---
abstract: 'We study the Hamiltonian of a two-dimensional log-gas with a confining potential $V$ satisfying the weak growth assumption – $V$ is of the same order than $2\log|x|$ near infinity – considered by Hardy and Kuijlaars \[J. Approx. Theory, 170(0) : 44-58, 2013\]. We prove an asymptotic expansion, as the number $n$ of points goes to infinity, for the minimum of this Hamiltonian using the Gamma-Convergence method of Sandier and Serfaty \[Ann. Proba., to appear, 2015\]. We show that the asymptotic expansion as $n\to +\infty$ of the minimal logarithmic energy of $n$ points on the unit sphere in ${\mathbb R}^3$ has a term of order $n$ thus proving a long standing conjecture of Rakhmanov, Saff and Zhou \[Math. Res. Letters, 1:647-662, 1994\]. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff \[Contemp. Math., 578:31-61,2012\] about the value of this term and the conjecture of Sandier and Serfaty \[Comm. Math. Phys., 313(3):635-743, 2012\] about the minimality of the triangular lattice for a “renormalized energy" $W$ among configurations of fixed asymptotic density.'
author:
- |
    Laurent Bétermin\
    Etienne Sandier
bibliography:
- 'allbiblio.bib'
title: Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere
---

**AMS Classification:** Primary 52A40, 82B05 ; Secondary 41A60, 82B21, 31C20.\
**Keywords:** Coulomb gas ; Abrikosov lattices ; Triangular lattice ; Renormalized energy ; Crystallization ; Logarithmic energy ; Number theory ; Logarithmic potential theory ; Weak confinement ; Gamma-convergence ; Ginzburg-Landau ; Vortices.\

Introduction
============

Let $(x_1,...,x_n)\in({\mathbb R}^2)^n$ be a configuration of $n$ points interacting through a logarithmic potential and confined by an external field $V$. The Hamiltonian of this system, also known as a Coulomb gas, is defined as $$w_n(x_1,...,x_n):=-\sum_{i\neq j}^n\log|x_i-x_j|+n\sum_{i=1}^nV(x_i)$$ where $|\cdot|$ is the Euclidean norm in ${\mathbb R}^2$. The minimization of $w_n$ is linked to the following classical problem of logarithmic potential theory : find a probability measure $\mu_V$ on ${\mathbb R}^2$ which minimizes $$\label{IV}
I_V(\mu):=\iint_{{\mathbb R}^2\times {\mathbb R}^2}\left(\frac{V(x)}{2}+\frac{V(y)}{2}-\log|x-y|\right)d\mu(x)d\mu(y)$$ amongst all probability measures $\mu$ on ${\mathbb R}^2$. This type of problem dates back to Gauss. More recent references are the thesis of Frostman [@Frost] and the monography of E.Saff and V.Totik [@SaffTotik]. The usual assumptions on $V:{\mathbb R}^2\to{\mathbb R}\cup\{+\infty\}$ are that it is lower semicontinuous, that it is finite on a set of nonzero capacity, and that it satisfies the growth assumption $$\label{classicgrowth}
\lim_{|x|\to +\infty}\left\{V(x)-2\log| x|\right\}=+\infty.$$ These assumptions ensure that a unique minimizer $\mu_V$ of $I_V$ exists and that it has compact support.

Recently, Hardy and Kuijlaars [@Hardy2] (see also [@Hardy]) proved that if one replaces by the so-called weak growth assumption $$\label{growthL}
\liminf_{|x|\to +\infty}\left\{V(x)-\log(1+|x|^2)  \right\}>-\infty,$$ then $I_V$ still admits a unique minimizer, which may no longer have compact support. Moreover Bloom, Levenberg and Wielonsky [@Bloom:2014qy] proved that the classical Frostman type inequalities still hold in this case.\
Coming back to the minimum of the discrete energy $w_n$, its relation to the minimum of $I_V$ is that as $n\to +\infty$, the minimum of $w_n$ is equivalent to $n^2 \min I_V$. The next term in the asymptotic expansion of $w_n$ was derived by Sandier and Serfaty [@2DSandier] in the classical case , it reads $$\min w_n =  n^2 \min I_V - \frac{n}{2} \log n + \alpha_V n +o(n),$$ where $\alpha_V$ is related to the minimum of a Coulombian renormalized energy studied in [@Sandier_Serfaty] which quantifies the discrete energy of infinitely many positive charges in the plane screened by a uniform negative background. Note that rather strict assumptions in addition to need to be made on $V$ for this expansion to hold, but they are satisfied in particular if $V$ is smooth and strictly convex.

Here, we show that such an asymptotic formula still holds when the classical growth assumption is replaced with the weak growth assumption . However it is no longer obvious that the minimum of $w_n$ is achieved in this case, as the weak growth assumption could allows one point to go to infinity.

 \[thintro\] Let $V$ be an admissible potential [^1]. Then the following asymptotic expansion holds. $$\label{expansion}
\inf_{({\mathbb R}^2)^n} w_n=I_V(\mu_V)n^2-\frac{n}{2}\log n +\left(\frac{1}{\pi}\min_{\mathcal{A}_1}W-\frac{1}{2}\int_{{\mathbb R}^2}m_V(x)\log m_V(x)dx\right)n + o(n),$$ where $\mu_V = m_V(x)\,dx$ is the unique minimizer of $I_V$ (see Section \[RE\] for precise definitions of $W$ and $\mathcal{A}_1$.)

This result is proved using the methods in [@Sandier_Serfaty; @2DSandier] suitably adapted to equilibrium measures with possibly non-compact support together with the compactification approach in [@Hardy2; @Hardy; @Bloom:2014qy]. This compactification allows also to connect the discrete energy problem for log gases in the plane with the discrete logarithmic energy problem for finitely many points on the unit sphere $\S^2$ in the Euclidean space ${\mathbb R}^3$.\
The logarithmic energy of a configuration $(y_1,...,y_n)\in (\S^2)^n$ is given by $$\textnormal{E}_{\log}(y_1,...,y_n):=-\sum_{i\neq j}^n\log\|y_i-y_j\|,$$ where $\|\cdot\|$ is the Euclidean norm in ${\mathbb R}^3$. Finding a minimizer of such an energy functional is a problem with many links and ramifications as discussed in the fundamental paper of Saff and Kuijlaars [@PointsSphere] (see also [@BrauchGrab]). For instance Smale’s $7^{th}$ problem [@231301] is to find, for any $n\geq 2$, a universal constant $c\in{\mathbb R}$ and a nearly optimal configuration $(y_1,...,y_n)\in (\S^2)^n$ such that, letting $\mathcal{E}_{\log}(n)$ denote the minimum of $\textnormal{E}_{\log}$ on $(\S^2)^n$, $$\textnormal{E}_{\log}(y_1,...,y_n)-\mathcal{E}_{\log}(n)\leq c\log n.$$ Identifying the term of order $n$ in the expansion of $\mathcal{E}_{\log}(n)$ can be seen as a modest step towards a better understanding of this problem.\
It was known (lower bound by Wagner [@57451281] and upper bound by Kuijlaars and Saff [@403978]), that $$\left(\frac{1}{2}-\log 2\right)n^2-\frac{1}{2}n\log n+c_1 n\leq \mathcal{E}_{\log}(n)\leq \left(\frac{1}{2}-\log 2\right)n^2-\frac{1}{2}n\log n+c_2 n$$ for some fixed constant $c_1$ and $c_2$. Thus one can naturally ask for the existence of the limit $$\lim_{n\to +\infty}\frac{1}{n}\left[\mathcal{E}_{\log}(n)-\left(\frac{1}{2}-\log2\right)n^2+\frac{n}{2}\log n  \right].$$

**CONJECTURE 1** (Rakhmanov, Saff and Zhou, [@2400418]): There exists a constant $C$ not depending on $n$ such that $$\mathcal{E}_{\log}(n)=\left(\frac{1}{2}-\log2  \right)n^2-\frac{n}{2}\log n+ Cn+o(n) \quad \text{ as } n\to +\infty.$$

**CONJECTURE 2** (Brauchart, Hardin and Saff, [@Brauchart]): The constant $C$ in Conjecture 1 is equal to $C_{BHS}$, where $$\label{CBHSdef}
C_{BHS}:=2\log2 +\frac{1}{2}\log\frac{2}{3}+3\log\frac{\sqrt{\pi}}{\Gamma(1/3)}.$$

As we will see, our results imply that the last conjecture is equivalent to one concerning the global optimizer of the renormalized energy $W$.

**CONJECTURE 3** (Sandier and Serfaty, [@Sandier_Serfaty], or see the review by Serfaty [@Serfaty:2013fk]) : The triangular lattice is a global minimizer of $W$ among discrete subsets of ${\mathbb R}^2$ with asymptotic density one.

The expansion in the particular case $V(x)=\log(1+|x|^2)$ transported to $\S^2$ using an inverse stereographic projection and appropriate rescaling gives an expansion for $\mathcal{E}_{\log}(n)$ and thus proves Conjecture 1. The constant $C$ in Conjecture 1 can moreover be expressed in terms of the minimum of the renormalized energy $W$. The value of $W$ for the triangular lattice obviously provides an upper bound for this minimum, and by using the Chowla-Selberg formula to compute the expression given in [@Sandier_Serfaty] for this quantity, we show that this upper bound is precisely $C_{BHS}$. This bound is of course sharp if and only if Conjecture 3 is true. Thus we deduce from the following.

There exists $C\neq 0$ independent of $n$ such that, as $n\to +\infty$, $$\mathcal{E}_{\log}(n)=\left(\frac{1}{2}-\log2  \right)n^2-\frac{n}{2}\log n+Cn+o(n),\quad  C= \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\log2.$$ Moreover $\displaystyle C\leq C_{BHS}$ where $C_{BHS}$ is given in , and equality holds iff $\displaystyle \min_{\mathcal{A}_1}W$ is achieved for the triangular lattice of density one.

The plan of the paper is as follows. In Section \[RE\] we recall the definition of $W$ and some of its properties from [@Sandier_Serfaty]. In Section \[EPWP\] we recall results about existence, uniqueness and variational Frostman inequalities for $\mu_V$. Moreover, we give the precise definition of an admissible potential $V$. In Sections \[SpF\] and \[UsLem\] we adapt the method of [@Sandier_Serfaty] to the case of equilibrium measures with noncompact support. The expansion is proved in Section \[AEOTH\]. Finally in Section \[CTLEOTS\] we prove Conjecture 1 about the existence of $C$, the upper bound $C\leq C_{BHS}$ and the equivalence between Conjectures 2 and 3.

