0 (x) , in particular

W (j) ≥ min A m 0 (x) W = m 0 (x) min A 1 W - π 2 log m 0 (x) .
Integrating with respect to P and noting that since only x appears on the right-hand side we may replace P by its first marginal there we find, in view of (1.36) that the lower bound (1.39) holds.

Proof of Theorem 2, completed

As mentioned above, Part B of the theorem is a direct consequence of Proposition 4.1, see Corollary 4.2. Part C follows from the comparison of Parts A and B and the fact that W and W K have the same minimizers and minimal value. This implies that, for minimizers, the chains of inequalities (1.39) and (1.40) are in fact equalities and that W dP must be minimized hence equal to α. Since (1.39) follows from (6.13), the inequality (6.13) must also be an equality and moreover we must have lim n→∞ ζ dν n = 0, which in view of (1.17), implies that lim i dist(x i , E) 2 = 0.

From the fact that F n (ν n ) = O(1), we deduce from Proposition 5.1, 3.1 and (3.7), (arguing exactly as in the proof of Theorem 3) that there exists C > 0 such that for every x, R > 1, we have

D(x, R) 2 min 1, |D(x, R)| R 2 ≤ Cn.
This completes the proof of Theorem 2.

Deviations: proof of Theorems 5 and Theorem 1

We start with the upper bound on log P β n . Let A n be a subset of C n . We identify points in C n with measures ν n of the form n i=1 δ x i . From (1.23), we have

P β n (A n ) = 1 K β n An e -1 2 βnFn( n i=1 δx i ) dx 1 . . . dx n hence (6.14) log P β n (A n ) n = - log K β n n + 1 n log An e -1 2 βnFn( n i=1 δx i ) dx 1 . . . dx n .
We deduce, writing

F n (ν) = F n (ν) -2 ζ dν, that (6.15) log P β n (A n ) n ≤ - log K β n n + 1 n log e -1 2 βn inf An Fn An e -βn ζ dνn dx 1 . . . dx n .
Let ν n such that F n (ν n ) ≤ inf An F n + 1/n. Then from (1.39) in Theorem 2 we have, using the notations there, lim inf n→∞ F n (ν n ) ≥ |E| π W (j) dP (x, j) where P = lim n P νn . Since

P νn ∈ i(A n ) by definition we have P ∈ A ∞ since A ∞ = ∩ n>0 ∪ m>n i(A m ) . We may thus write (6.16) lim inf n→+∞ F n (ν n ) ≥ |E| π inf P ∈A∞ W (j) dP (x, j).
Inserting into (6.15) we are led to (6.17

) log P β n (A n ) n ≤ - β|E| 2π inf P ∈A∞ W (j) dP (x, j) - log K β n n + 1 n log C n e -βn ζ dνn dx 1 . . . dx n + o(1)
thus in view of Lemma 3.5 and (4.4), we have established (1.47). An immediate corollary of (6.17), choosing A n to be the full space and using inf 1 π W (j) dP (j) = α and Lemma 3.5, is that (6.18) lim sup

n→∞ log K β n n ≤ - βα 2 + log |E|.
We next turn to the lower bound. Fix η > 0. Given A, let P ∈ Å be such that

(6.19) W (j) dP (x, j) ≤ inf P ∈ Å W (j) dP (j) + η 2 .
Since P ∈ Å, if η is chosen small enough (which we assume) then B(P, 2η) ⊂ A, where the ball is for a distance metrizing weak convergence as in Proposition 4.1.

We then apply Proposition 4.1 to P and η. We find δ > 0 and for any n large enough a set A n such that |A n | ≥ n!(πδ 2 /n) n and, rewriting (4.1) with (2.1), (6.20) lim sup

n→∞ sup An F n ≤ |E| π W K (j) dP (j) + η.
Moreover, for every (y 1 , . . . , y n ) ∈ A n and letting

{ν n = δ y 1 + • • • + δ yn } n , there exists {j n } n in L p loc (R 2 , R 2 ) such that curl j n = 2π(ν n -m 0 ) and such that the image P n of dx |E /|E| by the map x → (x, j n ( √ nx + •)) is such that (6.21) lim sup n→∞ dist(P n , P ) ≤ η.
In particular (1.49) holds. Moreover, inserting (6.20) and (6.19) into (1.23), we find that

log P β n (A n ) n ≥ - log K β n n - β|E| 2π inf P ∈ • A W (j) dP (j) - 1 2 βη + 1 n log A n √ n + o(1).
On the other hand, using |A n | ≥ n!(πδ 2 /n) n and Stirling's formula, we have log |A n | ≥ 2n log δ -Cn. Combining with (6.18), (1.48) follows, with

C η = -2 log δ + C + log |E|.
Theorem 1 immediately follows by combining (6.18), (4.4) and (1.24).

7 Proof of Proposition 4.1

The construction consists of the following. We are given ε > 0, which is the error we can afford. First we select a finite set of currents J 1 , . . . , J N (N will depend on ε) which will represent the probability P (x, j) with respect to its j dependence, and whose renormalized energies are well-controlled. Since P is T λ(x) -invariant, we need it to be well-approximated by measures supported on the orbits of the J i 's under translations. Secondly, we work in blownup coordinates and split the region E (whose diameter is order √ n) into many rectangles K with centers x K and sidelengths of order R large enough. Even though we choose R to be large, it will still be very small compared to the size of E , as n → ∞, so that the Diracs at x K / √ n approximate P (x, j) with respect to its x dependence. On each rectangle K, the weight m 0 is temporarily replaced by its average m K . Then we split each rectangle K into q 2 identical rectangles, with sidelengths of order 2R = R/q, where both R and q will be sufficiently large. We then select the proportion of the rectangles that corresponds to the weight that the orbit of each J i carries in the approximation of P . In these rectangles we paste a (translated) copy of J i at the scale m K and suitably modified near the boundary according to a construction of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] (Proposition 7.4 below) so that its tangential component on the boundary is 0 (this can be done while inducing only an error ε on W ). In the few rectangles that may remain unfilled, we paste a copy of an arbitrary J 0 whose renormalized energy is finite. We perform the construction above provided we are far enough from ∂E . The layer near the boundary must be treated separately, and there again an arbitrary (translated and rescaled) current can be pasted. Finally, we add a vector field to correct the discrepancy between m K and m 0 in each of the rectangles.

