Alexei Borodin
email: borodin@math.mit.edu

Sylvia Serfaty
email: serfaty@ann.jussieu.fr

Renormalized energy concentration in random matrices

Keywords: renormalized energy, random point processes, random matrices, sine processes, Ginibre ensemble, zeros of Gaussian analytic functions. MSC classification: 60G55, 60G10, 15B52

. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number of charges. The functional is expected to be a good measure of disorder of a configuration of points. We give certain formulas for its expectation for general stationary random point processes. For the random matrix β-sine processes on the real line (β = 1, 2, 4), and Ginibre point process and zeros of Gaussian analytic functions process in the plane, we compute the expectation explicitly. Moreover, we prove that for these processes the variance of the renormalized energy vanishes, which shows concentration near the expected value. We also prove that the β = 2 sine process minimizes the renormalized energy in the class of determinantal point processes with translation invariant correlation kernels.

Introduction

The aim of this paper is to introduce and compute a function, called the "renormalized energy", for some specific random point processes that arise in random matrix models, and in this way to associate to each of these processes a unique number, which is expected to measure its "disorder".

Our "renormalized energy", that we denote W, is defined over configurations of points lying either on the real line or on the plane, as the limit as N → ∞ of

W N ({a i }) = - 1 N i =j,a i ,a j ∈[0,N ] log 2 sin π(a i -a j ) N + log N in dimension 1, W N ({a i }) = 1 2πN 2 i =j,a i ,a j ∈[0,N ] 2 E N (a i -a j ) + log N 2πη(i) 2 in dimension 2,
where E N is an explicit Eisenstein series, and η is the Dedekind Eta function. This definition is inspired by that of the "renormalized energy", denoted W , introduced by Sandier and the second author in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] in the case of points in the plane and in [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] in the case of points on the real line. The definitions for W and W coincide when the point configuration has some periodicity (this is where our new definition originates), and in that case they amount to computing a sum of pairwise interactions

i =j G(a i -a j )
where a i are the points and G is a suitable logarithmic kernel (the Green's function on the underlying torus); however they are not identical in general. We will give more details on the connection in Section 2.

In [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] it was shown that in dimension 1, W is bounded below and its minimum is achieved at the perfect lattice Z. In dimension 2, the situation is more complex; it is also shown in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] that the minimum of W is achieved, but it is only conjectured that this minimum value is achieved at the perfect triangular lattice or "Abrikosov lattice" according to the terminology of the physics of superconductors (which was the first main motivation for W in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] where it was introduced). This conjecture is supported by the result that, among configurations of points which form a perfect lattice (of fixed volume), the renormalized energy is minimal if and only if the lattice is the perfect triangular lattice, i.e. with 60 • angles (that result is shown in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] based on the use of modular functions and results in number theory).

It is thus natural to think of W or W as a way to measure the disorder of a configuration of points. With this outlook, in dimension 1 the lattice Z is the most ordered configuration of points with prescribed density 1, while in dimension 2, it is expected to be the triangular lattice (which is better ordered than any other lattice, say the square lattice). In addition, due to its logarithmic nature, W has some nice scaling and additive properties, which we believe make it a very good object. A further motivation for choosing to study W as opposed to any other total pairwise interaction function is that, as seen in [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF] and [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF], W arises very naturally from the statistical mechanics of Coulomb or log gases, which contain as particular cases the Ginibre and GOE/GUE/GSE ensembles of random matrices. In [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], W was introduced and derived in the context of the minimization of the Ginzburg-Landau model of superconductivity. In [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF] it was derived as a sort of limiting interaction energy for two dimensional Coulomb gases, and similarly in [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] with log gases. These works were based on analysis and energy estimates (upper and lower bounds). Both the questions we pursue here and the methods we use are quite different: they aim at obtaining explicit formulas for specific random matrix models. In particular, this is a way to compute some interesting statistics over random matrix eigenvalues, as initiated by Dyson-Mehta [DM]. We will comment more on this just below.

Let us now briefly introduce the notion of a random point process. A (simple) random point process is a probability measure on the set of all locally finite collections of (mutually distinct) points in the space, cf. e.g. [DVJ]. It can also be viewed as a random measure of the form ξ(ω) = p∈Λ δ p , with the points p distinct and Λ discrete.

Random point processes are essentially characterized by their "k-point correlation functions" ρ k (x 1 , . . . , x k ), which give the probability densities of finding k points at the locations x 1 , . . . , x k . We will normalize our processes so that the average number of points per unit volume is always 1, which is equivalent to ρ 1 (x) ≡ 1.

Perhaps the most famous random point process is the Poisson process, characterized by the facts that the number of points in disjoint subsets are independent, and the number of points in any finite volume subset of the space follows a Poisson distribution with parameter equal to the volume of the set with respect to a reference measure. An important class of point processes is that of determinantal point processes, see [START_REF] Soshnikov | Determinantal random point fields[END_REF], [Ly], [Jo], [START_REF] König | Orthogonal polynomial ensembles in probability theory[END_REF], [HKPV], [START_REF] Soshnikov | Determinantal Random Fields[END_REF], [B] and references therein. That class is characterized by the fact that the k-point correlation functions are given by symmetric minors of a (correlation) kernel, cf. Section 3.1.

It is easy to see that lim N →∞ W N = +∞ for the translation invariant Poisson process, which means that it is "too chaotic" from the point of view of the renormalized energy. Let us list the (stationary) processes for which we show that W provides more meaningful information.

We will be interested in one dimension in the β-sine processes (β = 1, 2, 4) which arise as the local limit of the law of eigenvalues in random matrix ensembles with orthogonal, unitary, and symplectic symmetry groups (they are determinantal for β = 2 and Pfaffian otherwise). In two dimensions we will examine the "Ginibre" point process, which is also a determinantal process arising as the local limit of the law of eigenvalues of matrices from the complex Ginibre ensemble (i.e. square matrices with complex Gaussian iid entries), for further reference, see [Fo, AGZ, Me]; as well as the random process of zeros of random Gaussian analytic functions, often denoted GAF, whose description can be found in [HKPV].

As our processes are always translation invariant, the 2-point correlation function can always be written in the form ρ 2 (x, y) = 1 -T 2 (x -y) for the 2-point cluster function T 2 (we will come back to this notation in Sections 3.1 and 5).

The main results we obtain are the following:

• For general stationary processes we identify sufficient conditions on the process and its 2-point correlation function ρ 2 for the existence of lim N →∞ EW N , and give an explicit formula in terms of ρ 2 which is (up to constants) lim

N →∞ EW N = R d log |x|T 2 (x) dx,
with d = 1 or 2 according to the dimension.

• We apply this formula to the specific point processes mentioned above.

• For the specific point processes above, we explicitly compute the limit of the variance of W N and obtain that it is 0. This implies that for such processes, W N concentrates around its expected value, and converges in probability to lim EW N as N → ∞.

• We prove that in the class of determinantal point processes with translation invariant kernels in dimensions 1 and 2, lim N →∞ EW N is minimized by the processes whose correlation kernel is the Fourier transform of the characteristic function of the ball. In dimension 1 this gives the β = 2 sine process, which can be seen as a heuristic explanation of the fact that this process is the universal local limit of β = 2 random matrix ensembles. Indeed, such a local limit has to be translation invariant and determinantal, and it is natural to expect that it also minimizes an appropriate energy functional.

For our set of specific processes, we thus show that we can attribute to the process a unique number, which we compute explicitly. Whenever these numbers are distinct this implies that the processes are mutually singular (as measures on the space of point configurations). Moreover, we check that the processes that are expected to have the highest level of "order" or rigidity indeed have a lower value of lim N →∞ EW N . For example for the β-sine processes we find lim

N →∞ EW N = 2 -γ -log 2 for β = 1 lim N →∞ EW N = 1 -γ for β = 2 lim N →∞ EW N = 3 2 -log 2 -γ for β = 4,
where γ is the Euler constant. These three numbers form a decreasing sequence, as can be expected.

In two dimensions, we obtain that the Ginibre process has a higher lim N →∞ W N , hence less rigidity, than that of zeros of Gaussian analytic functions, in agreement with the results and philosophy of [GNPS].

The values of lim EW N for the β = 1, 2, 4 sine processes above coincide with twice the thermodynamic "energy per particle" or "mean internal energy" for the log-gas with infinitely many particles, as obtained by Dyson in 1962 [Dy, I.IX, III.VI]. This coincidence is not hard to explain (we are very grateful to Peter Forrester for the idea): If, instead of considering growing windows, one approximates the β-sine processes by the circular ensembles of particles on the unit circle with joint probability density i<j |z i -z j | β , one observes that in rescaled variables, W N is simply -2N -1 i =j log |z i -z j | + log N , and its characteristic function can be immediately obtained from Selberg's formula for the partition function:

E exp(it W circular N ) = N it Z(β -2it N ) Z(β) , Z(β) = Γ(1 + βN 2 ) (Γ(1 + β 2 )) N
, see e.g. [START_REF] Forrester | Log-gases and random matrices[END_REF]Section 4.7.2] for the formula for Z(β). Using Stirling's formula it is not hard to see that for any β > 0 and t ∈ R

lim N →∞ E exp itN 1/2 W circular N -u(β) = exp - v(β)t 2 2 with u(β) = Ψ 1 + β 2 -log β 2 , v(β) = 2 β -Ψ ′ 1 + β 2 , Ψ(z) = Γ ′ (z) Γ(z) ,
and the expression of u(β)/2 through the partition function Z(β) coincides with that of the thermodynamic energy per particle. By Lévy's continuity theorem this implies the central limit theorem lim

N →∞ Prob W circular N -u(β) v(β)/N ≤ s = 1 √ 2π s -∞ e -x 2 /2 dx, s ∈ R.
It is natural to conjecture that the same central limit theorem holds for W N on general β-sine processes (constructed in [VV]) as well, in particular providing asymptotic values lim EW N = u(β) and lim N • Var(W N ) = v(β) for any β > 0. However, at the moment we do not know how to prove it.

In the follow-up paper [DM], prompted by the wish to analyze experimental data, Dyson and Mehta looked for a way of approximating the energy per particle, in the case β = 1, by averaging a statistic of the form F (a i , a j ) over windows of increasing length in the random matrix spectrum. The concentration requirement of the asymptotically vanishing variance led them to the conclusion that with their choice of F , the statistic had to be corrected with further interaction terms [START_REF] Dyson | Statistical theory of the energy levels of complex systems, Part IV[END_REF]Eq. (109)]. Our renormalized energy W N seems to be a better solution to Dyson-Mehta's problem.

The paper is organized as follows: In Section 2 we give the precise definitions of W , the context about it and its connection to Coulomb energy, from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF][START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF]. This leads us to the definition of W N . In Section 3 we compute limits of expectations of W N : First on the real line and for general processes, then for the β = 1, 2, 4 sine processes, then in the plane and for general processes, finally for the explicit Ginibre and zeros of GAF processes. In Section 4 we find minimizers of lim W N among determinantal processes with translation-invariant correlation kernels. In Section 5 we compute the limit variance of W N for our specific processes (and show it is 0). In Section 6 we gather some miscellaneous computations: the effect on lim EW N of superposition and decimation of processes and the computation of expectation of lim EW N for the β = 2 discrete sine processes.

Definitions of W and W N

The aim of this section is to present a definition of W for a configuration of points either on the line or in the plane, which is directly deduced from that of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF][START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF], but depends only on the data of the points. We do not attempt here to fully and rigorously connect the next definition with that of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF][START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] (since we believe it presents some serious technical difficulties) but the link will be readily apparent.

We start by recalling the definitions from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], but rather in the form presented in [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF].