Renormalized Energy {#RE}
===================

Here we recall the definition of the renormalized energy $W$ (see [@2DSandier] for more details). For any $R>0$, $B_R$ denotes the ball centered at the origin with radius $R$.

Let $m$ be a nonnegative number and $E$ be a vector-field in ${\mathbb R}^2$. We say $E$ belongs to the **admissible class $\mathcal{A}_m$** if $$\label{div} {\mathop{\rm div}\nolimits}E=2\pi(\nu-m) \text{ and } {\mathop{\rm curl}\nolimits}E =0$$ where $\nu$ has the form $$\label{nu}
\nu=\sum_{p\in \Lambda}\delta_p \text{ , for some discrete set } \Lambda\subset{\mathbb R}^2,$$ and if $$\frac{\nu(B_R)}{|B_R|} \text{ is bounded by a constant independent of } R>1.$$

The real $m$ is the average density of the points of $\Lambda$ when $E\in\mathcal{A}_m$.

Let $m$ be a nonnegative number. For any continuous function $\chi$ and any vector-field $E$ in ${\mathbb R}^2$ satisfying where $\nu$ has the form we let $$\label{W}
W(E,\chi)=\lim_{\eta\to 0}\left(\frac{1}{2}\int_{{\mathbb R}^2\backslash\cup_{p\in\Lambda}B(p,\eta)}\chi(x) |E(x)|^2dx+\pi\log\eta\sum_{p\in \Lambda}\chi(p)   \right).$$

We use the notation $\chi_{B_R}$ for positive cutoff functions satisfying, for some constant $C$ independent of $R$

$$\label{chi}
|\nabla\chi_{B_R}|\leq C,\quad \text{Supp}(\chi_{B_R})\subset B_R,\quad \chi_{B_R}(x)=1 \text{ if } d(x,B_R^c)\geq 1.$$

where $d(x,A)$ is the Euclidean distance between $x$ and set $A$.

The **renormalized energy $W$** is defined, for $E\in \mathcal{A}_m$ and $\{\chi_{B_R}\}_R$ satisfying , by $$W(E)=\limsup_{R\to +\infty}\frac{W(E,\chi_{B_R})}{|B_R|}.$$

It is shown in [@Sandier_Serfaty Theorem 1] that the value of $W$ does not depend on the choice of cutoff functions satisfying , and that $W$ is bounded below and [*admits a minimizer over $\mathcal{A}_1$*]{}.

Moreover (see [@Sandier_Serfaty Eq. (1.9),(1.12)]), if $E\in\mathcal{A}_m$, $m>0$, then $$E'=\frac{1}{\sqrt{m}}E(./\sqrt{m})\in\mathcal{A}_1\quad \text{and} \quad W(E)=m\left(W(E')-\frac{\pi}{2}\log m  \right).$$ In particular $$\label{scaling}
\min_{\mathcal{A}_m} W=m\left( \min_{\mathcal{A}_1} W- \frac{\pi}{2}\log m  \right),$$ and $E$ is a minimizer of $W$ over $\mathcal{A}_m$ if and only if $E'$ minimizes $W$ over $\mathcal{A}_1$.

In the periodic case, we have the following result [@Sandier_Serfaty Theorem 2], which supports Conjecture 3 above:

\[Trioptimal\] The unique minimizer, up to rotation, of $W$ over Bravais lattices[^2] of fixed density $m$ is the triangular lattice $$\Lambda_m=\sqrt{\frac{2}{m\sqrt{3}}}\left({\mathbb Z}(1,0)\oplus {\mathbb Z}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\right).$$

This is proved in [@Sandier_Serfaty] using the result of Montgomery on minimal theta function [@Mont], we provide below an alternative proof.

For any Bravais lattice $\Lambda$ $$W(\Lambda)=ah(\Lambda)+b,$$ where $a>0$, $b\in{\mathbb R}$ and $h(\Lambda)$ is the height of the flat torus ${\mathbb C}/\Lambda$ (see [@OPS; @Chiu; @CoulLazzarini] for more details).

Indeed, Osgood, Phillips and Sarnak [@OPS Section 4, page 205] proved, for $\Lambda={\mathbb Z}\oplus \tau{\mathbb Z}$, $\tau=a+ib$, that $$h(\Lambda)=-\log(b|\eta(\tau)|^4)+C, \quad C\in {\mathbb R},$$ where $\eta$ is the Dedekind eta function[^3]. But from [@Sandier_Serfaty] we have $$W(\Lambda)=-\frac{1}{2}\log\left(\sqrt{2\pi b}|\eta(\tau)|^2\right)+C,$$ therefore $W(\Lambda)=ah(\Lambda)+b$.

Then from [@OPS Corollary 1(b)], the triangular lattice minimizes $h$ among Bravais lattices with fixed density, hence the same is true for $W$.

Equilibrium Problem in the Whole Plane {#EPWP}
======================================

In this section we recall results on existence, uniqueness and characterization of the equilibrium measure $\mu_V$ and we give the definition of the admissible potentials.

Equilibrium measure, Frostman inequalities and differentiation of $U^{\mu_V}$
-----------------------------------------------------------------------------

[([@Bloom:2014qy])]{.nodecor} Let $K\subset {\mathbb R}^2$ be a compact set and let $\mathcal{M}_1(K)$ be the family of probability measures supported on $K$. Then the **logarithmic potential** and the **logarithmic energy** of $\mu\in \mathcal{M}_1(K)$ are defined as $$U^{\mu}(x):=-\int_{K}\log|x-y|d\mu(y) \quad \text{and} \quad I_0(\mu):=-\iint_{K\times K} \log|x-y|d\mu(x)d\mu(y).$$ We say that $K$ is **log-polar** if $I_0(\mu)=+\infty$ for any $\mu\in \mathcal{M}_1(K)$ and we say that a Borel set $E$ is log-polar if every compact subset of $E$ is log-polar. Moreover, we say that an assertion holds **quasi-everywhere** (q.e.) on $A\subset {\mathbb R}^2$ if it holds on $A\backslash P$ where $P$ is log-polar.

We recall that the Lebesgue measure of a log-polar set is zero.

Now we recall results about the existence, the uniqueness and the characterization of the equilibrium measure $\mu_V$ proved in [@Frost; @SaffTotik] for the classical growth assumption , and by Hardy and Kuijlaars [@Hardy; @Hardy2] (for existence and uniqueness) and Bloom, Levenberg and Wielonsky [@Bloom:2014qy] (for Frostman type variational inequalities) for weak growth assumption .

\[equi\][([@Frost; @SaffTotik; @Hardy; @Hardy2; @Bloom:2014qy])]{.nodecor} Let $V$ be a lower semicontinuous function on ${\mathbb R}^2$ such that $\{x\in {\mathbb R}^2 ; V(x)<+\infty \}$ is a non log-polar subset of ${\mathbb R}^2$ satisfying $$\liminf_{|x|\to +\infty}\{V(x)-\log(1+|x|^2) \}>-\infty.$$ Then we have :

1.  $\displaystyle \inf_{\mu \in \mathcal{M}_1({\mathbb R}^2)}I_V(\mu)$ is finite, where $I_V$ is given by .

2.  There exists a unique equilibrium measure $\mu_V\in \mathcal{M}_1({\mathbb R}^2)$ with $$I_V(\mu_V)=\inf_{\mu \in \mathcal{M}_1({\mathbb R}^2)}I_V(\mu)$$ and the logarithmic energy $I_0(\mu_V)$ is finite.

3.  The support $\Sigma_V$ of $\mu_V$ is contained in $\{x\in {\mathbb R}^2 ; V(x)<+\infty\}$ and $\Sigma_V$ is not log-polar.

4.  Let $$\label{cvdef}
     c_V:=I_V(\mu_V)-\int_{{\mathbb R}^2}\frac{V(x)}{2}d\mu_V(x)$$ denote the Robin constant. Then we have the following Frostman variational inequalities : $$\begin{aligned}
     \label{Frost1}& U^{\mu_V}(x)+\frac{V(x)}{2}\geq c_V \quad \text{q.e. on } {\mathbb R}^2,\\
    \label{Frost2} & U^{\mu_V}(x)+\frac{V(x)}{2}\leq c_V \quad \text{for all } x\in\Sigma_V.\end{aligned}$$

In particular we have $\displaystyle U^{\mu_V}(x)+\frac{V(x)}{2}=c_V$ q.e. on $\Sigma_V$.

As explained in [@Hardy], the hypothesis of Theorem \[equi\] can be usefully transported to the sphere $\mathcal{S}$ in ${\mathbb R}^3$ centred at $(0,0,1/2)$ with radius $1/2$, by the inverse stereographic projection $T:{\mathbb R}^2\to \mathcal{S}$ defined by $$T(x_1,x_2)=\left(\frac{x_1}{1+|x|^2}, \frac{x_2}{1+|x|^2},\frac{|x|^2}{1+|x|^2} \right), \text{for any } x=(x_1,x_2)\in{\mathbb R}^2.$$ We know that $T$ is a conformal homeomorphism from ${\mathbb R}^2$ to $\mathcal{S}\backslash \{N\}$ where $N :=(0,0,1)$ is the North pole of $\mathcal{S}$.