To conclude the proof of Proposition 4.1, we collect all of the estimates on the constructed vector field to show that its energy w n is bounded above in terms of W dP and that the probability measures associated to the construction have remained close to P .

Estimates on distances between probabilities

First we choose distances which metrize the topologies of L p loc (R 2 , R 2 ) and B(X), the set of finite Borel measures on

X = E × L p loc (R 2 , R 2 ). For j 1 , j 2 ∈ L p loc (R 2 , R 2 ) we let d p (j 1 , j 2 ) = ∞ k=1 2 -k j 1 -j 2 L p (B(0,k)) 1 + j 1 -j 2 L p (B(0,k))
,

and on X we use the product of the Euclidean distance on E and d p , which we denote d X .

On B(X) we define a distance by choosing a sequence of bounded continuous functions {ϕ k } k which is dense in C b (X) and we let, for any µ 1 , µ 2 ∈ B(X),

d B (µ 1 , µ 2 ) = ∞ k=1 2 -k | ϕ k , µ 1 -µ 2 | 1 + | ϕ k , µ 1 -µ 2 | ,
where we have used the notation ϕ, µ = ϕ dµ.

We have the following general facts.

Lemma 7.1. For any ε > 0 there exists η 0 > 0 such that if P, Q ∈ B(X) and P -Q < η 0 , then d(P, Q) < ε. Here P -Q denotes the total variation of the signed measure P -Q, i.e. the supremum of ϕ, P -Q over measurable functions ϕ such that |ϕ| ≤ 1.

In particular, if

P = ∞ i=1 α i δ x i and Q = ∞ i=1 β i δ x i with i |α i -β i | < η 0 , then d B (P, Q) < ε.
Lemma 7.2. Let K ⊂ X be compact. For any ε > 0 there exists η 1 > 0 such that if x ∈ K, y ∈ X and d X (x, y) < η 1 then d B (δ x , δ y ) < ε.

Lemma 7.3. Let 0 < ε < 1. If µ is a probability measure on a set A and f, g : A → X are measurable and such that d B (δ f (x) , δ g(x) ) < ε for every x ∈ A, then

d B (f # µ, g # µ) < Cε(|log ε| + 1).
Proof. Take any bounded continuous function ϕ k defining the distance on B(X). Then if d B (δ f (x) , δ g(x) ) < ε for any x ∈ X we have in particular

|ϕ k (f (x)) -ϕ k (g(x))| 1 + |ϕ k (f (x)) -ϕ k (g(x))| ≤ 2 k ε.
It follows that

d B (f # µ, g # µ) ≤ k 2 -k min(ε2 k , 1) ≤ ε ([log 2 ε] + 1) + ∞ k=[log 2 ε]+1
2 -k ≤ Cε(|log ε| + 1).

Preliminary results

In what follows

E = √ nE, m 0 (x) = m 0 (x/ √ n):
we work in blown-up coordinates. We consider a probability measure P on E × L p loc (R 2 , R 2 ) which is as in the proposition. We let P be the probability measure on E × A 1 which is the image of P under (x, j) → (x, σ 1/m 0 (x) j), so that (7.1) P = δ x ⊗ δ σ 1/m 0 (x) j dP (x, j), P = δ x ⊗ δ σ m 0 (x) j d P (x, j)

It is easy to check that since P is T λ(x) -invariant, P is as well, and in particular it is translationinvariant.

The construction is based on the following statement which is a rewriting of Proposition 4.2 in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and the remark following it:

Proposition 7.4. Let K R = [-R, R] 2 , let {χ R } R satisfy (1.29
) and let W K be the renormalized energy relative to the family {K R } R .

Let G ⊂ A 1 be such that there exists C > 0 such that for any j ∈ G we have

(7.2) ∀R > 1, ν(K R ) |K R | < C,
for the associated ν's, and

(7.3) lim R→+∞ W (j, χ R ) |K R | = W K (j),
where the convergence is uniform w.r.t. j ∈ G.

Then for any ε > 0 there exists R 0 > 0, η 2 > 0 such that if R > R 0 and L is a rectangle centered at 0 whose sidelengths belong to [2R, 2R(1 + η 2 )] and such that |L| ∈ N, then for every j ∈ G there exists a j L ∈ L p loc (R 2 , R 2 ) such that the following hold i)

j L = 0 in L c , ii) There is a discrete subset Λ ⊂ L such that curl j L = 2π   p∈Λ δ p -1 L   . iii) If d(x, L c ) > R 3 4 then j L (x) = j(x) iv) (7.4) W (j L , 1 L ) |L| ≤ W K (j) + ε.
The next lemma explains how to partition E into rectangles.