Definition of W (j) in the plane

In [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], W was associated to a discrete set of points (with asymptotic density) m in the plane, via a vector field j: if j is a vector field in R 2 satisfying

(2.1) curl j = 2π(ν -m), div j = 0,
where m is a positive number and ν has the form

(2.2) ν = p∈Λ δ p for some discrete set Λ ⊂ R 2 ,
then for any function χ we define

(2.3) 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 2.1. Let m be a nonnegative number. Let j be a vector field in R 2 . We say j belongs to the admissible class A m if (2.1), (2.2) hold and

(2.4) ν(B R ) |B R | is bounded by a constant independent of R > 1.
Definition 2.2. The "renormalized energy" W (j) relative to the family of balls (centered at the origin) {B R } R>0 (and the number m) in R 2 is defined for j ∈ A m by

(2.5) W (j) := lim sup R→∞ W (j, χ R ) |B R | ,
where χ R denotes positive cutoff functions satisfying, for some constant C independent of R,

(2.6) |∇χ R | ≤ C, Supp(χ R ) ⊂ B R , χ R (x) = 1 if d(x, (B R ) c ) ≥ 1.
Note that by scaling, the density of points can be changed to any, and the rule for that change of scales is

(2.7) W (j) = m W (j ′ ) - π 2 log m ,
where ν has density m and j ′ = 1 √ m j( • √ m ) (hence the set ν ′ has density 1). This function was first introduced in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] and derived as a limiting interaction energy for vortices of Ginzburg-Landau configurations. Independently of Ginzburg-Landau it can be viewed as a Coulombian interaction energy for an infinite number of points in the plane, computed via a renormalization. Many of its properties are stated in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]. In [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF] it was directly connected to 2D log gases. We will give more details and properties of W in Section 2.3.

Definition of W (j) on the line

In [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] a one-dimensional analogue to the above definition was introduced for the study of 1D log gases, we now present it. The renormalized energy between points in 1D is obtained by "embedding" the real line in the plane and computing the renormalized energy in the plane, as defined in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]. More specifically, we introduce the following definitions:

R denotes the set of real numbers, but also the real line of the plane R 2 i.e. points of the form (x, 0) ∈ R 2 . For the sake of clarity, we denote points in R by the letter x and points in the plane by z = (x, y). For a function χ on R, we define its natural extension χ to a function on R 2 by χ(x, y) = χ(x). I R denotes the interval [-R/2, R/2] in R.

δ R denotes the measure of length on R seen as embedded in R 2 , that is

R 2 ϕδ R = R ϕ(x, 0) dx
for any smooth compactly supported test function ϕ in R 2 . This measure can be multiplied by bounded functions on the real-line.

Definition 2.3. For any function χ on R, and any function h in R 2 such that

(2.8) -∆h = 2π(ν -mδ R ) in R 2
where m is a nonnegative number and ν has the form (2.9) ν = p∈Λ δ p for some discrete set of points of R, we denote j = -∇ ⊥ h and

(2.10)

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)   ,
where χ is the natural extension of χ.

Definition 2.4. Let m be a nonnegative number. We say j = -∇ ⊥ h belongs to the admissible class A m if (2.8), (2.9) hold and

(2.11)

ν(I R ) R is bounded by a constant independent of R.
We use the notation χ R for positive cutoff functions over R satisfying, for some constant

C independent of R, (2.12) |∇χ R | ≤ C, Supp(χ R ) ⊂ I R , χ R (x) = 1 if |x| < R 2 -1.
Definition 2.5. The renormalized energy W is defined, for j ∈ A m , by

(2.13) W (j) := lim sup R→∞ W (j, χ R ) R .
Note that while W in 2D can be viewed as a renormalized way of computing j H 1 (R 2 ) , in 1D it amounts rather to a renormalized computation of j H 1/2 (R) .

In one dimension, the formula for change of scales is

(2.14) W (j) = m W (j ′ ) -π log m .
where j corresponds to density m and j ′ to m = 1.

Background

We recall here some properties and background from [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF][START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF].

-Since in the neighborhood of p ∈ Λ we have curl j = 2πδ p -m(x), div j = 0, we have near p the decomposition j(x) = ∇ ⊥ log |x -p| + f (x) where f is locally bounded, and it easily follows that the limits (2.3), (2.10) exists. 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, and thus W does not feel compact perturbations of the configuration of points. Even though the interactions are long-range, this is not difficult to justify rigorously.

-The cut-off function χ R cannot simply be replaced by the characteristic function of B R because for every p ∈ Λ lim

R→|p| R<|p| W (j, 1 B R ) = +∞, lim R→|p| R>|p| W (j, 1 B R ) = -∞.
-It would be tempting to define W as a function of ν only and not j by minimizing over j, however we do not know how to prove that the resulting object has good properties such as measurability.

An important question is that of characterizing the minimum and minimizers of W . For the case of dimension 1, it is proven in [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] that the value of W does not depend on the choice of χ R satisfying (2.12), that W is Borel-measurable over L p loc (R 2 , R 2 ) for p < 2, and that (2.15) min

A 1 W = -π log(2π)
is achieved for ν = p∈Z δ p .

For the case of dimension 2, it is proven in Theorem 1 of [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] that the value of W does not depend on {χ B R } R as long as it satisfies (2.6), that W is Borel-measurable over L p loc (R 2 , R 2 ), for p < 2 and that min Am W is achieved and finite. Moreover, there exists a minimizing sequence consisting of periodic vector fields. The value of the minimum of W is not known, it is conjectured (see [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]) that it is asymptotically equal to the value at the perfect triangular lattice. This conjecture is supported by the fact that the triangular lattice can be proved to be the minimizer among lattice configurations of fixed volume.

In addition to arising from the study of Ginzburg-Landau [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], W is naturally connected, as shown in [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF][START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF], to Vandermonde factors such as e -β 2 wn with (2.16)

w n (x 1 , . . . , x n ) = - i =j log |x i -x j | + n n i=1 V (x i ),
where x i , . . . , x n ∈ R d with d = 1 or 2, for any potential V (for example V quadratic). To explain this, let us introduce some more notation. We set (2.17)

I(µ) = R d ×R d -log |x -y| dµ(x) dµ(y) + R d V (x) dµ(x).
It is well known in potential theory (see [ST]) that, provided V (x)-log |x| → +∞ as |x| → ∞, I has a unique minimizer among probability measures, called the equilibrium measure -let us denote it µ 0 . It is characterized by the fact that there exists a constant c such that (2.18)

U µ 0 + V 2 = c
in the support of µ 0 , and U µ 0 + V 2 ≥ c everywhere where for any µ, U µ is the potential generated by µ, defined by

(2.19) U µ (x) = - R d log |x -y| dµ(y).
We may then set ζ = U µ 0 + V 2 -c where c is the constant in (2.18). It satisfies

(2.20) ζ = 0 in Supp(µ 0 ) ζ > 0 in R d \ Supp(µ 0 )
It is easy to check that ζ grows like V at infinity. The connection between w n and W is given by the following exact "splitting formula", valid in both 1 and 2 dimensions.

Lemma 2.6 ( [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF][START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF]). For any x 1 , . . . , x n ∈ R d , d = 1 or d = 2, the following holds

(2.21) w n (x 1 , . . . , x n ) = n 2 I(µ 0 ) - n d log n + 1 π W (-∇ ⊥ H, 1 R d ) + 2n n i=1 ζ(x i )
where W is defined in (2.3) or (2.10) respectively, and where

(2.22) H = -2π∆ -1 n i=1 δ x i -nµ 0 in R 2 , H ′ (n 1/d x) = H(x).
where the equation is solved in R 2 and µ 0 is naturally extended into a measure on

R 2 if d = 1. (Note that ∆ -1 is the convolution with 1/(2π) log | • |.)
Sketch of the proof. We start by writing

w n (x 1 , . . . , x n ) = △ c -log |x -y| dν(x) dν(y) + n V (x) dν(x)
where △ denotes the diagonal of (R d ) 2 and ν = n i=1 δ x i . The idea is to compute the righthand side by splitting ν as nµ 0 + ν -nµ 0 . This way, 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,
where we used that ζ = 0 on the support of µ 0 . Thus

(2.23) w(x 1 , . . . , x n ) = n 2 I(µ 0 ) + 2n ζ dν + △ c -log |x -y| d(ν -nµ 0 )(x) d(ν -nµ 0 )(y).
We then claim that the last term in the right-hand side is equal to

1 π W (-∇ ⊥ H, 1 R d ). We define 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 . It is then not difficult to deduce (2.24) △ 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).
On the other hand, by definition (2.3), (2.10) and using Green's formula, we have

1 2π R 2 \∪ i B(x i ,η) |∇H| 2 = 1 2π i ∂B(x i ,η) H ∂H ∂ν + 1 2 R 2 \∪ i B(x i ,η) H d(ν -nµ 0 ).
Using the decomposition H = H i -log |x -x i | near each x i , adding n log η and then letting η → 0, we arrive at the same result as the right-hand side of (2.24). This establishes the claim, and the final result then follows from the change of scales

x ′ = n 1/d x in W .

Calculation for points on a torus in dimension two

The reason why we need another definition here is that, as we already mentioned, we wish to define W based on the knowledge of ν, i.e. a set of points, alone. For a given ν, there is no uniqueness of the j satisfying (2.1) or (2.8) (in fact the indetermination is exactly a constant, see [START_REF] Sandier | 2D Coulomb gases and the renormalized energy[END_REF]Lemma 1.4]), and this makes it problematic to define W for points only in a measurable manner. However, if the configuration of points is periodic, then W can be computed nonambiguously from ν only.

The formula for this in dimension 2 is given in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF]. If the periodicity is with respect to a torus T, and the number of points in that torus is n (denote them a 1 , • • • , a n ) then there exists a unique (up to a constant) solution to

(2.25) -∆H {a i } = 2π n i=1 δ a i - n |T| on T (i.
e. a periodic solution) and we let

(2.26)

j {a i } = -∇ ⊥ H {a i } .
It was proved in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] that W (j {a i } ) is the smallest of the W (j) over all j satisfying (2.1) which are T-periodic, and it was established (see formula (1.10)) that

W (j {a i } ) = 1 2 i =j G(a i -a j ) + nc
where c is a constant, and G is the Green's function associated to the torus in the following way:

(2.27)

-∆G = 2π δ 0 - 1 |T| in T.
In [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], (1.11), an explicit formula is given for this:

(2.28) W (j

{a i } ) = 1 2 i =j p∈(Z 2 ) * \{0} e 2iπp•(a i -a j ) 4π 2 |p| 2 + n 2 lim x→0   p∈(Z 2 ) * \{0} e 2iπp•x 4π 2 |p| 2 + log |x|   .
which can in turn be expressed using Eisenstein series.