The procedure is as follows: Given $V:{\mathbb R}^2\to{\mathbb R}$, we may define (see [@Hardy]) $\mathcal V:\mathcal S\to {\mathbb R}$ by letting $$\label{transport}\mathcal V(T(x)) = V(x) - \log(1+|x|^2), \quad \mathcal V(N) = \liminf_{|x|\to +\infty} \{V(x) - \log(1+|x|^2)\}.$$ Then (see [@Hardy]), $V$ satisfies the hypothesis of Theorem \[equi\] if and only if $\mathcal V$ is a lowersemicontinuous function on $\mathcal S$ which is finite on a nonpolar set. Therefore, in this case, the minimum of $$I_{\mathcal V}(\mu) := \iint_{\mathcal S\times\mathcal S}\left( -\log\|x-y\| +\frac{\mathcal V(x)}2 + \frac{\mathcal V(y)}2 \right)\,d\mu(x)\,d\mu(y)$$ among probability measures on $\mathcal S$ is achieved. Here $\|x-y\|$ denotes the euclidean norm in ${\mathbb R}^3$. Moreover, still from [@Hardy], the minimizer $\mu_{\mathcal V}$ is related to $\mu_V$ by the following relation $$\label{pullback}\mu_{\mathcal V} = T\sharp\mu_V,$$ where $T\sharp\mu$ denotes the push-forward of the measure $\mu$ by the map $T$.

\[ad\]

\[admissiblef\] We say that $V:{\mathbb R}^2\to {\mathbb R}$ is **admissible** if it is of class $C^3$ and if, defining $\mathcal V$ as above,

1.  [**(H1)**]{.nodecor} : The set $\{x\in {\mathbb R}^2 ; V(x)<+\infty\}$ is not log-polar and $\displaystyle \liminf_{|x|\to +\infty}\{V(x)-\log(1+|x|^2)  \}>-\infty.$

2.  [**(H2)**]{.nodecor} : The equilibrium measure $\mu_{\mathcal V}$ is of the form $m_{\mathcal V}(x)\indic_{\Sigma_{{\mathcal V}}}(x) \,dx$, where $m_{\mathcal V}$ is a $C^1$ function on $\mathcal S$ and $dx$ denotes the surface element on $\mathcal S$, where the function $m_{\mathcal V}$ is bounded above and below by positive constants $\bm$ and $\mb$, and where $\Sigma_{{{\mathcal V}}}$ is a compact subset of ${{\mathcal S}}$ with $C^1$ boundary.

Using (H2) and , we find that $$d\mu_V(x) = m_V(x) \indic_{\Sigma_V} \,dx,$$ where $\Sigma_V = T^{-1}(\Sigma_{{\mathcal V}})$ and $$\label{mv}m_V(x) = \frac{m_{\mathcal V}(T(x))}{(1+|x|^2)^2}.$$ Note that $(1+|x|^2)^{-2}$ is the jacobian of the transformation $T$.

Splitting Formula {#SpF}
=================

Assume $V$ is admissible. We define as in [@2DSandier] the blown-up quantities: $$x'=\sqrt{n}x, \qquad m_V'(x')=m_V(x), \qquad d\mu_V'(x')=m_V'(x')dx'$$ and we define $$\label{zetadefinition}
\displaystyle \zeta(x):=U^{\mu_V}(x)+\frac{V(x)}{2}-c_V,$$ where $c_V$ is the Robin constant given in . Then by and , $\zeta(x)=0$ q.e. in $\Sigma_V$ and $\zeta(x)\geq 0$ q.e. in ${\mathbb R}^2\backslash \Sigma_V$.\
For $(x_1,...,x_n)\in ({\mathbb R}^2)^n$, we define $\displaystyle \nu_n=\sum_{i=1}^n\delta_{x_i}$ and $$H_n:=-2\pi \Delta^{-1}(\nu_n-n\mu_V)=-\int_{{\mathbb R}^2}\log|.-y|d(\nu_n-n\mu_V)(y)=-\sum_{i=1}^n\log|.-x_i|-n U^{\mu_V}$$ where $\Delta^{-1}$ is the convolution operator with $\frac{1}{2\pi}\log|\cdot|$, hence such that $\Delta\circ\Delta^{-1}=I_2$ where $\Delta$ denotes the usual laplacian. Moreover we set $$\label{hnp}
 H_n':=-2\pi\Delta^{-1}(\nu_n'-\mu_V')$$ where $\displaystyle \nu_n'=\sum_{i=1}^n\delta_{x_i'}$.

\[convHn\] Let $V$ be an admissible potential. Then we have $$\lim_{R\to +\infty} \int_{B_R} H_n(x)d\mu_V(x)=\int_{{\mathbb R}^2} H_n(x)d\mu_V(x) \quad \text{ and } \quad \lim_{R\to +\infty} W(\nabla H_n,\indic_{B_R})=W(\nabla H_n,\indic_{{\mathbb R}^2}).$$

From and the bounds on $m_{{\mathcal V}}$ we get[^4] $$\label{intlogfinite}
\int_{{\mathbb R}^2}\log(1+|y|)d\mu_V(y)<+\infty.$$ Therefore by the dominated convergence argument given in [@Mizuta Theorem 9.1, Chapter 5] (used for the continuity of $U^{\mu_V}$), we have $$\label{Hn}
 H_n(x)=\sum_{i=1}^n\int_{{\mathbb R}^2} \log\left( \frac{|x-y|}{|x-x_i|} \right)d\mu_V(y).$$ It follows that $H_n(x)=O\left( |x|^{-1}\right)$ as $|x|\to +\infty$ which implies the first equality using .

The second equality follows from the dominated convergence argument of Mizuta in [@MizutaContRiesz Theorem 1], because from we have $\nabla H_n(x)=O(|x|^{-2})$ as $|x|\to +\infty$ and thus $|\nabla H_n|^2$ in in $L^1({\mathbb R}^2)$.

Let $V$ be admissible. Then, for every configuration $(x_1,...x_n)\in ({\mathbb R}^2)^n$, $n\geq 2$, we have $$w_n(x_1,...,x_n)=n^2I_V(\mu_V)-\frac{n}{2}\log n +\frac{1}{\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})+2n\sum_{i=1}^n\zeta(x_i). \label{splitting}$$

We may proceed as in the proof of [@2DSandier Lemma 3.1] and make use of the Frostman type inequalities and and Lemma \[convHn\]. The important point is that, as shown in the proof of the previous lemma, we have $H_n(x)=O(|x|^{-1})$ and $\nabla H_n(x)=O(|x|^{-2})$ as $|x|\to +\infty$ which implies, exactly like in the compact support case, that $$\lim_{R\to +\infty} \int_{\partial B_R} H_n(x)\nabla H_n(x).\vec{\nu}(x)dx=0$$ where $\vec{\nu}(x)$ is the outer unit normal vector at $x\in \partial B_R$.

Lower bound {#UsLem}
===========

Here we follow the strategy of [@2DSandier], pointing out the required modifications in the noncompact case.

Mass spreading result and modified density $g$
----------------------------------------------

We have the following result from [@2DSandier Proposition 3.4]:

Let $V$ be admissible and assume $(\nu,E)$ are such that $\displaystyle \nu=\sum_{p\in\Lambda}\delta_p$ for some finite subset $\Lambda \subset {\mathbb R}^2$ and ${\mathop{\rm div}\nolimits}E=2\pi(\nu -m_V)$, ${\mathop{\rm curl}\nolimits}E=0$ in ${\mathbb R}^2$. Then, given any $\rho>0$ there exists a signed measure $g$ supported on ${\mathbb R}^2$ and such that:

-   there exists a family $\mathcal{B}_\rho$ of disjoint closed balls covering Supp$(\nu)$, with the sum of the radii of the balls in $\mathcal{B}_\rho$ intersecting with any ball of radius 1 bounded by $\rho$, and such that $$g(A)\geq -C(\|m_V\|_\infty+1)+\frac{1}{4}\int_A|E(x)|^2\indic_{\Omega\backslash\mathcal{B}_\rho}(x)dx, \quad  \text{ for any } A\subset{\mathbb R}^2,$$ where $C$ depends only on $\rho$;\

-   we have $$dg(x)=\frac{1}{2}|E(x)|^2dx \quad \text{ outside } \bigcup_{p\in\Lambda}B(p,\lambda)$$ where $\lambda$ depends only on $\rho$;\

-   for any function $\chi$ compactly supported in ${\mathbb R}^2$ we have $$\label{distW}
    \left|W(E,\chi)-\int\chi dg  \right|\leq CN(\log N +\|m_V\|_\infty)\|\nabla\chi\|_\infty$$ where $N=\string #\{p\in\Lambda ; B(p,\lambda)\cap Supp(\nabla\chi)\neq\emptyset  \}$ for some $\lambda$ and $C$ depending only on $\rho$;\

-   for any $U\subset \Omega$ $$\label{points} \string #(\Lambda \cap U)\leq C(1+\|m_V\|^2_\infty|\hat{U}|+g(\hat{U}))$$ where $\hat{U}:=\{x\in {\mathbb R}^2 ; d(x,U)<1\}$.

\[massdefi\] Assume $\displaystyle \nu_n=\sum_{i=1}^n\delta_{x_i}$. Letting $\displaystyle \nu'_n=\sum_{i=1}^n\delta_{x_i'}$ be the measure in blown-up coordinates and $E_{\nu_n}=\nabla H'_n$, we denote by $g_{\nu_n}$ the result of applying the previous proposition to $(\nu'_n,E_{\nu_n})$.

The following result [@2DSandier Lemma 3.7] connects $g$ and the renormalized energy.

[([@2DSandier])]{.nodecor} For any $\displaystyle \nu_n=\sum_{i=1}^n\delta_{x_i}$, we have $$\label{Wg} W(\nabla H'_n,\indic_{{\mathbb R}^2})=  \int_{{\mathbb R}^2}dg_{\nu_n}.$$

Ergodic Theorem
---------------

We adapt the abstract setting in [@2DSandier Section 4.1]. We are given a Polish space $X$, which is a space of functions, on which ${\mathbb R}^2$ acts continuously. We denote this action $(\lambda, u)\to \theta_\lambda u:=u(.+\lambda)$, for any $\lambda\in{\mathbb R}^2$ and $u\in X$. We assume it is continuous with respect to both $\lambda$ and $u$.

We also define $T^\varepsilon_\lambda$ and $T_\lambda$ acting on ${\mathbb R}^2\times X$, by $T^\varepsilon_\lambda(x,u):=(x+\varepsilon\lambda,\theta_\lambda u)$ and $T_\lambda(x,u):=(x,\theta_\lambda u)$.

For a probability measure $P$ on ${\mathbb R}^2\times X$ we say that $P$ is $T_{\lambda(x)}$-invariant if for every function $\lambda$ of class $C^1$, it is invariant under the mapping $(x,u)\mapsto (x,\theta_{\lambda(x)}u)$.

We let $\{f_\ep\}_\ep$, and $f$ be measurable functions defined on ${\mathbb R}^2\times X$ which satisfy the following properties. For any sequence $\{x_\varepsilon,u_\varepsilon\}_\varepsilon$ such that $x_\varepsilon \to x$ as $\varepsilon\to 0$ and such that for any $R>0$, $$\limsup_{\varepsilon \to 0} \int_{B_R}f_\varepsilon(x_\varepsilon+\varepsilon\lambda,\theta_\lambda u_\varepsilon)d\lambda <+\infty,$$ we have

1.  (Coercivity) $\{u_\varepsilon\}_\varepsilon$ has a convergent subsequence;

2.  ($\Gamma$-liminf) If $\{u_\varepsilon\}_\varepsilon$ converge to $u$, then $\displaystyle \liminf_{\varepsilon\to 0}f_\varepsilon (x_\varepsilon, u_\varepsilon)\geq f(x,u)$.