Lemma 7.5. There exists a constant C 0 > 0 such that, given any R > 1 and q ∈ N * , there exists for any n ∈ N * a collection K n of closed rectangles in E with disjoint interiors, whose sidelengths are between R = 2qR and R + C 0 R/R 2 , and which are such that

{x ∈ E : d(x, ∂E ) ≤ R} ⊂ E \ K∈Kn K ⊂ {x ∈ E : d(x, ∂E ) ≤ C 0 R}
and, for all K ∈ K n , (7.5)

K m 0 ∈ q 2 N.
Proof. For each j ∈ Z we let Then each strip {jR ≤ y < (j + 1)R} is cut into rectangles [t ij , t (i+1),j ] × [jR, (j + 1)R] where t 0j = -∞ and t i+1,j = min{t ≥ t ij + R : m j (t ij ) ∈ q 2 N}.

m j (t) = t x=-∞ (
Since by assumption (1.12) we have m 0 (x) ∈ [m, m] for any x ∈ E , it is not difficult to check that if such a rectangle is included in E then

t ij + R ≤ t i+1,j ≤ t ij + R + q 2 mR ,
and thus its sidelengths are between R and R+CR/R 2 since R/R 2 = 4q 2 /R. We let K n be the set of rectangles of the form [t ij , t (i+1),j ]×[jR, (j +1)R] which are included in {x : d(x, ∂E ) > R}.

From the above, it follows that these rectangles in fact cover the set {x : d(x, ∂E ) > CR} for some C > 0 independent of R > 1, q. By construction each K ∈ K n is such that

K m 0 = m j (t (i+1),j ) -m j (t ij ) ∈ q 2 N.
The next lemma explains how to select a good subset of L p loc (R 2 , R 2 ). Lemma 7.6. Let P be a translation invariant measure on X such that P -a.e. j ∈ A 1 . Then for any ε > 0, there exist subsets

H ε ⊂ G ε in L p loc (R 2 , R 2
) which are compact and such that, for any R ε > 0, i) η 0 being given by Lemma 7.1 we have

(7.6) P (E × G ε c ) < min(η 0 2 , η 0 ε), P (E × H c ε ) < min(η 0 , ε).
ii) For every j ∈ H ε , there exists Γ(j) ⊂ K mRε such that

(7.7) |Γ(j)| < CR ε 2 η 0 and λ / ∈ Γ(j) =⇒ θ λ j ∈ G ε .
iii) The convergence in the definition of W K (j) is uniform w.r.t. (x, j) ∈ G ε and, writing curl j = 2π(ν -1), (7.8) W K (j) and ν(K R ) R 2 are bounded uniformly w.r.t. (x, j) ∈ G ε and R > 1.

iv) We have (7.9) d B (P, P ) < Cε(|log ε| + 1), where

P = E×Hε 1 m 0 (x)|K Rε | √ m 0 (x)K Rε \Γ(j) δ x ⊗ δ σ m 0 (x) θµj dµ d P (x, j).
Moreover, there exists a partition of H ε into ∪ Nε i=1 H i ε satisfying diam (H i ε ) < η 3 , where η 3 is such that

(7.10) j ∈ H ε , d p (j, j ) < η 3 , m ∈ [m, m], µ ∈ √ mK Rε \Γ(j) =⇒ d B (δ σmθµj , δ σmθµj ) < ε;
and there exists for all i, J i ∈ H i ε such that (7.11)

W K (J i ) < inf H i ε W K + ε.
Proof.

Step 1: Choice of G ε . Since L p loc (R 2 , R 2 ) is Polish we can always find a compact set G ε satisfying (7.6) and P (G ε c ) < η 0 . Then from Lemma 7.1, P G ε (the restriction of P to G ε ) satisfies d B (P, P G ε ) < ε.

From the translation invariance of P and for any λ, we have P (E × θ λ G ε ) > 1 -η 0 and therefore d B ( P , P θ λ G ε ) < ε. In view of (7.1), it follows that for any λ ∈ R 2 we have P -P λ < η 0 and then d B (P, P λ ) < ε, where

P λ = E×θ λ Gε δ x ⊗ δ σ m 0 (x) j d P (x, j) = E×Gε δ x ⊗ δ θ λ σ m 0 (x) j d P (x, j).
Then using Lemma 7.3 we deduce that if A ⊂ R 2 is any measurable set of positive measure, then (7.12) d B (P, P ) < Cε(|log ε| + 1), where

P = E×Gε - A δ x ⊗ δ θ λ σ m 0 (x) j dλ d P (x, j).
Moreover, since P is T λ(x) -invariant, choosing χ to be a smooth positive function with integral 1 supported in B(0, 1), the ergodic theorem (as in Section 6.1 or see again [Be]) ensures that for P -almost every (x, j) the limit

lim R→+∞ 1 |K R | K R W (j(λ + •), χ(λ + •)) dλ
exists. Then 1 K R * χ is a family of functions which satisfies (1.29) with respect to the family of squares {K R } R , and from the definition of the renormalized energy relative to {K R } R we may rewrite the limit above as (7.13)

W K (j) = lim R→+∞ 1 |K R | W (j, 1 K R * χ).
By Egoroff's theorem we may choose the compact set G ε above to be such that, in addition to (7.12), the convergence in (7.13) is uniform on G ε . In fact, since W K (j) < +∞ and lim sup R ν(K R )/R 2 < +∞ for P -a.e. (x, j), where curl j = 2π(ν -1), we may choose G ε such that (7.8) holds.

The arguments above show that the properties (7.12), (7.8) can be satisfied for a compact set G ε of measure arbitrarily close to 1. We choose G ε such that (7.6) holds.

The next difficulty we have to face is that θ λ j need not belong to G ε if j does.