For simplicity we prefer to work with density 1 in a square torus, which is then necessarily [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], this will change the constants in the formulae. So for the sake of clarity we will include below a complete proof of the following:

T N := R 2 /(N Z) 2 where n = N 2 . Compared to
Lemma 2.7. Let a 1 , • • • , a n be n = N 2
points on T N . Let j {a i } be the T N -periodic vector field associated through (2.26) to the configuration of points {a i }, extended by periodicity to the whole plane. Then W (j

{a i } ) as defined in (2.5) is a function of a 1 , • • • , a n only, which is equal to (2.29) W N (a 1 , • • • , a n ) = π N 2 i =j G(a i -a j ) + πR
where G is the unique solution of

(2.30) -∆G = 2π δ 0 - 1 N 2 in T N , with -T N G(x) dx = 0; and R is a constant, equal to lim x→0 (G(x) + log |x|).
Proof. First, arguing as in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] Proposition 3.1, we have

(2.31) W (j {a i } ) = 1 |T N | lim η→0 1 2 T N \∪ n i=1 B(a i ,η) |j {a i } | 2 + πn log η ,
which means we reduce the computation in the plane to a computation on the torus. The next step is a renormalized computation à la [BBH], as skeched in the proof of Lemma 2.6. Solving for (2.25) via the Green function (2.30), in view of the translation invariance of the equation, we can choose (2.32)

H {a i } (x) = n i=1 G(x -a i ).
Let us denote

(2.33) R(x) = G(x) + log |x|, it is well-known that R(x) is a continuous function. We denote R := R(0). Inserting (2.26) into (2.31), we have (2.34) W (j {a i } ) = 1 |T N | lim η→0 1 2 T N \∪ n i=1 B(a i ,η) |∇H {a i } | 2 + πn log η .
Using Green's formula for integration by parts, we have

T N \∪ n i=1 B(a i ,η) |∇H {a i } | 2 = T N \∪ n i=1 B(a i ,η) (-∆H {a i } )H {a i } + n i=1 ∂B(a i ,η) H {a i } ∂H {a i } ∂ν .
Inserting (2.25) and (2.32), we have

T N \∪ n i=1 B(a i ,η) |∇H {a i } | 2 = - n j=1 T N \∪ n i=1 B(a i ,η) 2πG(x -a j ) dx + n i=1 n j=1 ∂B(a i ,η) G(x -a j ) ∂H {a i } ∂ν ,
with ν the inner unit normal to the circles. Using for the first term the fact that G is chosen to have average zero, and for the second term splitting G(x -a j ) as -log |x -

a j | + R(x -a j ),
using the continuity of R(x) and lim η→0 ∂B(a i ,η)

∂H {a i } ∂ν = 2π (from (2.25))
, and letting η → 0 we find

lim η→0 1 2 T N \∪ n i=1 B(a i ,η) |∇H {a i } | 2 + πn log η = π n i=1 R(0) + π i =j (-log |a i -a j | + R(a i -a j )).
Inserting into (2.34) yields (2.29).

It turns out that G can be expressed through an Eisenstein series. The Eisenstein series with parameters τ ∈ C and u, v ∈ R (cf. [La]) is defined by

(2.35) E u,v (τ ) := (m,n)∈Z 2 \{0} e 2iπ(mu+nv) Im(τ ) |mτ + n| 2 .
Let us now define, for x ∈ C,

(2.36)

E N (x) := E Re(x/N ),Im(x/N ) (i).
As in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF] we will also need another classical modular function: the Dedekind η function. It is defined, for any τ ∈ C, η(τ ), by

(2.37) η(τ ) = q 1/24 ∞ k=1
(1 -q k ) where q = e 2iπτ .

The following holds

Proposition 2.8. Let a 1 , • • • , a n be n = N 2 points on T N , and let W N be defined as in (2.29). E N being defined in (2.36), we have

(2.38) W N (a 1 , • • • , a n ) = 1 2N 2 i =j E N (a i -a j ) + π log N 2π -2π log η(i),
where points in the plane are identified with complex numbers.

Proof. As in [START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF], G, introduced in Lemma 2.7, can be computed from (2.30) using Fourier series and this yields:

(2.39)

G(x) = p∈Z 2 \{0} e 2iπ N p•x 2π|p| 2 .
We recognize here an Eisenstein series:

(2.40)

G(x) = 1 2π E Re( x N ),Im( x N ) (i) = 1 2π E N (x).
Note that the fact that G has zero average implies that (2.41)

T N E N (x) dx = 0.
There remains to compute the value of the constant R = lim x→0 (G(x, 0) + log |x|). For that we use the "second Kronecker limit formula" (see [La]) that asserts that

E u,v (τ ) = -2π log |f (u -vτ, τ )q v 2 /2 |,
where q = e 2iπτ , p = e 2iπz , z = u -vτ , and

(2.42)

f (z, τ ) = q 1/12 (p 1/2 -p -1/2 ) k≥1 (1 -q k p)(1 -q k /p).
We use this formula with

τ = i, u = Re( x N ), v = Im( x N ). In that case q = e -2π , z = x N
and p = e 2iπ x N (where the bar denotes complex conjugation), and the formula yields

(2.43) E N (x) = -2π log f x N , i e -π(Im x N ) 2 .
As x → 0 we have p → 1 and

p 1/2 -p -1/2 ∼ 2iπ
x N and also (see (2.37))

q 1/12 k≥1 (1 -q k p)(1 -q k /p) ∼ x→0 q 1/12 k≥1 (1 -q k ) 2 = η(i) 2 .
So as x → 0, we have

E N (x) ∼ -2π log η(i) 2 2iπ x N = -2π log 2π|x| N -4π log η(i).
Combining with (2.40), it follows that R = -log 2π N -2 log η(i). Inserting this and (2.40) into (2.29) we get the result.

We emphasize here that this formula is the exact value for W as defined in (2.5) provided we assume full periodicity with respect to T N .

Calculation for points on a torus in dimension one

We do here the analogue in dimension 1, i.e. we compute W given by (2.13) assuming that the point configuration is periodic with respect to T N := R/(N Z). We assume that there are

n = N points a 1 , • • • , a N in T N ,
hence W is computed with respect to the average density 1, i.e. m = 2π. There exists a unique (up to a constant)

H {a i } satisfying -∆H {a i } = 2π n i=1 δ (a i ,0) -δ R
in R 2 and which is N -periodic with respect to the first variable, i.e H {a i } (x + N, y) = H {a i } (x, y) for every (x, y) ∈ R 2 . We then set j {a i } = -∇ ⊥ H {a i } . Then, as in [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF] we have

Lemma 2.9. Let a 1 , • • • , a N be N points on T N = R/(N Z). Let j {a i } be as above. Then W (j {a i } ) as defined in (2.13) is a function of a 1 , • • • , a N only, which is equal to (2.44) W N (a 1 , • • • , a N ) = π N i =j G(a i -a j , 0) + πR
where G(z) is a restriction of the Green function of T N × R, more precisely the solution to

(2.45) -∆ z G(z) = 2π δ (0,0) - 1 N δ T N z ∈ T N × R with -T N G(x, 0) dx = 0, and R is a constant, equal to lim x→0 (G(x, 0) + log |x|) .
Here δ T N is the distribution defined over

T N × R by δ T N φ = T N φ(x, 0) dx.
Proof. The proof is analogous to that of Lemma 2.7. First we have

(2.46) W (j {a i } ) = 1 |T N | lim η→0 1 2 (T N ×R)\∪ N i=1 B((a i ,0),η) |∇H {a i } | 2 + πN log η .
We again write

(2.47)

H {a i } (z) = n i=1 G(z -(a i , 0)),
with G(z) = -log |z| + R(z), R a continuous function. Using Green's formula we have

(T N ×R)\∪ N i=1 B((a i ,0),η) |∇H {a i } | 2 = -2 (T N ×R)\∪ N i=1 B((a i ,0),η) H {a i } δ T N + N i=1 ∂B((a i ,0),η) H {a i } ∂H {a i } ∂ν .
From (2.47) we have lim

η→0 (T N ×R)\∪ N i=1 B((a i ,0),η) H {a i } δ T N = N j=1 T N G(x -a j , 0) dx = 0
by choice of G. The rest of the proof is exactly as in Lemma 2.7, inserting the splitting of G and (2.45).

On the other hand we shall see below that we have the following explicit formula for G restricted to the real axis:

(2.48) G(x, 0) = -log 2 sin πx N .
Note that a consequence of this and the zero average condition on G is that (2.49)

T N log 2 sin πv N dv = 0.
The previous lemma and (2.48) lead us to the following Proposition 2.10. Let a 1 , • • • , a N be N points on T N = R/(N Z) and let W N be defined as in (2.44). We have

(2.50) W N (a 1 , • • • , a N ) = - π N i =j log 2 sin π(a i -a j ) N -π log 2π N .
Proof. The first step is to prove (2.48). This is done by solving for (2.45) in Fourier series/transform. We choose the following normalization for Fourier transforms and series:

f (ξ) = f (x)e -2iπx•ξ dx c k (f ) = T N f (x)e -2iπkx N dx.
Then the Fourier inversion formula is f (x) = f (ξ)e 2iπx•ξ dξ and respectively

f (x) = 1 N k∈Z c k (f )e 2iπk N x .
Since G is T N -periodic in its first variable, we may define its Fourier transform as a Fourier series in the first variable and a Fourier transform in the second, i.e., for m ∈ Z and ξ ∈ R,

G(m, ξ) = x∈T N y∈R G(x, y)e -2iπmx N e -2iπyξ dx dy.
If G solves (2.45) then by Fourier transform, G has to satisfy

4π 2 m 2 N 2 + ξ 2 G(m, ξ) = 2π δ (0,0) - 2π N δ T N .
It is direct to establish that δ T N = N δ 0 m with δ 0 m by definition equal to 1 if m = 0 and 0 otherwise. Combining these facts, we obtain

G(m, ξ) = 1 -δ 0 m 2π m 2 N 2 + ξ 2 f or (m, ξ) = (0, 0).
The undetermination of G at (0, 0) corresponds to the fact that G is only determined by (2.45) up to a constant. By Fourier inversion, it follows that

G(x, y) = 1 N m∈Z ξ∈R 1 -δ 0 m 2π( m 2 N 2 + ξ 2 ) e 2iπ m N x+2iπyξ dξ + c = 1 N m∈Z * R e 2iπ m N x+2iπyξ 2π( m 2 N 2 + ξ 2 ) dξ + c.
Using the formula

∞ 0 cos(bx)
x 2 +z 2 dx = πe -|b|z 2z (cf. [START_REF] Prudnikov | Integrals and series[END_REF]) with b = 2πy and z = |m|/N , we arrive at

(2.51) G(x, y) = ∞ m=1 cos(2π m N x) m e -2π|y| |m| N + c.
We next particularize to the case y = 0, and use Clausen's formula (cf. [START_REF] Lewin | Polylogarithms and associated functions North-Holland[END_REF]Chap. 4])

(2.52)

∞ k=1 cos(kx) k = -log 2 sin x 2 for 0 < x < 2π,
and thus we find

G(x, 0) = -log 2 sin πx N + c.
The constant c should be chosen so that -T N G(x, 0) dx = 0, which imposes c = 0, in view of the relation T N log 2 sin πx N dx = 0, a direct consequence of (2.52). This establishes (2.48). In addition the value of R follows: R = lim x→0 G(x, 0) + log |x| = -log 2π

N and inserting into (2.44), the result follows.

New definition for a general point configuration in the plane/line

In order to define W for a general configuration of points, without referring to the current they generate, we choose to base ourselves on the formulas (2.38) and (2.50) of the previous subsection.

To get rid of some constants, we use however a different normalization1 and, for a given family of points {a i }, we define (2.53)

W N = - 1 N i =j,a i ,a j ∈[0,N ] log 2 sin π(a i -a j ) N + log N in dimension 1 respectively (2.54) W N = 1 2πN 2 i =j,a i ,a j ∈[0,N ] 2 E N (a i -a j ) + log N 2πη(i) 2 in dimension 2.
The results of the previous sections suggest trying to define W as the limit as N → ∞ of these quantities. More precisely, for a given random point process, W N becomes a random variable, and we try to see whether it has a limit as N → ∞. Again, we do not claim a complete rigorous connection between such a quantity (which only depends on the point configuration) and the original W , which we recall, was defined via a vector field j. We also emphasize that in (2.53) and (2.54) the number of points in [0, N ] resp. [0, N ] 2 is no longer necessarily equal to N resp. N 2 .