In contrast with the compact case we do not have the convergence of $\{x_\ep\}$.

Now let $V$ be an admissible potential on ${\mathbb R}^2$ and $\mu_V$ its associated equilibrium measure. We have

\[erg\] Let $V$, $X$, $(f_\varepsilon)_\varepsilon$ and $f$ be as above. We define $$\displaystyle F_\varepsilon(u):=\int_{{\mathbb R}^2}f_\varepsilon(x,\theta_{\frac{x}{\varepsilon}}u)d\mu_V(x)$$ Assume $(u_\varepsilon)_\varepsilon\in X$ is a sequence such that $F_\varepsilon(u_\varepsilon)\leq C$ for any $\varepsilon>0$. Let $P_\varepsilon$ be the image of $\mu_V$ by $x\mapsto (x,\theta_{\frac{x}{\varepsilon}}u_\varepsilon)$, then:

1.  $(P_\varepsilon)_\varepsilon$ admits a convergent subsequence to a probability measure $P$,

2.  the first marginal of $P$ is $\mu_V$,

3.  $P$ is $T_{\lambda(x)}$-invariant,

4.  for $P-a.e.$ $(x,u)$, $(x,u)$ is of the form $\displaystyle\lim_{\varepsilon\to 0}(x_\varepsilon,\theta_{\frac{x_\varepsilon}{\varepsilon}}u_\varepsilon)$,

5.  $\displaystyle \liminf_{\varepsilon\to 0} F_\varepsilon(u_\varepsilon)\geq \int_{{\mathbb R}^2\times X} f(x,u) dP(x,u)$.

6.  Moreover we have $$\int_{{\mathbb R}^2\times X} f(x,u)dP(x,u)=\int_{{\mathbb R}^2\times X}\left(\lim_{R\to +\infty}\dashint_{B_R}f(x,\theta_\lambda u)d\lambda \right)dP(x,u). \label{Ergoeq}$$ where $\dashint_{B_R}$ denote the integral average over $B_R$.

The proof follows [@Sandier_Serfaty; @2DSandier] but with $\mu_V$ replacing the normalized Lebesgue measure on a compact set $\Sigma_V$. We sketch it and detail the parts where modifications are needed. For any $R>0$ we let $\mu_V^R$ denote the restriction of $\mu_V$ to $B_R$, and $ P_\ep^R$ denote the image of $\mu_V^R$ by the map $x\mapsto (x,\theta_{\frac{x}{\varepsilon}}u_\varepsilon)$.

**STEP 1: Convergence of a subsequence of $(P_\varepsilon)$ to a probability measure $P$.** It suffices to prove that the sequence $\{P_\ep\}_\ep$ is tight. From [@Sandier_Serfaty; @2DSandier], which deals with the compact case, $\{P_\ep^R\}_\ep$ is tight, for any $R>0$.

Now take any $\delta>0$, we need to prove that there exists a compact subset $K_\delta$ of ${\mathbb R}^2\times X$ such that $P_\ep(K_\delta)>1-\delta$ for any $\ep>0$. For this we choose first $R>0$ large enough so that $(\mu_V - \mu_V^R)({\mathbb R}^2)<\delta/2$. This implies that $P_\ep^R$ has total measure at least $1-\delta/2$ and then we may use the tightness of $\{P_\ep^R\}_\ep$ to find that there exists a compact set $K_\delta$ such that $P_\ep^R(K_\delta)>1-\delta$. It follows that $P_\ep(K_\delta)>1-\delta$, and then that $\{P_\ep\}_\ep$ is tight.

**STEP 2: $P$ is $T_{\lambda(x)}$-invariant.** Let $\lambda$ be a function of class $C^1$ on ${\mathbb R}^2$, $\Phi$ be a bounded continuous function on ${\mathbb R}^2\times X$ and $P_\lambda$ be the image of $P$ by $(x,u)\mapsto (x,\theta_{\lambda(x)}u)$. By the change of variables $y=\varepsilon\lambda(x)+x=(\varepsilon\lambda+I_2)(x)$, we obtain $$\begin{aligned}
\int_{{\mathbb R}^2\times X}\Phi(x,u)dP_\lambda(x,u) &=\int_{{\mathbb R}^2\times X}\Phi(x,\theta_{\lambda(x)}u)dP(x,u)\\
&= \lim_{\varepsilon\to 0}\int_{{\mathbb R}^2\times X}\Phi(x,\theta_{\lambda(x)}u)dP_\varepsilon(x,u)\\
&= \lim_{\varepsilon\to 0}\int_{{\mathbb R}^2}\Phi(x,\theta_{\lambda(x)+\frac{x}{\varepsilon}}u_\varepsilon)d\mu_V(x)\\
&= \lim_{\varepsilon\to 0}\int_{{\mathbb R}^2}\Phi(x,\theta_{\frac{\varepsilon\lambda(x)+x}{\varepsilon}}u_\varepsilon)d\mu_V(x)\\
&= \lim_{\varepsilon\to 0}\int_{{\mathbb R}^2}\frac{\Phi\left((\varepsilon\lambda+I_2)^{-1}(y),\theta_{\frac{y}{\varepsilon}}u_\varepsilon  \right)m_V((\varepsilon\lambda+I_2)^{-1}(y))dy}{|\det(I_2+\varepsilon D\lambda((I_2+\varepsilon\lambda)^{-1}(y)))|},\end{aligned}$$ where $D\lambda$ is the differential of $\lambda$.

From the boundedness of $\Phi$ and the decay properties of $m_V$ (see ) it is straightforward to show that $$\begin{aligned}
\lim_{\varepsilon\to 0}\int_{{\mathbb R}^2} & \frac{\Phi\left((\varepsilon\lambda+I_2)^{-1}(y),\theta_{\frac{y}{\varepsilon}}u_\varepsilon  \right)m_V((\varepsilon\lambda+I_2)^{-1}(y))dy}{|\det(I_2+\varepsilon D\lambda((I_2+\varepsilon\lambda)^{-1}(y)))|} \\
&= \lim_{\varepsilon\to 0}\int_{{\mathbb R}^2}\Phi\left((\varepsilon\lambda+I_2)^{-1}(y),\theta_{\frac{y}{\varepsilon}}u_\varepsilon  \right) m_V(y)\,dy\\
&=\lim_{\varepsilon\to 0}\int_{{\mathbb R}^2\times X}\Phi\left((\varepsilon\lambda+I_2)^{-1}(y),u\right)dP_\varepsilon(y,u).\end{aligned}$$ Then, arguing as in [@2DSandier] using the tightness of $(P_\varepsilon)_\varepsilon$ we obtain $$\begin{aligned}
\lim_{\varepsilon\to 0}\int_{{\mathbb R}^2\times X}\Phi\left((\varepsilon\lambda+I_2)^{-1}(y),u\right)dP_\varepsilon(y,u)&=\lim_{\varepsilon\to 0}\int_{{\mathbb R}^2\times X}\Phi(y,u)dP_\varepsilon(y,u)\\
&=\int_{{\mathbb R}^2\times X}\Phi(x,u)dP(x,u),\end{aligned}$$ Which concludes the proof that $\displaystyle \int_{{\mathbb R}^2\times X}\Phi(x,u)dP_\lambda(x,u) = \int_{{\mathbb R}^2\times X}\Phi(x,u)dP(x,u)$, i.e. that $P$ is $T_{\lambda(x)}$-invariant.

Items 2 and 4 in the theorem are obvious consequences of the definition of $P$ and items $5$ and $6.$ require no modification from [@2DSandier]. We have proved above items 1 and 3.

Asymptotic Expansion of the Hamiltonian {#AEOTH}
=======================================

We define $$\label{alpha}\alpha_V:=\frac{1}{\pi}\int_{{\mathbb R}^2}\min_{\mathcal{A}_{m_V(x)}}Wdx=\frac{1}{\pi}\min_{\mathcal{A}_1}W-\frac{1}{2}\int_{{\mathbb R}^2}m_V(x)\log m_V(x)dx,$$ where the equality is a consequence of . The fact that $\alpha_V$ is finite follows from , which ensures that the integral converges.

We also let $$F_n(\nu)=
\left\{
\begin{array}{ll}
\displaystyle \frac{1}{n}\left(\frac{1}{\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})+2n\int \zeta d\nu  \right) &\mbox{if $\nu$ is of the form $\displaystyle \frac1n\sum_{i=1}^n\delta_{x_i}$,}\\
+\infty &\mbox{otherwise.}
\end{array}
\right.$$ For $\displaystyle \nu_n'=\sum_{i=1}^n\delta_{x_i'}$, let $E_{\nu_n}$ be a solution of ${\mathop{\rm div}\nolimits}{E_{\nu_n}}=2\pi(\nu_n'-m_V')$, ${\mathop{\rm curl}\nolimits}{E_{\nu_n}}=0$ and we set $$P_{\nu_n}:=\int_{{\mathbb R}^2}\delta_{(x,E_{\nu_n}(x\sqrt{n}+.))}d\mu_V(x).$$ The following result extends [@2DSandier Theorem 2] to a class of equilibrium measures with possibly unbounded support, which requires a restatement which makes it slightly different from its counterpart in [@2DSandier]. It is essentially a Gamma-Convergence (see [@BraidesGamma]) statement, consisting of a lower bound and an upper bound, the two implying the convergence of $\displaystyle \frac{1}{n}\left[w_n(x_1,...,x_n)-n^2I_V(\mu_V)+\frac{n}{2}\log n\right]$ to $\alpha_V$ for a minimizer $(x_1,...,x_n)$ of $w_n$.