Step 2: Choice of H ε . For j ∈ G ε , let Γ(j) be the set of λ's in √ mK Rε such that θ λ j ∈ G ε . Since, from (7.6) and the translation-invariance of P , for any λ ∈ R 2 we have

P (E×θ λ (G ε ) c ) < η 0 2 , it follows from Fubini's theorem that Gε |Γ(j)| d P (x, j) = √ mK Rε P (E × (θ λ G ε ) c ) dλ < 4mR ε 2 min(η 0 2 , η 0 ε).
Therefore, letting (7.14)

H ε = {j ∈ G ε : |Γ(j)| < 4mR ε 2 η 0 },
we have that (7.6) holds.

Combining (7.6) and (7.14) with Lemma 7.1, we deduce from (7.12) that (7.9) holds, where we have used the fact that θ λ σ m j = σ m θ √ mλ j to change the integration variable to µ = m 0 (x)λ in (7.12).

Step 3: Choice of J 1 , . . . , J Nε . We use the fact that G ε is relatively compact in L p loc (R 2 , R 2 ), Lemma 7.2, and the fact that (m, j) → σ m j is continuous to find that there exists η 4 > 0 such that for any m ∈ [m, m] and any j ∈ G ε it holds that (7.15)

d p (j, j ) < η 4 =⇒ d B δ σmj , δ σmj < ε.
Moreover, from the continuity of (µ, j) → θ µ j, there exists η 3 > 0 such that

(7.16) j ∈ G ε , d p (j, j ) < η 3 , µ ∈ √ mK Rε =⇒ d p θ µ j, θ µ j < η 4 .
If j ∈ H ε and µ ∈ K\... then θ µ j ∈ G ε hence applying (7.15), we get (7.10). Now we cover the relatively compact H ε by a finite number of balls B 1 , . . . , B Nε of radius η 3 /2, derive from it a partition of H ε by sets with diameter less than η 3 , by letting

H 1 ε = B 1 ∩ H ε and H i+1 ε = B i+1 ∩ H ε \ (B 1 ∪ • • • ∪ B i ) .
we then have (7.17)

H ε = Nε i=1 H i ε , diam (H i ε ) < η 3 ,
where the union is disjoint. Then we may choose J i ∈ H i ε such that (7.11) holds.

Completing the construction

Step 1: Choice of R ε . We apply Proposition 7.4 with G = G ε and √ mR, where m ∈ [m, m]. The proposition yields η 2 > 0, R ε > 1 such that for any j ∈ G ε and any m ∈ [m, m] and any rectangle L centered at 0 with sidelengths in [2 √ mR ε , 2 √ mR ε (1 + η 2 )], (7.4) is satisfied for some j L , with R replaced by √ mR ε . The reason for including √ m is that we will need to scale the construction to account for the varying weight m 0 (x).

Since our rectangles will be obtained from Lemma 7.5 and we wish to use the approximation by j L in them, we choose R ε large enough so that if m ∈ [m, m] and L is a rectangle centered at zero with sidelengths in [2

√ mR ε , 2 √ mR ε (1 + C 0 /R ε 2 )] then (7.18) C 0 R 2 ε < η 2 , C 1 R ε 2 < η 0 , K √ mRε(1-η 0 ) ⊂ {x : d(x, L c ) > √ mR ε 3 4 } ⊂ K √ mRε(1+η 0 ) ,
where C 0 is the constant in Lemma 7.5, C 1 ≥ 1 is to be determined later, and η 0 is the constant in Lemma 7.1.

If λ ∈ K √ mRε(1-η 0 ) and since j = j L if d(x, L c ) > √ mR ε 3 4
, we deduce from (7.18) that

θ λ j L = θ λ j in B(0, √ mR ε 3 4 ), so that from the definition of d p , taking R ε larger if necessary, (7.19) ∀j ∈ G ε , m ∈ [m, m], λ ∈ K √ mRε(1-η 0 ) , d p (θ λ σ m j, θ λ σ m j L ) < η 1 2 ,
where η 1 comes from Lemma 7.2 applied on {σ m j :

m ∈ [m, m], j ∈ G ε }, i.e. is such that (7.20) m ∈ [m, m], j ∈ G ε , j ∈ L p loc (R 2 , R 2 ) and d p (j, j ) < η 1 =⇒ d B (δ σmj , δ σmj ) < ε.
Step 2: Choice of q ε and the rectangles. We choose an integer q ε large enough so that (7.21)

N ε C 1 q ε 2 < η 0 , N ε q ε 2 × max 0≤i≤Nε m≤m≤m W K (σ m J i ) < ε
where C 1 > 1 is to be determined later. We apply Lemma 7.5 with R ε , q ε and N ε to obtain for any n a collection K n of rectangles (we omit to mention the ε dependence) which cover most of E , and we also apply Lemma 7.6. We rewrite P given by (7.9) as

(7.22) P = K∈Kn K √ n ×Hε 1 m 0 (x)|K Rε | √ m 0 (x)K Rε \Γ(j) δ x ⊗ δ σ m 0 (x) θµj dµ d P (x, j). Now we claim that if n is large enough and x ∈ K/ √ n, j ∈ H i ε , µ ∈ m 0 (x)K Rε \ Γ(j), then (7.23) d B δ x ⊗ δ σ m 0 (x) θµj , δ x K ⊗ δ σm K θµJ i < 2ε, where x K is the center of K/ √ n and m K is the average of m 0 over K/ √ m. Indeed, since m 0 is C 1 we have |x -x K | < C/ √ n, |m 0 (x) -m K | < C/ √ n thus if n is large enough, since θ µ j ∈ G ε we find d B δ x ⊗ δ σ m 0 (x) θµj , δ x K ⊗ δ σm K θµj < ε.
Moreover, since d p (j, J i ) < η 3 , we deduce from (7.10) that

d B δ x K ⊗ δ σm K θµj , δ x K ⊗ δ σm K θµJ i < ε,
which together with the previous estimate proves (7.23).