Let us comment a little on the minimization of W N . Since our definition has been relaxed, the question of whether W N achieves a minimum becomes unclear. However, in dimension 1, observing that still W N (Z) = 0, we have the following statement (in dimension 2 no such result is available):

Lemma 2.11. Assume a 1 , . . . , a k are k points in [0, N ]. Then (2.55) - 1 N i =j log 2 sin π(a i -a j ) N + log N ≥ 1 - k N log N + k N log N k .
Thus, for any point configuration

W N ≥ 1 -k N log N + k N log N k , where k is the number of points in [0, N ].
Proof. The proof is a simple adaptation of the proof of minimality of the perfect lattice in [START_REF] Sandier | 1D log-gases and the renormalized energy[END_REF]. Let a 1 , • • • , a k ∈ [0, N ], and assume a 1 < • • • < a k . Let us also denote u 1,i = a i+1 -a i , with the convention a k+1 = a 1 + N . We have k i=1 u 1,i = N . Similarly, let u p,i = a i+p -a i , with the convention a k+l = a l + N . We have k i=1 u p,i = pN . By periodicity of sin, we may view the points a i as living on the circle R/(N Z). When adding the terms in a i -a j in the sum of (2.55), we can split it according to the difference of p = j -i but modulo N . This way, there remains Inserting into (2.56) we obtain

(2.56) - 1 N i =j log 2 sin π(a i -a j ) N + log N = - 2 N [k/2]
(2.57) - 1 N i =j log 2 sin π(a i -a j ) N + log N ≥ - 2k N [k/2] p=1 log 2 sin pπ k + log N.
On the other hand, we know that W N (Z) = 0 which means that -2

N [N/2]
p=1 N log 2 sin pπ N + log N = 0, but also, since this is true for arbitrary integers N , (2.58)

-2

[k/2] p=1 log 2 sin pπ k + log k = 0.
Inserting into (2.57) we are led to (2.59)

- 1 N i =j log 2 sin π(a i -a j ) N + log N ≥ - k N log k + log N = 1 - k N log N + k N log N k .
3 Expectation of W N

Expectation and 2-point correlation functions

We now turn to evaluating W N for random point processes. In view of its form (2.53)-(2.54), the expectation of the random variable W N can be computed from the sole knowledge of the second correlation function of the process. Indeed, recall that for any k ≥ 1, the k-point correlation function ρ k of a random point process in R d is characterized by the property that

(3.1) E i 1 ,...,i k pairwise distinct F (x i 1 , . . . , x i k ) = (R d ) k F (x 1 , . . . , x k )ρ k (x 1 , . . . , x k ) dx 1 . . . dx k ,
where the expectation is with respect to our measure on locally finite subsets X = {x j } ⊂ R d , and F ranges over a suitable space of test functions, see e.g. [DVJ].

We note here that determinantal processes are a particular class of processes characterized by the fact that the correlation functions can be expressed as

(3.2) ρ k (x 1 , . . . , x n ) = det (K(x i , x j )) i,j∈[1,k]
for some kernel K(x, y), see [START_REF] Soshnikov | Determinantal random point fields[END_REF], [Ly], [Jo], [START_REF] König | Orthogonal polynomial ensembles in probability theory[END_REF], [HKPV], [START_REF] Soshnikov | Determinantal Random Fields[END_REF], [B] and references therein. This will be used later.

Here we need specifically the two-point correlation function. In addition, for our function the formula simplifies when the process is assumed to be stationary (i.e. translation invariant). From now on we make the basic assumption that we are dealing with a translation invariant multiplicity-free random point process in R or R 2 , of density 1 (i.e ρ 1 ≡ 1) with absolutely continuous correlation functions (hence, (3.1) holds). If ρ 2 (x, y) is the two-point correlation function of such a process, it is of the form r 2 (x -y) since the process is stationary. It is more convenient to work with the "second cluster function" T 2 = 1 -r 2 (we will give a general definition of the cluster functions in Section 5).

Our basic assumptions thus imply

(3.3) ρ 1 ≡ 1, ρ 2 (x, y) = 1 -T 2 (x -y) for some function T 2 .
By definition of ρ 2 , the expectation of W N is, in dimension 1,

(3.4) EW N = 1 N [0,N ] 2 log 2 sin π(x -y) N T 2 (x -y) dx dy + log N
(where we have used (2.53) and (2.49)) and respectively in dimension 2

(3.5)

EW N = - 1 2πN 2 [0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy + log N 2πη(i) 2
(where we have used (2.54) and (2.41)). The question is then whether these quantities have a limit as N → ∞. As we show below, this will only be true under additional assumptions which in particular ensure a sufficient decay of T 2 .

Finally, it would be most interesting to find natural conditions on the random points behavior (their spacing etc...) that would guarantee the existence of a limit to EW N .

Expectation for one-dimensional processes: theoretical formula

Theorem 1. Consider a random point process X on the real line with the one-point correlation function ρ 1 (x) and a two-point correlation function ρ 2 (x, y), satisfying (3.3). Under the following assumptions

1) sup v∈R |T 2 (v)| < ∞; 2) there exist a sequence {α N } N ≥1 such that log N ≪ α N ≪ N 1/2-ε as N → ∞ (for some ε > 0) and uniformly in A ∈ [α N , N -α N ] we have A α N T 2 (v) log 2 sin πv N dv = o(1) as N → ∞;
the following holds:

-if

∞ -∞ T 2 (v) log |v| dv < ∞ and ∞ -∞ T 2 (v) dx = c = 1, then EW N → ∞ as N → ∞; 2 -if ∞ -∞ T 2 (v) dx = 1 and 1 - α N -α N T 2 (v) dv = o (log N ) -1 for {α N } N ≥1 as
(3.6) lim N →∞ EW N = log 2π + ∞ -∞ log |v|T 2 (v) dv.
Remark 3.1. Condition 2) is satisfied by the stronger one:

(3.7) ∞ B |T 2 (v)| dv = o 1 log B as B → +∞.
To see this it suffices to observe that on

[α N , N -α N ] we have | sin πv N | ≥ | sin πα N N |. Proof of Theorem 1. In view of (3.4) we need to compute (3.8) lim N →∞ 1 N [0,N ] 2 log 2 sin π(x -y) N T 2 (x -y) dx dy + log N.
For (x, y) ∈ [0, N ] 2 we denote u = x + y and v = x -y. We then split [0, N ] 2 into the disjoint union of the following domains, see Figure 3.2, where α N is in condition 2):

• D 0 = {(x, y) ∈ [0, N ] 2 , |v| ≤ α N , α N ≤ u ≤ 2N -α N } • D 1 = {(x, y) ∈ [0, N ] 2 , v ≥ N -α N } • D 1 ′ = {(x, y) ∈ [0, N ] 2 , v ≤ -N + α N } • D 2 = {(x, y) ∈ [0, N ] 2 , u ≥ 2N -α N } • D 2 ′ = {(x, y) ∈ [0, N ] 2 , u ≤ α N } • D 3 = {(x, y) ∈ [0, N ] 2 , α N ≤ v ≤ N -α N } • D 3 ′ = {(x, y) ∈ [0, N ] 2 , -N + α N ≤ v ≤ -α N }
2 We believe that WN should be bounded below or at least that the value -∞ is in fact not taken. We evaluate the integral in (3.8) over each of these domains successively. We start with the contribution of D ′ 1 . Making the change of variables a = x N , b = N -y N we have

D ′ 1 log 2 sin π(x -y) N T 2 (x -y) dx dy = N 2 a≥0,b≥0,a+b≤ α N N log |2 sin π(a + b)|T 2 (N (a + b -1)) da db.
Using assumption 1) and noting that in

D 1 , sin π(a + b) = π(a + b) + O((a + b) 3
), and thus

| log |2 sin π(a + b)|| = | log |2π(a + b)|| + O( α 2 N N 2 ), we deduce D ′ 1 log 2 sin π(x -y) N T 2 (x -y) dx dy ≤ N 2 sup |T 2 | a≥0,b≥0,a+b≤ α N N | log |2π(a + b)|| da db + O α 4 N N 2 .
Using r 0 log s ds = O(s| log s|), it follows that (3.9)

D ′ 1 log 2 sin π(x -y) N T 2 (x -y) dx dy ≤ CN 2 sup |T 2 | α 2 N N 2 log α N N + O α 4 N N 2 = o(N ) using α N ≪ N 1/2-ε .
The estimate for the domains D 1 , D 2 , D 2 ′ is similar. For the contribution over D 3 , using the change of variables (x, y) → (u, v) we have

D 3 log 2 sin π(x -y) N T 2 (x -y) dx dy = 1 2 (2N -2α N ) N -α N α N T 2 (v) log 2 sin πv N dv.
This is o(N ) when assumption 2) holds. The estimate on D 3 ′ is completely analogous. We have thus found that all contributions over D 1 , D 1 ′ , D 2 , D 2 ′ , D 3 , D 3 ′ are negligible. The behavior of the integral will thus be determined by the contribution of D 0 .

Changing again the variables (x, y) into (u, v), we have

D 0 log 2 sin π(x -y) N T 2 (x -y) dx dy = 1 2 (2N -2α N ) α N -α N T 2 (v) log 2 sin πv N dv.
But in D 0 we have sin πv N = πv N (1 + O(

α 2 N N 2 )) as N → ∞, hence log | sin πv N | = log | 2πv N | + O( α 2 N N 2 ). Therefore, D 0 log 2 sin π(x -y) N T 2 (x -y) dx dy = (N -α N ) -log N α N -α N T 2 (v) dv + α N -α N log |2πv|T 2 (v) dv + sup |T 2 |O α 3 N N . Using α N ≪ N 1 2 , we easily deduce that if ∞ -∞ T 2 (v) dv = 1 and ∞ -∞ log |2πv|T 2 (v) dv < ∞, then 1 N D 0 log 2 sin π(x -y) N T 2 (x -y) dx dy + log N → ∞
as N → ∞, and we conclude as desired. If ∞ -∞ T 2 (v) dv = 1, then we may proceed and find

(3.10) 1 N D 0 log 2 sin π(x -y) N T 2 (x -y) dx dy + log N = α N -α N log |2πv|T 2 (v) dv + log N 1 - α N -α N T 2 (v) dv + O α N N log N + O α 3 N N
and we conclude (returning to (3.4) and (3.8), and using the assumptions) that (3.6) holds.

Specific computations on the line

In this subsection, we use the result from the previous theorem to compute explicit asymptotic values for EW N for some well-known point processes, namely the homogeneous Poisson process, and the β-sine processes with β = 1, 2, 4. (Recall that W N was defined in such a way that for the lattice Z we have W N = 0.) The homogeneous Poisson process satisfies ρ 1 (x) = 1 and ρ 2 (x, y) = 1, hence T 2 = 0. We immediately deduce from (3.5) that EW N → +∞. Hence, the Poisson process can be viewed as having the 'value of W ' equal to +∞.

The β-sine processes for β = 1, 2, 4 arise in random matrices as the local limits for random matrix ensembles with orthogonal, Hermitian, and symplectic symmetries, see [Me, AGZ, Fo] and references therein. These are stationary processes whose correlations can be computed as follows: introduce the kernels

K (2) (x, y) = sin π(x -y) π(x -y) for β = 2 (3.11) K (1) (x, y) =     sin π(x -y) π(x -y) ∂ ∂x sin π(x -y) π(x -y) 1 π π(x-y) 0 sin t t dt - 1 2 sgn(x -y) sin π(x -y) π(x -y)     for β = 1 (3.12) K (4) (x, y) =     sin 2π(x -y) 2π(x -y) ∂ ∂x sin 2π(x -y) 2π(x -y) 1 2π 2π(x-y) 0 sin t t dt sin 2π(x -y) 2π(x -y)     for β = 4 (3.13)
where all the indeterminacies 0/0 at x = y are resolved by continuity. The β = 2 sine process is a determinantal process with kernel K (2) (x, y), thus from (3.2), its 2-point correlation function is given by

(3.14) ρ 2 (x 1 , x 2 ) = det K (2) (x i , x j ) i,j∈[1,2]
.