Main result
-----------

\[THM\] Let $1<p<2$ and $X={\mathbb R}^2\times L_{loc}^p({\mathbb R}^2,{\mathbb R}^2)$. Let $V$ be an admissible function.\
[**A. Lower bound:**]{.nodecor} Let $(\nu_n)_n$ such that $F_n(\nu_n)\leq C$, then:

1.  $P_{\nu_n}$ is a probability measure on $X$ and admits a subsequence which converges to a probability measure $P$ on $X$,

2.  the first marginal of $P$ is $\mu_V$,

3.  $P$ is $T_{\lambda(x)}$-invariant,

4.  $E\in \mathcal{A}_{m_V(x)}$ $P$-a.e.,

5.  we have the lower bound $$\label{LB}
     \liminf_{n\to +\infty}\frac{1}{n\pi}W(\nabla H_n',\indic_{{\mathbb R}^2}) \geq \frac{1}{\pi}\int_{{\mathbb R}^2} \frac{W(E)}{m_V(x)}dP(x,E)\geq \alpha_V.$$

[**B. Upper bound.**]{.nodecor} Conversely, assume $P$ is a $T_{\lambda(x)}$-invariant probability measure on $X$ whose first marginal is $\mu_V$ and such that for $P$-almost every $(x,E)$ we have $E\in \mathcal{A}_{m_V(x)}$. Then there exist a sequence $\{\nu_n=\sum_{i=1}^n\delta_{x_i}\}_n$ of measures on ${\mathbb R}^2$ and a sequence $\{E_n\}_n$ in $L_{loc}^p({\mathbb R}^2,{\mathbb R}^2)$ such that ${\mathop{\rm div}\nolimits}E_n=2\pi(\nu'_n-m_V')$ and such that, defining $\displaystyle P_n=\int_{{\mathbb R}^2}\delta_{(x,E_n(x\sqrt{n}+.))}d\mu_V(x)$, we have $P_n \to P$ as $n\to +\infty$ and $$\label{UB}
\limsup_{n\to +\infty} F_n(\nu_n)\leq \frac{1}{\pi}\int_{{\mathbb R}^2} \frac{W(E)}{m_V(x)}dP(x,E).$$ [**C. Consequences for minimizers.**]{.nodecor} If $(x_1,...,x_n)$ minimizes $w_n$ for every $n$ and measure $\displaystyle \nu_n=\sum_{i=1}^n\delta_{x_i}$, then:

1.  for $P$-almost every $(x,E)$, $E$ minimizes $W$ over $\mathcal{A}_{m_V(x)}$;

2.  we have $$\label{minasymp}
    \lim_{n\to +\infty} F_n(\nu_n)=\lim_{n\to +\infty} \frac{1}{n\pi}W(\nabla H_n',\indic_{{\mathbb R}^2}) = \frac{1}{\pi}\int_{{\mathbb R}^2} \frac{W(E)}{m_V(x)}dP(x,E)= \alpha_V,$$ hence we obtain the following asymptotic expansion, as $n\to +\infty$: $$\label{asymptotic} \min_{({\mathbb R}^{2})^n} w_n=I_V(\mu_V)n^2-\frac{n}{2}\log n +\alpha_V n + o(n).$$

Proof of the lower bound
------------------------

We follow the same lines as in [@2DSandier Section 4.2]. Because $F_n(\nu_n)\leq C$ and , we have that $$\frac1{n^2} w_n(x_1,\dots x_n)\to I_V(\mu_V),$$ therefore $\nu_n$ converges to $\mu_V$ (this follows from the results in [@Hardy]).

We let $\nu'_n = \sum_i \delta_{x_i'}$, and $E_n$, $H_n'$, $g_n$ be as in Definition \[massdefi\].

Let $\chi$ be a $C^\infty$ cutoff function with support the unit ball $B_1$ and integral equal to 1. We define $$\mathbf{f}_n(x,\nu,E,g):=
\left\{ 
       \begin{array}{ll}
       \displaystyle\frac{1}{\pi}\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V(x)}dg(y) & \mbox{if $(\nu,E,g)=\theta_{\sqrt{n}x}({\nu}'_n,{E}_n,{g}_n)$,} \\
       +\infty & \mbox{otherwise.}
       \end{array}
       \right.$$ As in [@2DSandier Section 4.2, Step 1], if we let $$\label{Fn}
\mathbf{F}_n(\nu,E,g):=\int_{{\mathbb R}^2}\mathbf{f}_n\left(x, \theta_{x\sqrt{n}}(\nu,E,g)\right)d\mu_V(x),$$ then $$\begin{aligned}
\mathbf F_n(\nu_n', E_n,g_n) 
= \int_{{\mathbb R}^2}\frac{1}{\pi}\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V(x)}d(\theta_{x\sqrt{n}}\sharp g)d\mu_V(x) &= 
\frac{1}{\pi}\int_{{\mathbb R}^2}\int_{{\mathbb R}^2}\chi(y-x\sqrt{n})dxd{g}_n(y)\\
&\leq \frac{1}{n\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})+\frac{{g}_n^{-}(U^c)}{n\pi},\end{aligned}$$ by , where $\displaystyle U=\{x':d(x',{\mathbb R}^2\backslash \Sigma)\geq 1\}$. As in [@2DSandier], we have ${g}_n^{-}(U^c)=o(n)$. Hence, if $(\nu,E,g)=({\nu}'_n,{E}_n,{g}_n)$, as $n\to +\infty$: $$\mathbf{F}_n(\nu,E,g)\leq \frac{1}{n\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})+o(1),$$ and $\mathbf{F}_n(\nu,E,g)=+\infty$ otherwise.\
Now, as in [@2DSandier], we want to use Theorem \[erg\] with $\varepsilon=\frac{1}{\sqrt{n}}$ and $X=\mathcal{M}_+\times L_{loc}^p({\mathbb R}^2,{\mathbb R}^2)\times \mathcal{M}$ where $p\in]1,2[$, $\mathcal{M}_+$ is the set of nonnegative Radon measures on ${\mathbb R}^2$ and $\mathcal{M}$ the set of Radon measures bounded below by $-C_V:=-C(\|m_V\|_\infty^2+1)$. Let $Q_n$ be the image of $\mu_V$ by $x\mapsto (x,\theta_{x\sqrt{n}}(\bar{\nu}'_n,{E}_n,{g}_n))$. We have:\
1) The fact that $\textbf{f}_n$ is coercive is proved as in [@2DSandier Lemma 4.4]. Indeed, if $(x_n,\nu(n),E(n),g(n))_n$ is such that $x_n\to x$ and, for any $R>0$, $$\limsup_{n\to +\infty}\int_{B_R}\textbf{f}_n\left(x_n+\frac{\lambda}{\sqrt{n}},\theta_\lambda(\nu(n),E(n),g(n))\right)d\lambda <+\infty,$$ then the integrand is bounded for a.e. $\lambda$. By assumption on $\textbf{f}_n$, for any $n$, $$\theta_\lambda(\nu(n),E(n),g(n))=\theta_{x_n\sqrt{n}+\lambda}({\nu}_n',{E}_n,{g}_n),$$ hence it follows that $$(\nu(n),E(n),g(n))=\theta_{x_n\sqrt{n}}({\nu}_n',{E}_n,{g}_n).$$ For any $R>0$, there exists $C_R>0$ such that for any $n>0$, noting $B_R(x)$ the closed ball of radius $R$ centred at a point $x$, $$\begin{aligned}
\int_{B_R}\textbf{f}_n\left(x_n+\frac{\lambda}{\sqrt{n}},\theta_\lambda(\nu_n,E_n,g_n)\right)d\lambda &=\int_{B_R}\frac{1}{\pi}\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V\left(x_n+\frac{\lambda}{\sqrt{n}}\right)}d(\theta_{\lambda+x_n\sqrt{n}}\sharp {g}_n(y))d\lambda\\
&=\frac{1}{\pi}\int_{B_R}\int_{{\mathbb R}^2}\frac{\chi(y-x_n\sqrt{n}-\lambda)}{m_V\left(x_n+\frac{\lambda}{\sqrt{n}}\right)}d{g}_n(y)d\lambda\\
&=\frac{1}{\pi}\int_{{\mathbb R}^2}\chi\ast\left(\indic_{B_R(x_n\sqrt{n})}\frac{1}{m_V(./\sqrt{n})}  \right)(y)d{g}_n(y)<C_R.\end{aligned}$$ This, inequalities and the fact that ${g}_n$ is bounded below imply that ${g}_n(B_R(x_n\sqrt{n}))$ is bounded independently of $n$. Hence by the same argument as in [@2DSandier Lemma 4.4], we have the convergence of a subsequence of $(\nu_n,E_n,g_n)$.\
2) We have the $\Gamma$-liminf property: if $(x_n,\nu_n,E_n,g_n)\to(x,\nu,E,g)$ as $n\to +\infty$, then, by Fatou’s Lemma, $$\liminf_{n\to +\infty} \mathbf{f}_n(x_n,\nu_n,E_n,g_n)\geq f(x,\nu,E,g):=\frac{1}{\pi}\int\frac{\chi(y)}{m_V(x)} dg(y),$$ obviously if the left-hand side is finite. Therefore, Theorem \[erg\] applies and implies that:\

1.  The measure $Q_n$ admits a subsequence which converges to a measure $Q$ which has $\mu_V$ as first marginal.

2.  It holds that $Q$-almost every $\displaystyle (x,\nu,E,g)$ is of the form $\displaystyle \lim_{n\to +\infty}(x_n, \theta_{x_n\sqrt{n}}({\nu}'_n,{E}_n,{g}_n))$.

3.  The measure $Q$ is $T_{\lambda(x)}$-invariant.

4.  We have $\displaystyle \liminf_{n\to +\infty} \mathbf{F}_n({\nu}'_n,{E}_n,{g}_n)\geq \frac{1}{\pi} \int_{{\mathbb R}^2}\left(\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V(x)} dg(y)\right) dQ(x,\nu,E,g)$.

5.  $\displaystyle \frac{1}{\pi}\int \int \frac{\chi(y)}{m_V(x)}dg(y) dQ(x,\nu,E,g)=\int \left( \lim_{R\to +\infty}\dashint_{B_R} \int \frac{\chi(y-\lambda)}{m_V(x)}dg(y)d\lambda\right) dQ(x,\nu,E,g)$.