Using (7.23) together with Lemmas 7.2, 7.1, and (7.7), we deduce from (7.9) and (7.22) that d B (P, P ) < Cε(|log ε| + 1), where

P = K∈Kn 1≤i≤Nε K √ n ×H i ε -√ m K K Rε δ x K ⊗ δ σm K θµJ i dµ d P (x, j) = K∈Kn 1≤i≤Nε p i,K -√ m K K Rε δ x K ⊗ δ σm K θµJ i dµ, (7.24)
where (7.25)

p i,K = P K √ n × H i ε .
Step 3: Choice of subrectangles and current j n . We now replace p i,K in the definition (7.24) by

(7.26) |K| q ε 2 |E | n i,K , where n i,K = q ε 2 |E | |K| p i,K .
We have, since

P ( K √ n × L p loc (R 2 , R 2 )) = |K|/|E |, (7.27) Nε k=1 n i,K ≤ q ε 2 |E | |K| P K √ n × L p loc (R 2 , R 2 ) = q ε 2 and |K Rε| |E | n i,K -p i,K < C |K| q ε 2 |E | + n i,K R 2 ε |E | .
Summing with respect to i and K, using the facts that K∈Kn |K| < |E |, (7.27), and the fact that the cardinal of

K n is |E | 4q 2 ε R 2 ε , we find 1≤i≤Nε,K∈Kn |K Rε| |E | n i,K -p i,K < C N ε q ε 2 + 1 R 4 ε .
We may always choose C 1 large enough in (7.18) and (7.21) so that the right-hand side is < η 0 . Then Lemma 7.1 implies that d B (P, P

P (4) = 1 |E | K∈Kn 1≤i≤Nε n i,K m K √ m K K Rε δ x K ⊗ δ σm K θµJ i dµ.
Next, we divide each K ∈ K n into a collection L K of q ε 2 identical subrectangles in the obvious way and we partition L K into collections L K,i , 0 ≤ i ≤ N ε such that if k ≥ 1 then L K,i contains n i,K subrectangles. This is clearly possible from (7.27). If the inequality is strict we put the extra subrectangles in L K,0 , there will be n 0,K of them and then (7.29)

Nε k=0 n k,K = q ε 2 .
We rewrite (7.28) as (7.30)

P (4) = 1 |E | K∈Kn 1≤i≤Nε L∈L K,i 1 m K √ m K K Rε δ x K ⊗ δ σm K θµJ i dµ. Now, for L ∈ L K,i , let L = √ m K ( L -x L)
, where x L denotes the center of L. From Lemma 7.5, a rectangle K ∈ K n has sidelengths between 2q ε R ε and 2q ε R ε (1+C 0 /R ε 2 ). Therefore L is a rectangle centered at zero with sidelengths between 2 and (7.19) holds. This, and the results of Lemma 7.6, allow us to apply Proposition 7.4 on L to any J i , 1 ≤ i ≤ N ε . Note that |L| ∈ N follows from the fact that

√ m K R ε and 2 √ m K R ε (1 + C 0 /R ε 2 ),
|L| = m K | L| = - K m 0 |K| q ε 2 = 1 q ε 2 K m 0
and (7.5). In this way, we define currents J i,L which satisfy (7.4) and (7.19). We claim that, as a consequence of the latter, we have (7.31)

j = J i,L on L =⇒ d B -√ m K K Rε δ x K ⊗ δ σm K θµJ i dµ, 1 m K |K Rε | L δ x K ⊗ δ σm K θµj dµ < Cε.
This goes as follows: (i) Using Lemma 7.1 and (7.7),(7.18), we find that integrating on 4) ) < Cε(|log ε| + 1), using Lemma 7.3 we find d B (P, P (5) ) < Cε(|log ε| + 1), where (7.32)

√ m K K (1-η 0 )Rε \ Γ(J i ) instead of √ m K K Rε and
P (5) = 1 |E | K∈Kn 1≤i≤Nε L∈L K,i 1 m K L δ x K ⊗ δ σm K θµ Ji,L dµ = 1 |E | K∈Kn 1≤i≤Nε L∈L K,i L/ √ m K δ x K ⊗ δ θ λ σm K Ji,L dλ,
where the last equality follows by changing variables to λ = µ/ √ m K , and where Ji,L denotes an arbitrarily chosen element of L p loc (R 2 , R 2 ) such that Ji,L = J i,L on L, the constant C being independent of this choice.

If we choose an arbitrary J 0 in A 1 and let the sum in (7.32) range over 0 ≤ i ≤ N ε instead of 1 ≤ i ≤ N ε we obtain a measure P (6) such that, by (7.21),

P (5) -P (6) ≤ 1 |E | K∈Kn N ε |K| q ε 2 ≤ η 0 ,
hence using Lemma 7.1 we have d B (P (5) , P (6) ) < ε and then d B (P, P (6) ) < Cε(|log ε| + 1). We now define the vector field j int n : R 2 → R 2 by letting j

int n (x) = σ m K J i,L (x -x L) on L = x L + L/ √ m K , for every K ∈ K n , 0 ≤ i ≤ N ε and L ∈ L K,i . Then, for every L ∈ L K,i we have j int n (x L + •) = σ m K J i,L
on L, therefore we may choose Ji,L = σ 1/m K j int n (x L + •) in (7.32) and then then we may summarize the above by writing (7.33) d B (P, P (6) ) < Cε(|log ε| + 1),

P (6) = 1 |E | K∈Kn K δ x K ⊗ δ θ λ j int n dλ.
Note that since J i,L = 0 outside L, we also have (7.34)

j int n = K∈Kn 1≤i≤Nε L∈L K,i σ m K J i,L (• -x L), curl j int n = 2π K∈Kn p∈Λ K (δ p -m K ),
where Λ K is a finite subset of the interior of K. The second equation is satisfied in the sense of distributions on R 2 .