The correlation functions for the β = 1, 4 sine processes have the form

(3.15) ρ n (x 1 , . . . , x n ) = qdet K (β) (x i , x j ) i,j∈[1,n] , n = 1, 2, . . .
where qdet denotes the quaternion determinant, see e.g. [START_REF] Forrester | Log-gases and random matrices[END_REF]Section 6.1.1] for a definition. Alternatively, the right-hand side of (3.15) can be expressed as the Pfaffian of a closely related matrix, cf. [START_REF] Forrester | Log-gases and random matrices[END_REF]Proposition 6.1.5]. Random point processes with correlation functions of such form are often called Pfaffian, see [START_REF] Borodin | Determinantal point processes[END_REF]Section 10] and references therein. The three processes above satisfy ρ 1 ≡ 1, and their second cluster functions can be easily seen to be given by

T (2) 2 (v) = sin πv πv 2 for β = 2 (3.16) T (1) 2 (v) = sin πv πv 2 - 1 π ∂ ∂v sin πv πv πv 0 sin t t dt - π 2 sgn(v) for β = 1 (3.17) T (4) 2 (v) = sin 2πv 2πv 2 - 1 2π ∂ ∂v sin 2πv 2πv 2πv 0 sin t t dt for β = 4. (3.18)
Proposition 3.2. The β-sine processes for β = 1, 2, 4 satisfy the assumptions of Theorem 1, and

lim N →∞ EW N = 1 -γ for β = 2 lim N →∞ EW N = 2 -γ -log 2 for β = 1 lim N →∞ EW N = 3 2 -log 2 -γ for β = 4,
where γ is the Euler constant.

Before stating the proof, we recall some integrals of classical functions that we will need. We state them without a proof and refer to [START_REF] Prudnikov | Integrals and series[END_REF][START_REF] Prudnikov | Integrals and series[END_REF].

+∞ 0 sin v v dv = π 2 (3.19) +∞ 0 sin v v log v dv = π 2 γ (3.20) +∞ 0 sin v v 2 dv = π 2 (3.21) +∞ 0 sin v v 2 log v dv = - π 2 (γ + log 2 -1). (3.22)
These formulas can be found in [START_REF] Prudnikov | Integrals and series[END_REF], they are respectively (2.5.3.12), (2.6.32.3), (2.5.6.9), and (2.6.32.7). Finally we need a few more integrals that are based on

Si(x) = x 0 sin t t dt, si(x) = Si(x) - π 2 = - +∞ x sin t t dt and Ei(x) = x -∞ e t t dt.
These are

+∞ 0 si(v) sin v v 2 + z 2 dv = π 2z sinh(z)Ei(-z) (3.23) +∞ 0 v si(v) v 2 + z 2 dv = π 2 Ei(-z) (3.24) +∞ 0 Si(v) sin v v 2 + z 2 dv = π 4 e -z Ei(z) -Ei(-z) z , (3.25)
cf. respectively (2.6.4.11), (2.6.2.10) and (2.6.4.16) in [START_REF] Prudnikov | Integrals and series[END_REF].

Proof Proposition 3.2. We start with the simplest case. Case β = 2: First it is easy to check that assumptions (3.3) as well as assumption 1) of Theorem 1 are verified. Also, we have

T 2 (v) = O(|v| -2 ) as v → ∞, and 1 - α N -α N T 2 = O((α N ) -1 ) = o((log N ) -1
), thus assumption 2) is implied by (3.7). According to Theorem 1 we then have lim

N →∞ EW N = ∞ -∞ log |2πv|
sin πv πv ). Assumptions (3.3) and 1), 2) are then verified as in the β = 2 case. Next we check that T

(1) 2 = 1. By evenness and integration by parts

∞ -∞ T (1) 2 (v) dv = 2 ∞ 0 sin πv πv 2 - 1 π ∂ ∂v sin πv πv πv 0 sin t t dt - π 2 dv = 4 ∞ 0 sin πv πv 2 dv -1 = 1.
The convergence is also fast enough, since

T 2 (v) = O(|v| -2 ).
According to Theorem 1 we thus have lim

N →∞ EW N = ∞ -∞ log |2πv| sin πv πv 2 - 1 π ∂ ∂v sin πv πv πv 0 sin t t dt - π 2 sgn(v) dv = 2 ∞ 0 log |2πv| sin πv πv 2 - 1 π ∂ ∂v sin πv πv -1 πv 0 sin t t dt - π 2 dv.
Integrating by parts, we are led to

(3.27) lim N →∞ EW N = 4 ∞ 0 log(2πv) sin πv πv 2 dv + 2 π ∞ 0 sin πv πv -1 πv 0 sin t t - π 2 dv v - 2 π +∞ 0 log(2πv) sin πv v dv.
In view of (3.26) the first term on the right-hand side is equal to 2 -2γ. The second is equal to

2 π +∞ 0 sin πv πv -1 si(πv) dv v = 2 π +∞ 0 sin u u -1 si(u) du u = 2 π lim z→0 ∞ 0 si(u) sin u u 2 + z 2 - si(u)u u 2 + z 2 du.
With (3.23) and (3.24),

2 π +∞ 0 sin πv πv -1 si(πv) dv v = lim z→0 Ei(-z) sinh(z) z -1 .
On the other hand, near x = 0 one has

(3.28) Ei(x) = γ + log |x| + ∞ n=1
x n nn! , hence the above right-hand side limit is 0. The second term in (3.27) thus vanishes. From (3.20) and (3.19), the third one is equal to γ -log 2. This concludes the case β = 1. Case β = 4: Assumptions (3.3) and 1) are easily verified, we proceed to 2). We may write

T 2 (v) = sin 2πv 2πv 2 - 1 2π ∂ ∂v sin 2πv 2πv 2πv 0 sin t t dt - π 2 sgn(v) - 1 4 ∂ ∂v sin 2πv 2πv sgn(v).
The first part is O(|v| -2 ) just as in the case above, so it remains to check that

A α N ∂ ∂v sin 2πv 2πv log 2 sin πv N dv = o(1) uniformly in A ∈ [α N , N -α N ].
Integrating by parts, we have

A α N ∂ ∂v sin 2πv 2πv log 2 sin πv N dv = sin 2πA 2πA log 2 sin πA N - sin 2πα N 2πα N log 2 sin πα N N - A α N sin 2πv 2v cos πv N N sin πv N dv.
In [α N , A] we may bound from below sin πv N by sin πα N N , which is asymptotically equivalent to

πα N N as N → ∞.
Hence the integral on the right hand side may by bounded by 1) in view of assumption 4). The other terms are also easily found to be o(1) by a similar argument.

C α N N α N dv v ≤ C log N α N = o(
We next check that T 2 = 1 with fast enough convergence. By evenness and integration by parts

α N -α N T (4) 2 (v) dv = 2 α N 0 sin 2πv 2πv 2 - 1 2π ∂ ∂v sin 2πv 2πv 2πv 0 sin t t dt dv = 4 α N 0 sin 2πv 2πv 2 dv - 1 2π sin 2πα N α N 2πα N 0 sin t t dt = 1 + O((α N ) -1 ).
According to Theorem 1 we thus have lim

N →∞ EW N = ∞ -∞ log |2πv| sin 2πv 2πv 2 - 1 2π ∂ ∂v sin 2πv 2πv 2πv 0 sin t t dt dv.
Using evenness and integration by parts as above, we find lim

N →∞ EW N = 4 ∞ 0 sin 2πv 2πv 2 log |2πv| dv + 1 π ∞ 0 sin 2πv 2πv 2 2πv 0 sin t t dt dv.
By (3.22) the first term on the right-hand side is equal to -γ -log 2 + 1. By change of variables, the second term is equal to

1 π +∞ 0 Si(u) sin u u 2 du = 1 π lim z→0 +∞ 0 Si(u) sin u u 2 + z 2 du = 1 4 lim z→0 e -z Ei(z) -Ei(-z) z
by (3.25). Combining with (3.28) we find that the second term is equal to 1 2 and we conclude the proof.

As expected lim N →∞ EW N decreases as β = 1, 2, 4 increases.

Expectation for two-dimensional processes : theoretical formula

In the plane, the computations are easier because we can take advantage of the fast (exponential) decay of the correlation kernels.

Theorem 2. Consider a random point process X in the plane, with a one point correlation function ρ 1 (x) and a two-point correlation function ρ 2 (x, y), satisfying (3.3). We identify the plane with the complex plane C. Under the assumption

R 2 |v| k |T 2 (v)| dv < ∞ for k = 1, 2, 3;
the following holds:

-if

R 2 T 2 (v) dv = c = 1, we have EW N → ∞ as N → ∞; -if R 2 T 2 (v) dv = 1 and 1 -[-N,N ] 2 T 2 (v) dv = o((log N ) -1
), then lim N →∞ EW N exists and is finite if and only if R 2 T 2 (v) log |v| dv converges, and if so, then

(3.29) lim N →∞ EW N = R 2 log |v| T 2 (v) dv.
Proof. Returning to (3.5) we have to compute

lim N →∞ - 1 2πN 2 [0,N ] 2 ×[0,N ] 2 E(x -y)T 2 (x -y) dx dy + log N 2πη(i) 2 .
Making the change of variables (u, v) = (x + y, x -y), we have

[0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy = 1 4 v∈[-N,N ] 2 u∈S N (v) E(v)T 2 (v) du dv, where S N (v) = {x + y : x ∈ [0, N ], y ∈ [0, N ] 2 , x -y = v}. We may compute that |S N (v)| = 4N 2 -2N |v 1 | -2N |v 2 | + |v 1 ||v 2 |, so (3.30) [0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy = 1 4 [-N,N ] 2 4N 2 -2N |v 1 | -2N |v 2 | + |v 1 ||v 2 | E N (v)T 2 (v) dv.
Next we return to (2.43) where f is given by (2.42) and perform an asymptotic analysis as N → ∞. We have

p 1/2 -p -1/2 = e iπ x N -e -iπ x N = 2iπ x N + O |x| 2 N 2 ,
while, since p = 1 + O( |x| N ) we may write (with q = e -2π )

q 1/12 k≥1 (1 -q k p)(1 -q k /p) =   q 1/12 k≥1 (1 -q k )   (1 + O(p -1)) = η(i) 2 + O |x| N , hence f x N , i = 2iπ x N η(i) 2 1 + O |x| N .
Inserting into (2.43) and combining with e -π(

Im x N ) 2 = 1 + O( |x| 2 N 2 ) we obtain (3.31) E N (x) = -2π log |x| -2π log 2πη(i) 2 N + O |x| N as N → ∞.
Inserting this into (3.30), we are led to

[0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy = [-N,N ] 2 N 2 - 1 2 N |v 1 | - 1 2 N |v 2 | + 1 4 |v 1 ||v 2 | -2π log |v| -2π log 2πη(i) 2 N + O |v| N T 2 (v) dv. Therefore - 1 2πN 2 [0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy + log N 2πη(i) 2 = log N 2πη(i) 2 1 - [-N,N ] 2 1 + O |v| N + O |v| 2 N 2 T 2 (v) dv + [-N,N ] 2 log |v| + O |v| N 1 + O |v| N + O |v| 2 N 2 T 2 (v) dv.
Using the assumption we have

[-N,N ] 2 |v| N T 2 (v) dv = o(1), [-N,N ] 2 |v| 2 N 2 T 2 (v) dv = o(1), [-N,N ] 2 |v| 3 N 3 T 2 (v) dv = o(1), [-N,N ] 2 |v| N log |v|T 2 (v) dv = o(1), [-N,N ] 2 |v| 2 N 2 log |v|T 2 (v) dv = o(1). It follows that, as N → ∞, - 1 2πN 2 [0,N ] 2 ×[0,N ] 2 E N (x -y)T 2 (x -y) dx dy + log N 2πη(i) 2 = log N 2πη(i) 2 1 - [-N,N ] 2 T 2 (v) dv + [-N,N ] 2 log |v|T 2 (v) dv + o(1).
The result then easily follows.