Now we can follow exactly the lines of [@2DSandier Section 4.2, Step 3] to deduce from $4)$, after noticing that $P_n$ is the marginal of $Q_n$ corresponding to the variables $(x,E)$ which converge to a $T_{\lambda(x)}$-invariant probability measure. Moreover $$\begin{aligned}
\liminf_{n\to+\infty} \frac{1}{n\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})&\geq\int \left( \int \chi dg \right)\frac{dQ(x,\nu,E,g)}{m_V(x)} \\
&= \int\lim_{R\to +\infty} \left( \frac{1}{\pi R^2}\int \chi\ast \indic_{B_R} dg\right)\frac{dQ(x,\nu,E,g)}{m_V(x)}\\
&\geq \frac{1}{\pi}\int W(E) \frac{dQ(x,\nu,E,g)}{m_V(x)}=\frac{1}{\pi}\int \frac{W(E)}{m_V(x)} dP(x,E).\end{aligned}$$ Thus the lower bound is proved. The fact that the right-hand side is larger than $\alpha_V$ is obvious because the first marginal of $\displaystyle \frac{dP}{m_V}$ is the Lebesgue measure.

Proof of the upper bound, the case $\supp(\mu_V)\neq{\mathbb R}^2$
------------------------------------------------------------------

The discussion following Theorem \[equi\] permits to immediately reduce the case of $V$’s such that $\supp (\mu_V)\neq {\mathbb R}^2$ to the case of a compact support. Indeed in this case there exists $y\in{{\mathcal S}}$ which does not belong to the support of $\mu_{{\mathcal V}}$. Let $R$ be a rotation such that $R(N) = y$, then the minimum of $I_{{{\mathcal V}}\circ R}$ is $\mu_{{{\mathcal V}}\circ R} = R^{-1}\sharp\mu_{{\mathcal V}}$ hence $N$ does not belong to its support.

Letting $\varphi = T^{-1} R T$, we have that $\varphi$ is of the form $z\to\ds \frac{az+b}{cz+d}$ with $ad-bc = 1$, and applying , to ${{\mathcal V}}\circ R$ we have that $$\mu_{V_\varphi} = T^{-1}\sharp\mu_{{{\mathcal V}}\circ R},$$ where $$\mathcal V\circ R(T(x)) = V_\varphi(x) - \log(1+|x|^2).$$ This implies that $\mu_{V_\varphi}$ has compact support since $N$ does not belong to the support of $\mu_{{{\mathcal V}}\circ R} $. Moreover, using again to evaluate ${{\mathcal V}}(R T(x))$ we find for any $x$ such that $RT(x)\neq N$, i.e. $x\neq -d/c$, $$\begin{aligned}
&V_\varphi(x)  = V(T^{-1}RT(x)) - \log(1+|T^{-1}RT(x)|^2)  + \log(1+|x|^2),\\
& V_\varphi(-d/c) = {{\mathcal V}}(N) + \log(1+|d/c|^2)=\log(1+|d/c|^2) +\liminf_{|x|\to +\infty} \{V(x) - \log(1+|x|^2)\}.\end{aligned}$$ Finally we find that $$\label{vp} V_\varphi(x)  = V(\varphi(x)) - \log(1+|\varphi(x)|^2) + \log(1+|x|^2), \quad V_\varphi(-d/c) = \liminf_{y\to-d/c} V_\varphi(y).$$

Now we rewrite the discrete energy by changing variables, to find that, writing $w_{n,V}$ instead of $w_n$ to clarify the dependence on $V$, $$\label{disc1}w_{n,V}(x_1,\dots,x_n) = -\sum_{i\neq j}^n\log|\varphi(y_i) - \varphi(y_j)|+ n\sum_{i=1}^nV(\varphi(y_i)),$$ where $x_i = \varphi(y_i)$. Now we use the identity (see [@Hardy]) $$\|T(x)-T(y)\|=\frac{|x-y|}{\sqrt{1+|x|^2}\sqrt{1+|y|^2}}$$ applied to $\varphi( x)$, $\varphi(y)$ together with the fact that $\varphi = T^{-1} R T$ and that $R$ is a rotation to get $$\|T(x)-T(y)\|=\frac{|\varphi(x)-\varphi(y)|}{\sqrt{1+|\varphi(x)|^2}\sqrt{1+|\varphi(y)|^2}}.$$ The two together imply that $$\log|\varphi(x) - \varphi(y)| = \log|x-y| + \frac12 \log (1+|\varphi(x)|^2) + \frac12 \log (1+|\varphi(y)|^2) - \frac12 \log (1+|x|^2)- \frac12 \log (1+|y|^2).$$ Replacing in shows that $$\label{chvar}w_{n,V}(x_1,\dots, x_n) = w_{n,V_\varphi}(y_1,\dots, y_n) + \sum_i \log (1+|\varphi(y_i)|^2)  - \sum_i \log (1+|y_i|^2),\quad x_i = \varphi(y_i).$$

It follows from that an upper bound for $\min w_{n,V}$ can be computed by using a minimizer for $w_{n,V_\varphi}$ as a test function. But now we recall that $\mu_{V_\varphi}$ has compact support, hence the results of [@2DSandier] apply and we find, using the fact that for such a minimizer $\frac1n\sum_i\delta_{y_i}$ converges to $\mu_{V_\varphi}$, $$\label{mm}\min w_{n,V} \le  n^2 I_{V_\varphi}(\mu_{V_\varphi}) - \hal n \log n + \left(\alpha_{V_\varphi}  + \int \log \left(\frac{1+|\varphi(x)|^2}{1+|x|^2}\right)\,d\mu_{V_\varphi}(x)\right)n + o(n),$$ where $$\alpha_{V_\varphi} = \frac{\alpha_1}{\pi}-\hal \int_{\Sigma_{V_\varphi}} m_{V_\varphi}(x)\log  m_{V_\varphi}(x)\, dx, \quad  \alpha_1:=\min_{\mathcal A_1} W.$$ We remark that $I_{V_\varphi}(\mu_{V_\varphi})=I_V(\mu_V)$ because $\mu_{V_\varphi}=\varphi^{-1}\sharp \mu_V$. Moreover, it follows from that $$m_{V_\varphi}(x) = m_V(\varphi(x))\left(\frac{1+|\varphi(x)|^2}{1+|x|^2}\right)^2,$$ which plugged in the expression for $\alpha_{V_\varphi}$ and then in yields, $$\min w_{n,V} \le n^2 I_V(\mu_V) - \hal n \log n + \alpha_{V} n +o(n),$$ which matches the lower-bound we already obtained and thus proves Theorem \[thintro\] in the case where the support of $\mu_V$ is not the full plane.

Proof of the upper bound by compactification and conclusion
-----------------------------------------------------------

Here we assume that $\Sigma_V={\mathbb R}^2$. Let $$\varphi(z):=-\frac{1}{z}=\varphi^{-1}(z).$$ Then, using the notations of the previous section, we deduce from that $$V_\varphi(z)=V(\varphi(z))+2\log|z|.$$ To simplify exposition and notation, we assume that $\mu_V(B_1)=\mu_V(B_1^c)=1/2$, otherwise there would exist $R$ such that $\mu_V(B_R)=\mu_V(B_R^c)=1/2$ and we should use the transformation $\varphi_R(z)=\varphi_R^{-1}(z)=-Rz^{-1}$ instead.

Our idea is to cut $\Sigma_V={\mathbb R}^2$ into two parts in order to construct a sequence of $2n$ points associated to a sequence of vector-fields. We will only construct test configurations with an even number of points, again to simplify exposition and avoid unessential technicalities.

**STEP 1: Reminder of the compact case and notations.** We need [@2DSandier Corollary 4.6] when $K$ is a compact set of ${\mathbb R}^2$:

\[UpBound\] [([@2DSandier])]{.nodecor} Let $P$ be a $T_{\lambda(x)}$-invariant probability measure on $X=K\times L_{loc}^p({\mathbb R}^2,{\mathbb R}^2)$, where $K$ is a compact subset of ${\mathbb R}^2$ with $C^1$ boundary.

We assume that $P$ has first marginal $dx_{|K}/|K|$ and that for $P$-almost every $(x,E)$ we have $E\in\mathcal{A}_{m(x)}$, where $m$ is a smooth function on $K$ bounded above and below by positive constants. Then there exists a sequence $\{\nu_n=\sum_{i=1}^n\delta_{x_i}\}_n$ of empirical measures on $K$ and a sequence $\{E_n\}_n$ in $L_{loc}^p({\mathbb R}^2,{\mathbb R}^2)$ such that ${\mathop{\rm div}\nolimits}E_n=2\pi(\nu'_n-m')$, such that $E_n = 0$ outside $K$ and such that $\displaystyle P_{n}:=\dashint_{K}\delta_{(x,E_n(\sqrt{n}x+.))}dx  \to P$ as $n\to +\infty$. Moreover $$\limsup_{n\to +\infty}      \frac{1}{n\pi} W(E_n,\indic_{{\mathbb R}^2})\leq \frac{|K|}{\pi}\int W(E)dP(x,E).\\$$

We write $\mu_V=\mu_V^1+\mu_V^2$ where $\mu_V^1:={\mu_V}_{|B_1}$ and $\mu_V^2:={\mu_V}_{|\bar{B_1^c}}$, where $\bar{A}$ denotes the closure of set $A$ in ${\mathbb R}^2$. Let $\tilde{\mu}_V^2:=\varphi\sharp\mu_V^2$, then we have $$d\mu_V^1(x)=m_V(x)\indic_{B_1}(x)dx=:m_V^1(x)dx \quad \text{ and } \quad d\tilde{\mu}_V^2(x)=m_{V_\varphi}(x)\indic_{B_1}(x)dx=:m_{V_\varphi}^2(x)dx,$$ where $m_{V_\varphi}(x)=m_V(\varphi^{-1}(x))| \det(D\varphi_x^{-1})|$.\
Note that, by assumption **(H2)** and we have that there exists positive constants $\bm$ and $\mb$ such that, for any $x\in B_1$, $$0<\mb\leq m_V(x)\leq \bm\quad \text{ and } \quad 0<\mb\leq m_{V_\varphi}(x)\leq \bm.$$ Moreover the boundary $\partial B_1$ is $C^1$.