Step 4: Treating the boundary. Let Ê := E \ ∪ K∈Kn K. We let t ∈ [0, √ n] denote arclength on ∂E -where is the length of ∂E -and s denote the distance to ∂E , so that (t, s) same equation with B l replacing K and m l replacing m K . By elliptic regularity, we deduce for any q > 1 that ∇H

K L q (K) (resp. ∇H l L q (B l ) ) is bounded by C q,ε m 0 -m K L ∞ (K) (resp. C q,ε m 0 -m l L ∞ (B l ) .) Since m 0 is C 1 we have |∇m 0 | ≤ C/ √ n, therefore m 0 -m K L ∞ (K) ≤ CR ε / √ n, while m 0 -m l L ∞ (B l ) ≤ C. We deduce that (7.37) ∇H K L q ≤ C q,ε √ n , ∇H l L q ≤ C q,ε .
We let (7.38)

j K = j int n |K , and
(7.39) j n = j int n + K∈Kn -∇ ⊥ H K + kε i=1 -∇ ⊥ H l = K∈Kn j K -∇ ⊥ H K + kε l=1 j l -∇ ⊥ H l , Λ n = ∪ K∈K Λ K ∪ kε l=1 Λ l ,
where j K and H K are set to 0 outside K and similarly for j l , H l outside B l . Then curl j = 2π( p∈Λn δ p -m 0 ) in R 2 . This completes the construction of j n .

Estimating the energy

Step 1: Energy estimate. We have

W (j K , 1 K ) = 0≤i≤Nε L∈L K,i W (σ m K J i,L (• -x L), L).
From (7.4) we find, letting L = √ m K ( L -x L), using (7.29) and |L| = |K|/q ε 2 , that (7.40)

W (j K , 1 K ) = 0≤i≤Nε L∈L K,i W (σ m K J i,L , 1 L-x L ) ≤ |K| Nε i=0 n i,K q ε 2 W (σ m K J i ) + Cε .
We estimate the integral of |j n | 2 on R 2 \ ∪ p∈Λn B(p, η). From (7.39), this integral involves on the one hand the square terms (7.41)

kε l=1 (B i )η |j l -∇ ⊥ H l | 2 + K∈Kn Kη |j K -∇ ⊥ H K | 2 ,
where K η = K \ ∪ p∈Λn B(p, η) and similarly for (B l ) η , and on the other hand the rectangle terms

K,K ∈Kn K =K Kη∩K η (j K -∇ ⊥ H K ) • (j K -∇ ⊥ H K ) + 1≤l =i≤kε • • • + K∈Kn 1≤l≤kε
. . . We estimate the latter as follows: Since the rectangles in K n do not overlap, the first sum is equal to zero. A nonzero rectangle term must involve some B l , and moreover a given B l can only be present in a number of terms bounded independently of n, ε because the overlap of the balls B l and the rectangles K is bounded. Thus from (7.35) we have at most C √ n/R ε nonzero rectangle terms. Moreover, since the K l 's are disjoint, and disjoint from the K's, in a rectangle term involving B l ∩ K the integral can be taken over K \ K l , and in a term involving B l ∩ B i it can be taken over (B l ∩ B i \ K i ) ∪ (B i ∩ B l \ K l ).

In any case we use Hölder's inequality and the bound j l -∇ ⊥ H l L q (B l \K l ) ≤ C ε,q for some q > 2, which follows from (7.36), (7.37), together with the bound j l -∇ ⊥ H l L q (B l ) , j K -∇ ⊥ H K L q (K) ≤ C ε,q , which follows from (7.40), (7.36) using Lemma 4.7 in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], to conclude that each rectangle term is bounded by C ε and then that their sum is O( √ n), meaning a quantity bounded by a constant depending on ε times √ n. The limit as η → 0 of the terms in (7.41) is estimated as above by expanding the squares and using Hölder's inequality with (7.37), (7.36), (7.40), together with the bound (7.35) to show that

lim η→0 1 2 kε l=1 (B l )η |j l -∇ ⊥ H l | 2 + K∈Kn Kη |j K -∇ ⊥ H K | 2 + π#Λ n log η ≤ K∈Kn W (j K , 1 K ) + O( √ n).
In view of the bound O( √ n) for the rectangle terms and (7.40) we find using (7.29) that (7.42)

W (j n , 1 R 2 ) ≤ K∈Kn 0≤i≤Nε |K| n i,K q ε 2 W (σ m K J i ) + Cnε + O( √ n).
Step 2: We proceed to estimating W (j n , 1 R 2 ). We have, using (7.26), (7.25), (7.21), then the fact that m 0 -m K ≤ CR ε / √ n on K, then (7.11) with (1.33), then (7.11) and finally (7.1), that Nε i=1 (7.43) Here we have used the fact that W K is bounded below by some (negative) constant, a fact proved in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] that we use below several times.