Specific computations in the plane

We turn to computing that limit for two specific processes. The first one is the determinantal random point process with correlation kernel

(3.32) K(x, y) = e -π 2 (|x| 2 +|y| 2 -2xy) .
This process arises in random matrices as the local limit of the complex Ginibre ensemble, see e.g. [START_REF] Forrester | Log-gases and random matrices[END_REF]Proposition 15.2.3], and thus it is sometimes called the Ginibre point process.

From the determinantal structure of the correlation functions, cf. (3.2), we have ρ 2 (x, y) = 1 -|K(x, y)| 2 = 1 -e -π|x-y| 2 and T 2 (v) = e -π|v| 2 . This easily satisfies all the assumptions of Theorem 2 (in particular T 2 = 1) and we obtain Proposition 3.3. The determinantal process with kernel (3.32) satisfies lim

N →∞ EW N = 1 2 (γ -log π) .
This statement can be compared to a similar computation done in [Ja], see also [START_REF] Forrester | Log-gases and random matrices[END_REF]Ex.15.3.1(iv)].

Proof. According to Theorem 2 it suffices to compute

R 2 log |v|e -π|v| 2 dv = ∞ 0 log r e -πr 2 2πr dr = ∞ 0 1 2
(log s -log π)e -s ds, using the change of variables s = πr 2 . We have ∞ 0 e -s log s ds = γ, and the result follows.

The second one is the process of the zeros of a Gaussian analytic function (often denoted GAF). It consists of the random zeros of the analytic function ∞ n=0 ξn √ n! z n when the ξ n are i.i.d Gaussians suitably normalized, and it is a stationary process in the plane. The general background can be found e.g. in [HKPV].

The second cluster function for the process, when the density ρ 1 is taken to be 1, is given according to [FH] by

T 2 (x) = 1 -h π|x| 2 2 where h(x) = 1 + 1 2 d 2 dx 2 x 2 (coth x -1)
. It is easy to check that the assumptions of Theorem 2 are satisfied, and we deduce Proposition 3.4. The "zeros of Gaussian analytic functions" process satisfies

lim N →∞ EW N = - 1 2 (1 + log π).
Proof. To check that we may apply Theorem 2 we first compute

R 2 T 2 (v) dv = R 2 1 -h π|v| 2 2 dv = ∞ 0 1 -h πr 2 2 2πr dr = 2 ∞ 0 (1 -h(u)) du = - d dx (x 2 (coth x -1)) ∞ 0 = -2x(coth x -1) + x 2 (1 -coth 2 x) ∞ 0 = 1
where we have used the change of variables u = πr 2 /2 and the asymptotic relation coth x ∼ 1

x as x → 0. It is also easy to check that the convergence of T 2 is exponential, hence fast enough, and we may apply Theorem 2. This yields lim

N →∞ EW N = R 2 log |v|T 2 (v) dv = 2π ∞ 0 log r 1 -h πr 2 2 r dr = 2 ∞ 0 log 2u π (1 -h(u)) du = - 1 2 ∞ 0 log 2u π d 2 du 2 (u 2 (coth u -1)) du.
Let us now compute

∞ ε log 2u π d 2 du 2 (u 2 (coth u -1)) du = log 2x π d dx (x 2 (coth x -1)) ∞ ε - ∞ ε 1 u d du (u 2 (coth u -1)) du = -log 2ε π 2ε(coth ε -1) + ε 2 (1 -coth 2 ε) -[x(coth x -1)] ∞ ε - ∞ ε (coth u -1) du = -log 2ε π (1 + O(ε)) + 1 + O(ε) -[log sinh x -x] ∞ ε .
Taking the limit ε → 0, we conclude lim

N →∞ EW N = - 1 2 (1 + log π).
Thus the "GAF process" has more order than the "Ginibre point process". This is in agreement with the findings of [GNPS] where the authors show via a different approach that the GAF process is more "rigid" than the Ginibre one.

Optimization over determinantal processes

As explained in the introduction and Section 2.3, the question of minimizing W is an important one, and open in dimension 2. It thus seems interesting to try to minimize lim N →∞ EW N as expressed in (3.6) and (3.29), over a subclass of processes.

In this section we show that we can characterize the minimizer of this expression over the class of determinantal random point processes whose correlation kernel K(x, y) is Hermitian and translation invariant, i.e. K(x, y) = k(x -y) for some function k. For those processes, we have T 2 = k 2 . Note, however, that the important determinantal process with kernel (3.32) is not in this class: while all its correlation functions are translation invariant, the correlation kernel is not.

We prove the following statement. The proof relies on a rearrangement inequality.

Theorem 3. Let K be the class of determinantal processes on the real line, respectively the plane, with self-adjoint translation-invariant kernels K(x, y) = k(x -y), and

k(v) ∈ L 2 (R d ), d = 1 or d = 2, such that 1) ρ 1 (x) = k(0) = 1 and R d k 2 (x) dx = 1; 2) R d log |x|k 2 (x) dx < ∞. Let F(k) = R d log |x|k 2 (x) dx.
Then for any process from K with correlation kernel K(x, y) = k(x -y), we have

F(k) ≥ F( 1 B ),
where B is the ball centered at 0 of volume one, and • is the Fourier transform. Thus, on the real line F(k) is minimized over K by the β = 2 sine process, while on the plane the minimizing determinantal process has the kernel given by k

(v) = 1 B (v) = J 1 (2 √ π|v|)
√ π|v| , where J 1 is a Bessel function of the first kind.

Remark 4.1. Condition 1) says that our processes have density 1, which we have assumed throughout, and that lim N →∞ EW N is finite, cf. Theorems 1 and 2. The functional F coincides with lim N →∞ EW N , given that the decay assumptions are satisfied.

Proof of Theorem 3. Let us denote by f the inverse Fourier transform of k, and by T K the integral operator corresponding to K via T K (ϕ) = K(x, y)ϕ(y) dy. We have

T K (ϕ) = k(x -y)ϕ(y) dy = k * ϕ.
By Macchi-Soshnikov's theorem, see [START_REF] Soshnikov | Determinantal random point fields[END_REF], any self-adjoint translation-invariant correlation kernel of a determinantal process gives rise to an integral operator with spectrum between 0 and 1. Hence, the spectrum of

T K is in [0, 1]. Since T K is also the convolution by k, this implies that f = ǩ takes values in [0, 1]. Moreover f ∈ L 2 with |f | 2 = k 2 =
1, and f = k(0) = 1 from assumption 1). A function f with values in [0, 1] which satisfies f = f 2 = 1 can only be a characteristic function of a set A, denoted f = 1 A , with A measurable of measure 1. Writing k A for the corresponding k, we may write

k A (x) = e -2iπξ•x 1 A (ξ) dξ.
There remains to optimize over A, measurable set of measure 1, (4.1)

I(A) := R d k 2 A (x) log |x| dx.
Let A be measurable of measure 1, such that I(A) < +∞. Noting that k A is an L ∞ (and also continuous) function, for every α > 0 the integrals

R d k 2 A (x) 1 -|x| -α α dx
then also converge, by comparison. Given any τ ∈ R, (e τ h -1)/h converges to τ monotonically as h → 0 (it suffices to check that this function is increasing in h). It then follows that in each of the domains |x| < 1 and |x| ≥ 1, we have 1-|x| -α α → log |x| as α → 0, monotonically. Splitting the integrals as sums over these two regions, it follows by monotone convergence theorem that (4.2)

I(A) = lim α→0 R d k 2 A (x) 1 -|x| -α α dx.
We then remark (in view of the formula

f * g = f ĝ) that k 2 A (x) is the Fourier transform of f A (x) = R d 1 A (y)1 A (y -x) dy = |(A + x) ∩ A|.
We next claim that f A is a continuous function. First, consider the case where A is an open set. Then, as x → x 0 we have 1 A (y)1 A (y -x) → 1 A (y)1 A (y -x 0 ) almost everywhere, by openness of A, while |1 A (y)1 A (y -x)| ≤ 1 A (y) and 1 A ∈ L 1 . The claim is thus true by dominated convergence theorem. Second, if A is a general measurable set, by outer regularity of the measure we may approximate it by an open set U such that A ⊂ U and |U \A| < ε.

Then it is immediate that for any x,

|f A (x)-f U (x)| = ||(A+x)∩A|-|(U +x)∩U || ≤ 2ε.
Since f U is continuous and ε is arbitrary, it follows that f A has to be continuous too as a uniform limit of continuous functions. The claim is proved. We also note that f A (0

) = |A| = 1. The next ingredient is that in dimension 1 (4.3) (1 -|x| -α ) = δ -2(-α)! sin πα 2 (2π|ξ|) α-1 , while in dimension 2 (4.4) (1 -|x| -α ) = δ -π α-1 Γ( 2-α 2 ) Γ( α 2 ) |ξ| α-2
For a reference see e.g. [START_REF] Edwards | Functional Analysis[END_REF]Chapter 5], or [START_REF] Schwartz | Théorie des distributions[END_REF]page 113].

Let us continue with the one-dimensional case. We deduce from the above facts that (4.5)

I α (A) := R k 2 A (x) 1 -|x| -α α dx = 1 α R f A (ξ) δ -2(-α)! sin πα 2 (2π|ξ|) α-1 .
The relation can be justified by convoluting δ -2(-α)! sin πα 2 (2π|ξ|) α-1 with a Gaussian kernel approximating δ at scale ε, using the fact that f g = f ĝ in the Schwartz class, the continuity of f A and then letting ε → 0 on both sides. Moreover, this argument shows that f A (ξ)|ξ| α-1 dξ is convergent. Using f A (0) = 1 and Fubini's theorem, we may rewrite (4.5) as

(4.6) I α (A) = 1 α 1 -2(-α)! sin πα 2 R 2 1 A (y)1 A (y -x)(2π|x|) α-1 dx dy = 1 α - 2 α (-α)! sin πα 2 R 2 1 A (y)1 A (z)(2π|y -z|) α-1 dz dy.
Notice that 1 α (-α)! sin πα 2 ∼ π 2 as α → 0 so for α small enough, -2 α (-α)! sin πα 2 (2π|y -z|) α-1 is increasing in |y -z|. Now Riesz's rearrangement inequality (see [START_REF] Lieb | Loss, Analysis[END_REF]Theorem 3.7]) asserts that a quantity of the form of the right-hand side of (4.6) is always decreased by changing 1 A into its symmetric rearrangement (1 A ) * = 1 A * . This means that for all α small enough, I α (A) ≥ I α (A * ). But I(A) = lim α→0 I α (A) hence the same is true also for I i.e. I(A) ≥ I(A * ). The symmetric rearrangement A * of A is the ball centered at 0 and of volume |A| = 1. We have thus found that k 2 A (x) log |x| dx is minimal when A is the ball centered at 0 and of volume 1. In dimension one, the Fourier transform of 1 [-1 2 , 1 2 ] is sin πx πx , which corresponds to the determinantal process with K(x, y) = sin π(x-y) π(x-y) , that is the sine process (for β = 2). In dimension 2, the argument is exactly parallel, starting again from (4.4).

Computations of variance of W N

In Section 3 we dealt with the expectation of W . In this section we turn to examining its variance in the sense of computing lim N →∞ Var(W N ) for the same specific random point processes.

In what follows we will need the formalism of (higher) cluster functions to efficiently deal with the k-point correlation functions for k = 2, 3, 4; we refer to [TW, Fa] for details and further references on this formalism.