Now let $P$ be a $T_{\lambda(x)}$-invariant probability measure on $X$ whose first marginal is $\mu_V$ and be such that for $P$-almost very $(x,E)$, we have $E\in \mathcal{A}_{m_V(x)}$. We can write $$P=P^1+P^2,$$ where $P^1$ is the restriction of $P$ to $B_1\times L^p_{loc}(B_1,{\mathbb R}^2)$ with first marginal $\mu_V^1$, and $P^2$ is the restriction of $P$ to $B_1^c\times L^p_{loc}(B_1^c,{\mathbb R}^2)$ with first marginal $\mu_V^2$. We define $\tilde P^1$ by the relation $$dP^1(x,u)=m_V(x)|B_1|d\tilde{P}^1(x,u),$$ and then $\tilde{P}^1$ is a $T_{\lambda(x)}$-invariant probability measure on $B_1\times L^p_{loc}(B_1,{\mathbb R}^2)$ with first marginal $dx_{|B_1}/|B_1|$ and such that, for $\tilde{P}^1$-a.e. $(x,E)$, $E\in \mathcal{A}_{m_V^1(x)}$. We denote by $\varphi\sharp P^2$ the pushforward of $P^2$ by $$\label{chgtvar}
(x,E)\mapsto \left(y, \tilde{E}\right),\ \text{ where } \ y:=\varphi(x) \text{ and } \tilde{E}:=(D\varphi_y)^T E(D\varphi_y \cdot),$$ where $D\varphi_x = \lambda(x) I_2$ is the differential of $\varphi$ at point $x$. Then if $\div E   = 2\pi(\nu - m_V(x)dx)$ then $\div \tilde{E} = 2\pi(\vp\sharp\nu -\lambda^2(y) m_V(\varphi(y)))$ so that for $\varphi\sharp P^2$-a.e. $(y,\tilde{E})$ the vector field $\tilde{E}$ belongs to $ \mathcal{A}_{m_{V_\varphi}(y)}$, since $$m_{V_\varphi}(y)\,dy = m_V(\varphi(y)) \,d(\varphi(y)) = m_V(\varphi(y))\lambda(y)^2\,dy.$$ We define $\tilde P^2$ by the relation $$d(\varphi\sharp P^2)(y,\tilde{E})=m_{V_\varphi}(y)|B_1|d\tilde{P}^2(y,\tilde{E}),$$ and then $\tilde{P}^2$ is a $T_{\lambda(x)}$-invariant probability measure on $B_1\times L^p_{loc}(B_1,{\mathbb R}^2)$ with first marginal $dy_{|B_1}/|B_1|$ and such that, for $\tilde{P}^2$ a.e. $(y,\tilde{E})$, $\tilde{E}\in \mathcal{A}_{m_{V_\varphi}(y)}$.

**STEP 2: Application of Theorem \[UpBound\].** We may now apply Theorem \[UpBound\] to $\tilde{P}^1$ and $\tilde{P}^2$. We thus construct a sequence $\{\nu_n^1:=\sum_{i=1}^n\delta_{x_i^1}\}$ of empirical measures on $B_1$ and a sequence $\{E_n^1\}_n$ in $L^p_{loc}(B_1,{\mathbb R}^2)$ such that $${\mathop{\rm div}\nolimits}E_n^1=2\pi((\nu_n^1)'-(m_V^1)')\quad \text{ and } \quad \tilde{P}_n^1:=\dashint_{B_1}\delta_{(x,E_n^1(\sqrt{n}x+.))}dx  \to \tilde{P}^1,$$ as $n\to +\infty$. Moreover, we have $$\label{F1}
\limsup_{n\to +\infty} \frac{1}{n\pi} W(E_n^1,\indic_{{\mathbb R}^2})\leq \frac{|B_1|}{\pi}\int W(E)d\tilde{P}^1(x,E).$$ Applying now the same Theorem to $\tilde{P}^2$, we construct a sequence $\{\tilde{\nu}_n^2:=\sum_{i=1}^n\delta_{\tilde{x}_i^2}\}$ of empirical measures on $B_1$ and a sequence $\{\tilde{E}_n^2\}_n$ in $L^p_{loc}(B_1,{\mathbb R}^2)$ such that $${\mathop{\rm div}\nolimits}\tilde{E}_n^2=2\pi((\tilde{\nu}_n^2)'-(m_{V_\varphi}^2)') \quad \text{ and } \tilde{P}_n^2:=\dashint_{B_1}\delta_{(x,\tilde{E}_n^2(\sqrt{n}x+.))}dx  \to \tilde{P}^2,$$ as $n\to +\infty$. Moreover, we have $$\label{Ftilde2}
\limsup_{n\to +\infty} \frac{1}{n\pi} W(\tilde E_n^2,\indic_{{\mathbb R}^2})\leq \frac{|B_1|}{\pi}\int W(\tilde{E})d\tilde{P}^2(y,\tilde{E}).$$

**STEP 3: Construction of sequences and conclusion.** It is not difficult to see that we can assume $\tilde{x}_j^2\neq 0$ for any $j$ and any $n\geq 2$ (otherwise we translate a little bit the point). Now we set $x_j^2 := \vp(\tilde x_j^2)$ and in view of , for each $n$ we define $$\nu_n^2:=\varphi\sharp \tilde{\nu}_n^2=\sum_{j=1}^n \delta_{x_j^2} \quad \text{ and } \quad E_n^2(x):=(D\varphi_{n^{-1/2}x})^T \tilde{E}_n^2(n^{1/2}\varphi(n^{-1/2}x)).$$ Hence, we have a sequence of vector-fields $E_n^2$ of $L^p_{loc}(B_1^c,{\mathbb R}^2)$ such that $${\mathop{\rm div}\nolimits}E_n^2=2\pi((\nu_n^2)'-(m_V^2)')$$ where $m_V^2(x)=m_V(x)\indic_{\bar{B_1^c}}(x)$ is the density of $\mu_V^2$.

We have, for sufficiently small $\eta$ such that $0\not\in B(\tilde{x}_i^2,\eta)$ for every $i$, $$\begin{aligned}
&W(E_n^2,\indic_{{\mathbb R}^2}) \\
&=\lim_{\eta\to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i=1}^n B(x_i^2,\eta)} |E_n^2(x')|^2dx' +\pi n\log\eta   \right)\\
&=\lim_{\eta\to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i=1}^n B(x_i^2,\eta)} |(D\varphi_{n^{-1/2}x'})^T \tilde{E}_n^2(n^{1/2}\varphi(n^{-1/2}x'))|^2dx' +\pi n\log\eta   \right)\\
&=\lim_{\eta\to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i=1}^n B(y_i^2,|\varphi'(x_i^2)|\eta)} |\tilde{E}_n^2(y')|^2dy' +\pi n\log\eta   \right)\\
&=\lim_{\eta\to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i=1}^n B(y_i^2,|\varphi'(x_i^2)|\eta)} |\tilde{E}_n^2(y')|^2dy' +\pi\sum_{i=1}^n \log|\varphi'(x_i)|\eta  -\pi \sum_{i=1}^n \log |\varphi '(x_i)| \right)\\
&=W(\tilde{E}_n^2,\indic_{{\mathbb R}^2})-\pi \sum_{i=1}^n \log |\varphi '(x_i)|,\end{aligned}$$ where the change of variable is $y'=n^{1/2}\varphi(n^{-1/2}x')$.\
Furthermore, we have $$\int W(\tilde{E})d \tilde{P}^2(y,\tilde{E})=\frac{1}{|B_1|}\int W(\tilde{E})\frac{d(\varphi \sharp P^2)(y,\tilde{E})}{m_{V_\varphi}(y)}=\frac{1}{|B_1|}\int W\left(D\varphi_y^T E(D\varphi_y .)  \right)\frac{dP^2(x,E)}{m_{V_\varphi}(y)}$$ by change of variable $y=\varphi(x)$ and $\tilde{E}=D\varphi_y^T E(D\varphi_y \cdot)$.\
Now we remark that, for $\lambda>0$ and $E\in \mathcal{A}_m$, $$\begin{aligned}
W(\lambda E(\lambda .)) &=\lim_{R\to +\infty}\frac{1}{\pi R^2} \lim_{\eta \to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i} B(y_i,\eta)} \chi_R(y) \lambda^2 |E(\lambda y)|^2 dy+\pi \sum_{i}\chi_R(y_i)\log\eta\right)\\
&=\lim_{R\to +\infty}\frac{1}{\pi R^2} \lim_{\eta \to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i} B(x_i,\lambda\eta)} \chi_R(x/\lambda)|E(x)|^2dx +\pi \sum_i \chi_R(x_i/\lambda)\log\eta\right)\end{aligned}$$ where $x=\lambda y$. Thus, setting $R'=R\lambda$ and $\eta' =\eta \lambda$, we get $$\begin{aligned}
&W(\lambda E(\lambda .)) \\
&=\lim_{R'\to +\infty}\frac{\lambda^2}{\pi R'^2} \lim_{\eta' \to 0} \left(\frac{1}{2}\int_{{\mathbb R}^2\backslash \bigcup_{i} B(x_i,\eta')} \chi_{R'}(x)|E(x)|^2dx +\pi \sum_i \chi_{R'}(x_i)\left(\log\eta'-\log\eta \right)\right)\\
&=\lambda^2\left(W(E)-m\log\lambda   \right).\end{aligned}$$ Applying this equality with $\lambda=|\varphi'(x)|^{-1}=|\varphi'^{-1}(y)|$, we obtain $$\begin{aligned}
&\limsup_{n\to +\infty} \frac{1}{n\pi}\left(W(E_n^2,\indic_{{\mathbb R}^2}) +\pi\sum_{i=1}^n\log|\varphi'(x_i)| \right)\\
&\leq \frac{1}{\pi}\int \frac{1}{|\varphi'(x)|^2}\left( W(E)+\log|\varphi'(x)|m_V^2(x) \right)\frac{d P^2(x,E)}{m_{V_\varphi}(y)},\end{aligned}$$ that is to say, because $m_V^2$ is the density of points $\{x_i\}$ as $n\to +\infty$, $$\begin{aligned}
&\limsup_{n\to +\infty} \frac{1}{n\pi}W(E_n^2,\indic_{{\mathbb R}^2})+\int_{B_1^c} \log|\varphi '(x)|d\mu_V^2(x)\\
&\leq \frac{1}{\pi}\int W(E) \frac{d P^2(x,E)}{m_V(x)}+\int_{B_1^c} \log|\varphi '(x)|dP^2(x).\end{aligned}$$ As $\int_{B_1^c} \log|\varphi '(x)|d P^2(x)=\int_{B_1^c} \log|\varphi '(x)|d\mu_V^2(x)$, it follows that $$\label{F2}
\limsup_{n\to +\infty} \frac{1}{n\pi}W(E_n^2,\indic_{{\mathbb R}^2})\leq \frac{1}{\pi}\int W(E) \frac{d P^2(x,E)}{m_V(x)}.$$ Finally, we set $$\nu_{2n}:=\nu_n^1+\nu_n^2 \quad \text{ and } \quad E_{2n}:=E_n^1+E_n^2,$$ and by and , we have, since $E_n^1$ and $E_n^2$ have disjoint supports, $$\begin{aligned}
\limsup_{n\to +\infty} \frac{1}{n\pi} W(E_n,\indic_{{\mathbb R}^2})&\leq \frac{1}{\pi}\int \frac{W(E)}{m_V(x)}dP^1(x,E)+\frac{1}{\pi}\int \frac{W(E)}{m_V(x)}dP^2(x,E)\\
&=\frac{1}{\pi}\int \frac{W(E)}{m_V(x)}dP(x,E)\end{aligned}$$ which proves . Furthermore, by changes of variable, $$P_n^1:=\int_{B_1}\delta_{(x,E_n^1(x\sqrt{n}+.))}d\mu_V(x)\to P^1 \quad \text{and} \quad P_n^2:=\int_{B_1^c}\delta_{(x,E_n^2(x\sqrt{n}+.))}d\mu_V(x)\to P^2$$ in the weak sense of measure, and it follows that $$P_n=P_n^1+P_n^2\to P^1+P^2= P.\\$$ Part **C** follows from **A** and **B** as in [@2DSandier].