|K|n i,K q ε 2 W K (σ m K J i ) ≤ |E | Nε i=1 P K √ n × H i ε W (σ m K J i ) + |K|ε ≤ |E | Nε i=1 K √ n ×H i ε W K (σ m 0 (x) J i ) d P (x, j) + |K| C √ n + ε ≤ |E | Nε i=1 K √ n ×H i ε W K (σ m 0 (x) j) d P (x, j) + |K| C √ n + Cε = |E | K √ n ×Hε W K (j) dP (x, j) + |K| C √ n + Cε .
We proceed by estimating n 0,K . From (7.26) we deduce that Nε i=1

(n i,K + 1) ≥ q ε 2 |E | |K| P K √ n × H ε ≥ q ε 2 |E | |K| |K| |E | -P K √ n × H ε c
, and then it follows from (7.29) that

n 0,K = q ε 2 - Nε i=1 n i,K ≤ N ε + q ε 2 |E | |K| P K √ n × H ε c .
Summing over K ∈ K n , using the fact that (7.44) |E \ ∪ Kn K| < C ε √ n and then (7.21), (7.6), we find that

K∈K |K| q ε 2 n 0,K W K (σ m K J 0 ) ≤ C|E | P (E × H ε c ) + 1 √ n + ε ≤ Cn C ε √ n + ε .
Summing (7.43) with respect to K ∈ K n and adding the above estimate we find, in view of (7.44), (7.42) and (7.6), that (7.45)

W (j n , 1 R 2 ) ≤ n|E| E×L p loc W K (j) dP (x, j) + Cn ε + C ε √ n .
Step 3: Energy bound for (x 1 , . . . , x n ). From (7.45), the constructed currents {j n } and points {Λ n } n satisfy curl j n = 2π p∈Λn δ p -m 0 in R 2 with #Λ n = n and (7.46) lim sup n W (j n , 1 R 2 ) n ≤ |E| W K (j) dP (x, j) + Cε. Now let {x i } i = {p/ √ n} p∈Λn be the points in Λ n in the initial scale, and let ν n = i δ x i . Since Λ n ⊂ E by construction, we have Supp(ν n ) ⊂ E and thus ζ dν = 0. Moreover, defining H n by (1.19), we have that -∆H n = curl j n and we may thus write j n = -∇ ⊥ H n -∇f n for some function f n . But j n = 0 outside of E, by construction, while H n decays fast at infinity by its definition (1.19) and the fact that the right-hand side of (1.19) has integral 0. Letting U η = ∪ p∈Λn B(p, η), we first have

B R \Uη |j n | 2 - B R \Uη |j n + ∇f n | 2 = -2 B R \Uη (j n + ∇f n ) • ∇f n + B R \Uη |∇f n | 2 .
Since j n ∈ L q loc for any q < 2 and since f n ∈ W 1,q loc (R 2 ) for all q, the last two terms on the right-hand side converge as η → 0 to the integrals over B R . Also integrating by parts, using the Jacobian structure and the decay of f n and j n , we have B R (j n + ∇f n ) • ∇f n = B R -∇ ⊥ H n • ∇f n → 0 as R → +∞. Therefore, letting η → 0 then R → +∞ in the above yields

W (j n , 1 R 2 ) -W (-∇ ⊥ H n , 1 R 2 ) ≥ |∇f n | 2 ≥ 0.
Together with (7.46), we deduce in view of (2.1) that (7.47) lim sup n→∞ 1 n w n (x 1 , . . . , x n ) -n 2 I(µ 0 ) + n 2 log n ≤ |E| π W K (j) dP (x, j) + Cε.

Step 4: Existence of A n . We claim that if n is large enough and if j ∈ L p loc (R 2 , R 2 ) is such that This would follow immediately from Lemmas 7.1, 7.2 and 7.3 if θ √ nx j int n belonged to some compact set independent of x / ∈ F and n. In our case we note that if x belongs to some L ∈ L K,i , where K ∈ K n and 0 ≤ i ≤ N ε , then

j int n (x + •) = σ m K J i,L (• + x -x L).
Moreover, since J i ∈ H ε , from (7.14) it follows that if x -x L / ∈ Γ(J i )/ √ m K then j := To prove the existence of the set A n , we note that the currents J i used in constructing j int n depend on ε but are independent on n. Then they are truncated to obtain J i,K where the sidelengths of L are in [R ε /C, CR ε ], i.e. in an interval independent of n. It follows at once that there exists δ > 0 such that the points in L may be perturbed by an amount δ so that for every i, K and L ∈ L K,i the perturbed J p i,L is at a distance at most η 1 /4 of J i,K , for every n. Then in view of (7.44) and (7.37) it follows that for n large enough the resulting j p n will satisfy (7.48) for x far enough from ∂E , i.e. outside a set of proportion relative to |E | tending to 0 as n → ∞. We deduce that j p n satisfies (7.49), hence if n is large enough d B (P j p n , P ) < Cε(|log ε| + 1). The same reasoning implies that if we let {x 1 , . . . , x n } be the points in Λ n in original coordinates, then perturbing the points in Λ n by an amount δ > 0 small enough, i.e. perturbing the x i 's by an amount δ/ √ n at most we obtain points y i such that w n (y i ) ≤ w n (x i ) + ε. Since the ordering of the points is irrelevant, we let S n denote the set of permutations of 1 . . . n and define

σ m K J i (• + x -x L) ∈ G ε . If
A n = {(y 1 , . . . , y n ) : ∃ σ ∈ S n |x i -y σ(j) | < δ.