For any nonempty subset S = {i 1 , . . . , i k } of {1, . . . , N } we write ρ S = ρ k (x i 1 , . . . , x i k ), where ρ k is the k-point correlation function, and define the n-point cluster function as (5.1)

T n (x 1 , . . . , x n ) = (-1) n-m (m -1)!ρ S 1 . . . ρ Sm with the sum running over all partitions of {1, . . . , n} into nonempty subsets S 1 , . . . , S m . From the T n , the ρ n can be recovered through the reciprocal formula

(5.2)

ρ n = (-1) n-m T S 1 . . . T Sm .
If a random point process is determinantal (cf. (3.2)) with correlation kernel K, then (see e.g. [TW]) for any k ≥ 1

(5.3)

T k (x 1 , . . . , x k ) = 1 k σ∈S k K(x σ(1) , x σ(2) ), . . . , K(x σ(k) , x σ(1) ),
where S k denotes the symmetric group on k symbols.

If the correlation functions of a point process are given by quaternion determinants (cf. (3.15)) then (see e.g. [TW]) for any k ≥ 1 (5.4)

T k (x 1 , . . . , x k ) = 1 2k T r σ∈S k K(x σ(1) , x σ(2) ), . . . , K(x σ(k) , x σ(1) ).
Lemma 5.1. We have

Var   i =j G N (a i -a j )   = I 1 + • • • + I 6
where

I 1 = 2 [0,N ] 2
G N (x -y) 2 dx dy (5.5)

I 2 = -4 [0,N ] 3 G N (x -y)G N (x -z)T 2 (y, z) dx dy dz (5.6) I 3 = 2 [0,N ] 4 G N (x -y)G N (z -t)T 2 (x, z)T 2 (
y, t) dx dy dz dt (5.7)

I 4 = -2 [0,N ] 2
G N (x -y) 2 T 2 (x, y) dx dy (5.8)

I 5 = 4 [0,N ] 3 G N (x -y)G N (x -z)T 3 (x,
y, z) dx dy dz (5.9)

I 6 = - [0,N ] 4 G N (x -y)G N (z -t)T 4 (x, y, z, t) dx dy dz dt, (5.10) where G N (x) = -log 2 sin πx N in dimension 1, resp. G N (x) = 1 2π E N (x) defined in (2.36) in dimension 2.
Proof. Expanding the square, we have (5.14) where the sums are still taken over points in [0, N ], and p.d. stands for "pairwise distinct". Since G N is even, it is clear that all the sums in (5.12) and (5.13) are equal, and the sums in (5.14) as well. Using k-point correlation functions (cf. (3.1)), we thus may write

  i =j,a i ,a j ∈[0,N ] G N (a i -a j )   2 = i,j,k,l p.d. G N (a i -a j )G N (a k -a l ) (5.11) + i,j,l p.d. G N (a i -a j )G N (a i -a l ) + i,j,k p.d. G N (a i -a j )G N (a k -a i ) (5.12) + i,j,l p.d. G N (a i -a j )G N (a j -a l ) + i,j,k p.d. G N (a i -a j )G N (a k -a j ) (5.13) + i =j G N (a i -a j ) 2 + i =j G N (a i -a j )G N (a j -a i ),
(5.15) E   i =j,a i ,a j ∈[0,N ] G N (a i -a j )   2 = [0,N ] 4 G N (x -y)G N (z -t)ρ 4 (x, y, z, t) dx dy dz dt + 4 [0,N ] 3 G N (x -y)ρ 3 (x, y, z) dx dy dz + 2 [0,N ] 2 G N (x -y) 2 ρ 2 (x -y) dx dy.
It is now convenient to express this in terms of the cluster functions T k , using (5.2), which yields

ρ 2 (x, y) = T 1 (x)T 1 (y) -T 2 (x, y) = 1 -T 2 (x, y) ρ 3 (x, y, z) = 1 -T 2 (x, y) -T 2 (x, z) -T 2 (y, z) + T 3 (x, y, z) ρ 4 (x, y, z, t) = 1 -T 2 (x, y) -T 2 (x, z) -T 2 (x, t) -T 2 (y, z) -T 2 (y, t) -T 2 (z, t) +T 3 (x, y, z) + T 3 (x, y, t) + T 3 (x, z, t) + T 3 (y, z, t) +T 2 (z, y)T 2 (z, t) + T 2 (x, z)T 2 (y, t) + T 2 (x, t)T 2 (y, z) -T 4 (x, y, z, t).
Substituting these relations into (5.15) and using that N 0 G N = 0, we obtain (writing the terms in the same order)

E   i =j,a i ,a j ∈[0,N ] G N (a i -a j )   2 = [0,N ] 2 G N (x -y)T 2 (x, y) dx dy 2 (5.16) + 2 [0,N ] 4 ] G N (x -y)G N (z -t)T 2 (x, z)T 2 (y, t) dx dy dz dt - [0,N ] 4 G N (x -y)G N (z -t)T 4 (x, y, z, t) dx dy dz dt + 4 [0,N ] 3 G N (x -y)G N (x -z)T 3 (x, y, z) dx dy dz -4 [0,N ] 4 G N (x -y)G N (x -z)T 2 (y, z) dx dy dz + 2 [0,N ] 2 G N (x -y) 2 dx dy -2 [0,N ] 2 G N (x -y) 2 T 2 (x, y) dx dy.
Similarly (and as we have seen in the proof of Theorem 1) we have

E i =j G N (a i -a j ) = - [0,N ] 2
G N (x -y)T 2 (x, y) dx dy and the result follows.

The one-dimensional case

Theorem 4. For the sine-β processes with β = 1, 2, 4 as described above, we have

lim N →∞ Var(W N ) = 0.
Proof. Since we already know from Proposition 3.2 that lim N →∞ EW N exists, and in view of (2.53), it suffices to show that (5.17) lim

N →∞ 1 N 2   E   i =j,a i ,a j ∈[0,N ] G N (a i -a j )   2 -   E i =j,a i ,a j ∈[0,N ] G N (a i -a j )   2   = 0.
We apply Lemma 5.1 and now deal with all the terms I 1 to I 6 in (5.5)-(5.10). First, we have (5.18)

I 1 = 2 [0,N ] 2 G N (x -y) 2 dx dy = 2N 2 [0,1] 2 (log |2 sin π(x -y)|) 2 dx dy.
For I 2 , using the explicit expression for G N , making the change of variables x ′ = x/N, y ′ = y/N, t = y -z, and recalling that T 2 is translation-invariant since the process is,3 we find

(5.19) I 2 = -4 [0,N ] 4 G N (x -y)G N (x -z)T 2 (y -z) dx dy dz = -4N 2 [0,1] 2 [N (y-1),N y] log |2 sin π(x -y)| log 2 sin π x -y + t N T 2 (t) dx dy dt.
One may like to think that f (x -y + t N )T 2 (t) dt → f (x -y) since T 2 = 1. However, log |2 sin •| is not regular enough to apply this reasoning and T 2 may converge only conditionally. First notice that the cluster functions given in (3.16)-(3.17)-(3.18) satisfy (5.20)

|T 2 (v)| = O 1 |v| , (5.21) |v|>M T 2 (v) dv = O 1 M .
Pick two exponents a, b > 0 with a+b < 1. Let us examine the t integral in the right-hand side of (5.19). Assume first that [-N b , N b ] ⊂ [N (y -1), N y] and that |x -y| > N -a . Note that for this to be satisfied it suffices that (5.22) (x, y) ∈ S N,a := {(x, y) ∈ R 2 : N -a < y < 1 -N -a , |x -y| > N -a }.

By the mean value formula, we may write for some

|θ| < N b log 2 sin π x -y + t N -log |2 sin π(x -y)| = π cos π(x -y + θ N ) N sin π(x -y + θ N ) = O(N a-1 )
since we assumed |x -y| > N -a . Thus

(5.23)

[-N b ,N b ] log |2 sin π(x -y)| log 2 sin π x -y + t N T 2 (t) dt = [-N b ,N b ] log 2 |2 sin π(x -y)|T 2 (t) dt + O(N a-1 log N ) = log |2 sin π(x -y)| 1 -O(N -b ) + O(N a-1 log N ) = log |2 sin π(x -y)| + O(N -b log N ).
where we have used (5.20), then (5.21) and |x-y| > N -a . We then claim that if |x-y| > N -a we have

(5.24)

|t|>N b log |2 sin π(x -y)| log 2 sin π x -y + t N T 2 (t) dt = o(1).
Assuming this, and combining with (5.23) we obtain that

(x,y)∈[0,1] 2 ∩S N,a [N (y-1),N y] log |2 sin π(x -y)| log 2 sin π x -y + t N T 2 (t) dx dy dt = (x,y)∈[0,1] 2 ∩S N,a log 2 |2 sin π(x -y)| dx dy + o(1).
But it is easy to check, since the integrals converge and (5.20) holds, that the contributions of the set where (5.22) does not hold are o(1) as N → ∞. We may thus conclude that (5.25)

I 2 = -4N 2 (x,y)∈[0,1] 2 log 2 |2 sin π(x -y)| dx dy + o(N 2 ).
To finish with this I 2 term, it remains to prove (5.24). For β = 1, 2 this is immediately true since T 2 (v) = O(|v| -2 ). For β = 4, we notice that the same argument that was used above to restrict to |x -y| > N -a can be used to restrict to |x -y + t N | > N -c (note that the initial integral is symmetric in y and z). Inserting the formula for T 2 (3.18), and neglecting the O(1/t 2 ) part of T 2 , we thus have to prove that

|t|>N b ,|x-y+ t N |>N -c - ∂ ∂t sin 2πt 2πt 1 2π 2πt 0 sin t t log 2 sin π x -y + t N = o(1).
We integrate by parts and find that the boundary terms are negligible, and there remains to show that

|t|>N b ,|x-y+ t N |>N -c sin 2πt 2πt ∂ ∂t log 2 sin π x -y + t N 1 2π 2πt 0 sin t t dt = o(1).
If the derivative falls on the second factor, we are back to the O(1/t 2 ) situation which gives a negligible term, and for the other term we use

∂ ∂t log 2 sin π x -y + t N = O(N c-1 )
by explicit computation, which gives that the integral is O(N c-1 log N ) = o(1). This completes the treatment of I 2 . We turn to I 3 . Using a similar change of variables, we may write this term

I 3 = 2N 2 [0,1] 2 [N (x-1),N x] [N (y-1),N y] log |2 sin π(x -y)| log 2 sin π x -y + v -u N × T 2 (u)T 2 (v) dx dy du dv.
Very similar manipulations to those above show that log 2 sin π x -y + v-u N can be replaced by log |2 sin π(x -y)| with a o(N 2 ) correction. This leads us to

I 3 = 2N 2 [0,1] 2 log 2 |2 sin π(x -y)| dx dy + o(N 2 ).
For I 4 , we have

I 4 = [0,N ] 2 log 2 2 sin π(x -y) N T 2 (x -y) dx dy = o(N 2 ).
Indeed, we may take away a δN -neighborhood of the diagonal, outside of which log 2 2 sin π(x-y) N is bounded by log 2 N and |T 2 | is controlled by log N , using (5.20). Thus the whole integral is controlled by N log 3 N = o(N 2 ).

Adding the above results we find that

I 1 + I 2 + I 3 + I 4 = o(N 2 ).
It remains to show that I 5 and I 6 also give o(N 2 ) contributions. The expressions I 5 and I 6 are estimated using explicit formulas for cluster functions of the sine-β processes. First returning to (3.11)-(3.12) we see that for β = 1, 2, the entries of K (2) and K (1) are O 1 1+|x-y| . Combining with (5.3)-( 5.4), it follows that T 3 (x, y, z) = O 1 1+|(x-y)(y-z)(z-x)| in both these cases. For (3.13), we have (5.26)

K (4) (x, y) =   O 1 1+|x-y| O 1 1+|x-y| O(1) O 1 1+|x-y|   .
We thus obtain

T 3 (x, y, z) = O 1 1 + |(x -y)(y -z)| +O 1 1 + |(y -z)(z -x)| +O 1 1 + |(x -y)(z -x)| .
In (5.9) we may first (as above) remove a small neighborhood of the diagonals, off of which Replacing T 3 by its above estimates, and changing variables to x + y + z and successively two out of x -y, y -z, and z -x, we find [0,N ] 3 |T 3 (x, y, z)| dx dy dz ≤ O(N log 3 N ), and

I 5 = o(N 2 ).
We finally turn to I 6 . The formula for T 4 is given by (5.3)-(5.4). Comparing to (3.11)-(3.12), we see that for β = 1, 2, we have

T 4 (x, y, z, t) = O 1 1 + |(x -y)(y -z)(z -t)(t -z)| .
The same reasoning as for I 5 gives I 6 = o(N 2 ). For β = 4, in the formula for T 4 obtained with (3.13), in view of (5.26), there are terms which a priori have insufficient decay: they are terms of the form . This leads again to o(N 2 ) contributions.

∂ ∂x 1 sin π(x 1 -x 2 ) 2π(x 1 -x 2 ) ∂ ∂x 3 sin 2π(x 3 -x 4 ) 2π(x 3 -x 4 ) 2π(x 2 -x 3 ) 0 sin t t
For (5.28), we may first restrict the integral to |x -z| > N a and |y -t| > N a , using arguments as above. Then, we may replace 2π(x-z) 0 sin s s ds and 2π(y-t) 0 sin s s ds by π 2 sgn(x-z) and π 2 sgn(y -t) respectively, making only a o(N 2 ) error. Then note that the integrand in x -y is a locally odd function, so we can remove domains |x -y| < N b , |z -t| < N b from the integration domain. Finally, integration by parts in x gives additional decay, yielding o(N 2 ) contribution. We conclude that I 6 = o(N 2 ), and the result follows.

Corollary 5.2. For all point processes with finite lim N →∞ E(W N ) and lim

N →∞ Var(W N ) = 0, W N → lim N →∞ EW N as N → ∞ in L 2 (Ω)
and thus in probability.

The proof is immediate. Processes satisfying these assumptions and having different values for lim N →∞ EW N are thus mutually singular, such as β-sine processes with different β ∈ {1, 2, 4}. Proof. The starting point is again Lemma 5.1. We note that in view of (3.32), (3.2) and (5.1), all the cluster functions for that process are exponentially decreasing when viewed as functions of pairwise distances between arguments. We start with the term (5.6). Replacing G N by 1 2π E N , using the translation invariance, and changing variables as before (y -z = u), we find E 1 (x -y) (E 1 (x -y)) -E 1 (x -y + u/N )) T 2 (u)dx dy du < η.

The two-dimensional case

I 2 + 2I 1 = 2N 2 π [0,
On the other hand, still in view of the definition (2.35)-(2.36), E 1 is uniformly continuous away from Z 2 , hence we may write, as N → ∞, Since this is true for any η, it follows that

I 2 + 2I 1 = o(N 2 ).
Similarly,

I 3 -I 1 = [0,1] 2 N x-[0,N ] 2 N y-[0,N ] 2 E 1 (x-y) E 1 x -y + u -v N -E 1 (x -y) T 1 (u)T 2 (v) dx dy du dv.
The same reasoning shows that this is o(N 2 ). For I 4 , the change of variables x ′ = x/N and y ′ = y/N yields

I 4 = -2N 2 [0,1] 2
E 1 (x -y) 2 T 2 (N (x -y)) dx dy.

We may take out a δ neighborhood of the diagonal and its translates by Z 2 , off of which E 1 (x -y) can be bounded by C log |x -y| and T 2 ( x-y N ) by e -CN 2 δ 2 . The whole term is thus o(N 2 ).

We turn to (5.9). From (5.3) and (3.32) we find that |T 3 (x, y, z)| ≤ e -C(|x-y| 2 +|y-z| 2 +|x-z| 2 ) . As above we have

I 5 = 4
[0,1] 3 E 1 (x -y)E 1 (x -z)T 2 (N x, N y, N z) dx dy dz and as above we may take out δ-neighborhoods |x -y| < δ or |x -z| < δ or |y -z| < δ (and their translates by Z 2 ), outside of which the E 1 terms are bounded by log's and T 3 by e -CN 2 δ 2 . The whole term is thus o(N 2 ).

A very similar reasoning applies to I 6 . Combining all these, we find the result.

Remark 5.3. In view of the proof above, the same result holds for any process such that the cluster functions decay sufficiently fast away from diagonals, for example exponentially.

Miscellaneous computations

In this section we gather various additional computations of expectations and additional facts.

Operations on processes

In this subsection, we examine the effect on lim N →∞ EW N of two common operations on independent processes: superposition and decimation (see [DVJ]).

Proposition 6.1. Let X 1 , • • • , X M be M independent translation invariant point processes with density 1 and two-point correlation functions ρ (i) 2

1≤i≤M , satisfying the assumptions of Theorem 1.

Assume that lim N →∞ EW N (X i ) < ∞ for i = 1, . . . , M . Let X denote the superposition of independent processes Xi , where Xi denotes the image of the process X i under the dilation by factor M of the line. Then, with obvious notation,

lim N →∞ EW N (X ) = log M + 1 M M i=1 lim N →∞
EW N (X i ).

Proof. Let T (i) 2 be the second cluster functions corresponding to X i . Let ρ(i) 2 now denote the second correlation function for the process Xi , which has density 1/M . We have ρ(i) 2 (x, y) =

The result follows easily using Theorem 1.

In dimension 2, reproducing the proof, but replacing the dilations by factor M by dilations by factor √ M , we obtain instead the result:

lim N →∞ EW N (X ) = 1 2 log M + 1 M M i=1 lim N →∞
EW N (X i ).

For example, superposing two independent processes with the same second correlation function leads to an increase of lim N →∞ EW N by log 2 in dimension 1 and 1 2 log 2 in dimension 2. Superposition can thus be seen as "increasing the disorder".

We next turn to decimation. One can define a 'random decimation' of a process by erasing points at random with probability 1/2. The second correlation function then transforms into R 2 (x, y) = 1 4 ρ 2 (x, y).

But the space needs to be rescaled by a factor 2 in order to maintain a density one, so the correlation function after that is ρ ′ 2 (x, y) = ρ 2 (2x, 2y).

It is clear in view of Theorems 1 and 2 that if a process has finite lim N →∞ EW N , its decimation will not, since the condition T 2 = 1 will be destroyed by this operation. On the other hand, one can define a 'deterministic decimation' by erasing every even (or odd) point of an (ordered) random point configuration followed by rescaling of the space to keep the density at 1. While we cannot say anything about this operation in general, one can observe what it does in a couple of cases.

It is known, see e.g. [START_REF] Anderson | An introduction to random matrices[END_REF]page 66], that the β = 2 sine process is the deterministic decimation of superposition of two β = 1 sine processes, or symbolically (sine β = 2) = decimation((sine β = 1) ⊔ (sine β = 1)).

From Proposition 3.2 we know that lim N →∞ EW N is 1 -γ for the left-hand side, and Proposition 6.1 says that lim N →∞ EW N is 2-γ for (sine β = 1)⊔(sine β = 1). Thus, the deterministic decimation decreased the value of lim N →∞ EW N by 1.

Similarly, (sine β = 4) = decimation((sine β = 1)), and we see that the decimation decreased the value of lim N →∞ EW N from 2 -γ -log 2 to 3 2 -γ -log 2.

Discrete β = 2 sine process

The β = 2 discrete sine process was first obtained in [BOO] as the bulk scaling limit of the Plancherel measure for symmetric groups, and it was shown in [BKMM] to be the a universal local scaling limit for a broad family of discrete probabilistic models of random matrix type with β = 2. The goal of this section is to compute lim N →∞ EW N for the suitably scaled discrete sine process embedded into the real line. By the construction, this provides an interpolation between the case of the perfect lattice, for which W N ≡ 0, and the case of the continuous β = 2 sine process treated in the previous sections. Let ρ ∈ (0, 1). The discrete sine process with density ρ is a random point process on Z with the correlation functions (k ≥ 1)

ρ k (x 1 , . . . , x k ) = det sin πρ(x i -x j ) π(x i -x j ) k i,j=1
. Proposition 6.2. Embed Z into R via n → ρn; this turns the discrete sine process of density ρ into a random point process on R with density 1. For the latter process we have Proof. For the calculation, we assume that N ρ is an integer (the same argument however should hold without this assumption by examining more carefully error terms). The calculation is

  denotes the integer part. But the function log |2 sin x| is stricly concave on [0, π].

Figure 1 :

 1 Figure 1: Splitting of the domain of integration

  = 1 -γ, hence the result. Case β = 1: Similarly to the case of β = 2, one checks that T 2 (v) = O(|v| -2 ). Indeed, note that π|v| 0 sin t t dt -π 2 = +∞ π|v| sin t t dt = cos(π|v|) π|v| -+∞ π|v| cos t t 2 = O(|v| -1

  |G N (x -y)| and |G N (y -z)| are bounded by O(log N ). It then remains to estimate [0,N ] 3 |T 3 (x, y, z)| dx dy dz.

Theorem 5 .

 5 For the determinantal random point process with kernel (3.32) we have lim N →∞ Var(W N ) = 0.

  1] 2 N y-[0,N ] 2 E N (N (x-y)) (E N (N (x -y)) -E N (N (x -y + u/N ))) T 2 (u)dx dy du.Noting that in view of the definition of E (cf. (2.35)-(2.36)) we have E N (N x) = E 1 (x) and from (3.31), E 1 behaves like C log |x| near x = 0 (and similarly near points of the lattice Z 2 ). Given η > 0 there thus exists δ > 0 such that [0,1] 2 ∩{|x-y-Z 2 |<δ} N y-[0,N ] 2

  2 \{|x-y-Z 2 |<δ} N y-[0,N ] 2 E 1 (x -y) (E 1 (x -y)) -E 1 (x -y + u/N )) T 2 (u)dx dy du = o(1).

  above, then lim N →∞ EW N exists and is finite if and only if

∞

-∞ T 2 (v) log |v| dv converges, and if so then

the choice of normalization in dimension 1 is based on (2.15)

hence by abuse of notation we write T2(x, y) = T2(xy) as for Theorem 1

Acknowledgements: We are very grateful to Peter Forrester for his idea of using circular ensembles for the heuristics above, and also for drawing our attention to [Ja]. A. B. thanks Alan Edelman for help with numerics and S.S. thanks C. Sinan Güntürk for helpful discussions related to Section 4. A. B. was partially supported by NSF grant DMS-1056390, S.S. was supported by a EURYI award. We also thank the MSRI for support to attend a workshop where this work was initiated.

then can be viewed as a discrete version of that of Theorem 1. First, by definition (2.53) of W N , we have

ρ 2 (i, j) log 2 sin πρ(i -j) N + log N and since ρ 2 (i, j) = ρ 2 -sin πρ(i-j) π(i-j)

2

we find

Let us first examine the contribution of the last sum. From the knowledge of W N (Z) = 0, we know that for K ∈ Z we have

Applying this to K = N/ρ we find that (6.4)

We next turn to the first two terms in (6.3). As in Theorem 1, we expect only the near diagonal terms to contribute, so that log 2 sin

Inserting this and (6.4) into (6.3) we find (6.5)

We first focus on

Indeed, we can bound

) and this multiplied by log N N is negligible as N → +∞. The last sum then appears as a Fourier series and can be computed explicitly, which leads to (6.7) We next turn to (6.8)

Again, the term containing u can be neglected since it is bounded by 1

. Combining (6.5)-(6.8) and letting N → ∞, we finally arrive at (6.2).

The graph of lim N →∞ EW N in (6.2) is presented in Fig. 6.2, it shows a function decreasing from 1 -γ at ρ = 0 to 0 at ρ = 1, as expected.