Then, given η > 0, from the previous discussion and choosing ε > 0 small enough we have for any n and any (y 1 , . . . , y n ) ∈ A n that (4.1) is satisfied and the existence of j n such that curl j n = 2π i δ y i -m 0 and such that {P jn } n satisfies (4.2). This concludes the proof of Proposition 4.1.

  x 1 ,...,xn) dx 1 . . . dx n

  j+1)R y=jR m 0 (x, y) dy dx.

  )) < η 1 /2 for any x ∈ E \ F for some set F satisfying |F | < η 0 |E|, then (7.49) d B -E δ x ⊗ δ θ √ nx j dx, -E δ x ⊗ δ θ √ nx j intn dx < Cε(|log ε| + 1).

n

  dx < Cε(|log ε| + 1), where F is the union of F and of the union with respect to 0≤ i ≤ N ε , K ∈ K n and L ∈ L K,i of 1 √ n x L + Γ(J i )/ √ m K , of 1 √ n {x ∈ L : dist(x, ∂ L) ≤ η 0 R ε }, and of E \ ∪ Kn K √ n . It turns out that | F | < Cη 0 if n is large enough, Cbeing of course independent of ε, and thus using Lemma 7.1 we deduce (7.49). The claim is proved.

  4) ) < Cε(|log ε| + 1) is still true after replacing p i,K by

	|K Rε | |E | n i,K in (7.24), i.e. where
	(7.28)

  L induces an error of Cε. (ii)From (7.19), and (7.20) applied to θ µ J i and θ µ j we have d B (δ θµJ i , δ θµj ) < ε and thus in view of Lemma 7.3 we may replace θ µ J i by θ µ j in the integral with an error of Cε|log ε| at most. (iii) Using (7.18), (7.7) and Lemma 7.1 again, we may integrate back on √ m K K Rε and L rather than on K (1-η 0 )Rε \ Γ(J i ), with an additional error of Cε. this proves (7.31).Combining (7.31) with (7.30) and d B (P, P

  in addition, dist(x, ∂ L) > η 0 R ε , then we deduce from (7.19) thatd p j int n (x + •), j < η 1 /2 and d p (j(x + •), j ) < η 1 . Lemma 7.2 then yields d B (δ j int n (x+•) , δ j ) < ε and d B (δ j(x+•) , δ j ) < ε thus d B (δ j int n (x+•) , δ j(x+•) ) < 2ε.In view of Lemma 7.3 we find

	d B	1 |E| E\	F δ

x ⊗ δ θ √ nx j dx, 1 |E| E\ F δ x ⊗ δ θ √ nx j int

For photos one can see http://www.fys.uio.no/super/vortex/

The obstacle problem is a free-boundary problem and a much-studied classical problem in the calculus of variations, for general reference see[Fri, KS].

Note that this definition slightly differs from[START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]: Am here corresponds to A2πm there.

In fact R0 could be any positive constant, and then c, C would depend on R0 as well, but this requires to adjust ρ accordingly and we omit for simplicity to prove this fact.

Acknowledgements: We are grateful to Alexei Borodin, as well as Gérard Ben Arous, Amir Dembo, Percy Deift, and Alice Guionnet for their interest and helpful discussions. We thank Peter Forrester for useful comments and references. E. S. was supported by the Institut Universitaire de France and S.S. by a EURYI award.

is a C 1 coordinate system on {x ∈ E : d(x, (E ) c ) < c √ n}, if c > 0 is small enough, since the boundary of E is C 1 by (1.13). We let C t denote the curvilinear rectangle of points with coordinates in [0, t] × [0, CR ε ], where R ε = q ε R ε and C is large enough so that Ê ⊂ {x ∈ E : d(x, ∂E ) < CR ε }, and define m(t) = Ct∩ Ê m 0 . Since the distance of ∪ K∈Kn K to a given x ∈ ∂E is between R ε and C 0 R ε from Lemma 7.5 and since m 0 is bounded above and below by (1.15), the derivative of t → m(t) is between R ε /C and CR ε for some C > 0 large enough.

We let

and choose 0 = t 0 , . . . , t kε = √ n to be such that

We note that indeed t kε = √ n : Since the integral of m 0 on each square K ∈ K n is an integer as well as the integral on E , we have Ê m 0 ∈ N and therefore m(

From the above remark about the derivative of t → m(t), we deduce that m(

This means that the sidelengths of the curvilinear rectangle

and that the number of points n l to put there in is of order R ε 2 .

We may then include each of the sets K l := Ê ∩ (C t l+1 \ C t l ) in a ball B l with radius in [R ε /C, CR ε ] and we may also choose a set of n l points Λ l which are at distance at least 1/C from each other and the complement of K l . Let j l = -∇ ⊥ H, where H solves -∆H = 2π( p∈Λ l δ p -m l ) in B l and ∂ ν H • τ = 0 on ∂B l , where

Then we have curl j l = 2π( p∈Λ l δ p -m l ) in B l and j l • τ = 0 on ∂B l and we claim that for any q ≥ 1,

where the constants do not depend on n, but do depend on ε through R ε . This is proved by noting that these quantities are finite, and that a compactness argument shows that the bound is uniform for any choice of points which are at distance at least 1/C from each other and the complement of some K l ⊂ B l , using for instance the explicit formulas for W in [LR]. Note that because the sets {K l } and the rectangles {K} are disjoint, have measure between R ε 2 /C and CR ε 2 and diameter between R ε /C and CR ε , we know that their overlap is bounded by a constant C independent of ε, n.

Step 5: Rectification of the weight. We rectify the weights m K , m l : For K ∈ K n we let H K solve -∆H K = 2π(m 0 -m K ) on K and ∂ ν H K = 0 on ∂K. Similarly we let H l solve the