Consequence: the Logarithmic Energy on the Sphere {#CTLEOTS}
=================================================

As we have an asymptotic expansion of the minimum of the Hamiltonian $w_n$ where minimizers can be in the whole plane – not only in a compact set as in the classical case – we will use the inverse stereographic projection from ${\mathbb R}^2$ to a sphere in order to determine the asymptotic expansion of optimal logarithmic energy on sphere.

Inverse stereographic projection
--------------------------------

Here we recall properties of the inverse stereographic projection used by Hardy and Kuijlaars [@Hardy; @Hardy2] and by Bloom, Levenberg and Wielonsky [@Bloom:2014qy] in order to prove Theorem \[equi\].\
Let $\mathcal{S}$ be the sphere of ${\mathbb R}^3$ centred at $(0,0,1/2)$ of radius $1/2$, $\Sigma$ be an unbounded closed set of ${\mathbb R}^2$ and $T:{\mathbb R}^2\to \mathcal{S}$ be the associated inverse stereographic projection defined by $$T(x_1,x_2)=\left(\frac{x_1}{1+|x|^2}, \frac{x_2}{1+|x|^2},\frac{|x|^2}{1+|x|^2} \right), \quad \text{for any $x=(x_1,x_2)\in{\mathbb R}^2$},$$ where ${\mathbb R}^2:=\{(x_1,x_2,0);x_1,x_2\in{\mathbb R}\}$. We know that $T$ is a conformal homeomorphism from ${\mathbb C}$ to $\mathcal{S}\backslash \{N\}$ where $N :=(0,0,1)$ is the North pole of $\mathcal{S}$.\
We have the following identity: $$\label{distanceT}
\|T(x)-T(y)\|=\frac{|x-y|}{\sqrt{1+|x|^2}\sqrt{1+|y|^2}}, \text{ for any } x,y\in {\mathbb R}^2.$$ Furthermore, if $|y|\to +\infty$, we obtain, for any $x\in{\mathbb R}^2$: $$\label{interinf} \|T(x)-N\|=\frac{1}{\sqrt{1+|x|^2}}.$$

We note $\Sigma_{\mathcal{S}}=T(\Sigma)\cup \{N\}$ the closure of $T(\Sigma)$ in $\mathcal{S}$. Let $\mathcal{M}_1(\Sigma)$ be the set of probability measures on $\Sigma$. For $\mu \in  \mathcal{M}_1(\Sigma)$, we denote by $T\sharp\mu$ its push-forward measure by $T$ characterized by $$\int_{\Sigma_{\mathcal{S}}}f(z)dT\sharp\mu(z)=\int_\Sigma f(T(x))d\mu(x),$$ for every Borel function $f:\Sigma_{\mathcal{S}}\to {\mathbb R}$. The following result is proved in [@Hardy]:

\[THomeo\] The correspondance $\mu\to T\sharp\mu$ is a homeomorphism from the space $\mathcal{M}_1(\Sigma)$ to the set of $ \mu\in \mathcal{M}_1(\Sigma_S)$ such that $ \mu(\{N\})=0$.

Asymptotic expansion of the optimal logarithmic energy on the unit sphere
-------------------------------------------------------------------------

An important case is the equilibrium measure associated to the potential $$V(x)=\log(1+|x|^2)$$ corresponding to the external field $\mathcal{V}\equiv 0$ on $\mathcal{S}$ and where $T\sharp\mu_V$ is the uniform probability measure on $\mathcal{S}$ (see [@Hardy]). Hence $V$ is an admissible potential and from we find $$\displaystyle d\mu_V(x)=\frac{dx}{\pi(1+|x|^2)^2}\quad \text{and} \quad \Sigma_V={\mathbb R}^2.$$

We define $$\overline{w}_n(x_1,...,x_n):=-\sum_{i\neq j}^n\log|x_i-x_j|+(n-1)\sum_{i=1}^n\log(1+|x_i|^2),$$ and we recall that the logarithmic energy of a configuration $(y_1,...,y_n)\in \mathcal{S}^n$ is given by $$\textnormal{E}_{\log}(y_1,....,y_n):=-\sum_{i\neq j}^n\log\|y_i-y_j\|.$$ Furthermore, we recall that $\mathcal{E}_{\log}(n)$ denotes the minimal logarithmic energy of $n$ points on $\S^2$.

\[wminimizer\] For any $(x_1,...,x_n)\in ({\mathbb R}^2)^n$, we have the following equalities: $$\overline{w}_n(x_1,...,x_n)=\textnormal{E}_{\log}(T(x_1),...,T(x_n)) \quad \text{ and } \quad w_n(x_1,...,x_n)=\textnormal{E}_{\log}(T(x_1),...,T(x_n), N),$$ which imply that $$\begin{aligned}
&(x_1,...,x_n) \text{ minimizes } \overline{w}_n \iff (T(x_1),...,T(x_n)) \text{ minimizes } \textnormal{E}_{\log}\\
&(x_1,...,x_n) \text{ minimizes } w_n \iff (T(x_1),...,T(x_n),N) \text{ minimizes } \textnormal{E}_{\log}.\end{aligned}$$

For any $1\leq i \leq n$, we set $y_i:=T(x_i)$, hence we get, by , $$\begin{aligned}
\textnormal{E}_{\log}(y_1,....,y_n)&:=-\sum_{i\neq j}^n\log \|y_i-y_j\|\\
&=-\sum_{i\neq j}^n\log\|T(x_i)-T(x_j)\| \\
&=-\sum_{i\neq j}^n\log\left(  \frac{|x_i-x_j|}{\sqrt{1+|x_i|^2}\sqrt{1+|x_j|^2}} \right) \\
&=-\sum_{i\neq j}^n\log|x_i-x_j|+(n-1)\sum_{i=1}^n\log(1+|x_i|^2)\\
&=\overline{w}_n(x_1,...,x_n).\end{aligned}$$ Furthermore, by , we obtain $$\begin{aligned}
w_n(x_1,...,x_n)&=\overline{w}_n(x_1,...,x_n)+\sum_{i=1}^n\log(1+|x_i|^2)\\
&=-\sum_{i\neq j}\log\|y_i-y_j\|-2\sum_{i=1}^n\log\|y_i-N\|=\textnormal{E}_{\log}(y_1,....,y_n,N).\end{aligned}$$

\[weakmeas\] If $(x_1,...,x_n)$ minimizes $w_n$ or $\overline{w}_n$, then, for $\displaystyle \nu_n:=\sum_{i=1}^n\delta_{x_i}$, we have $$\frac{\nu_n}{n}\to \mu_V, \quad \text{ as } n\to +\infty,$$ in the weak sense of measures.

Let $(x_1,...,x_n)$ be a minimizer of $\bar{w}_n$, then $(T(x_1),...,T(x_n))$ is a minimizer of $\textnormal{E}_{\log}$. Brauchart, Dragnev and Saff proved in [@EquilSphere Proposition 11] that $$\frac{1}{n}\sum_{i=1}^n\delta_{T(x_i)}\to T\sharp\mu_V.$$ As $T\sharp\mu_V(N)=0$, by Lemma \[THomeo\] we get the result.\
If $(x_1,...,x_n)$ is a minimizer of $w_n$, then $(T(x_1),...,T(x_n), N)$ minimizes $\textnormal{E}_{\log}$ and we can use our previous argument because $$\frac{1}{n+1}\left(\sum_{i=1}^n \delta_{T(x_i)}+\delta_N\right)=\frac{1}{n}\sum_{i=1}^n\delta_{T(x_i)}\left( \frac{n}{n+1} \right)+\frac{\delta_N}{n+1}\to T\sharp \mu_V,$$ in the weak sense of measures, and we have the same conclusion.

\[CVlog\] If $(x_1,...,x_n)$ is a minimizer of $w_n$ and if $\displaystyle \nu_n:=\sum_{i=1}^n\delta_{x_i}$ then $$\lim_{n\to +\infty}\frac{1}{n}\int_{{\mathbb R}^2}\log(1+|x|^2)d\nu_n(x)=\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).$$ There exists minimizers of $\overline{w}_n$ for which the same is true.

Let $(x_1,...,x_n)$ be a minimizer of $\overline{w}_n$. We define $y_i:=T(x_i)$ for any $1\leq i\leq n$ and we notice that $$\begin{aligned}
\frac{1}{n}\int_{{\mathbb R}^2}\log(1+|x|^2)d\nu_n(x)&=-\frac{2}{n}\int_{{\mathbb R}^2}\log\left(\frac{1}{\sqrt{1+|x|^2}}  \right)d\nu_n(x)\\
&=-\frac{2}{n}\int_{\mathcal{S}}\log\|y-N\|dT\sharp\nu_n(y),\end{aligned}$$ and by the previous Lemma, $(y_1,...,y_n)$ is a minimizer of $\textnormal{E}_{\log}$ on $\mathcal{S}$. For any rotation $R$ of ${{\mathcal S}}$ the rotated configuration of points is still a minimizer, and it is clear that the average over rotations $R$ of $$-\frac{2}{n}\sum_i \log\|Ry_i-N\|$$ is equal to $$-2\dashint_{{{\mathcal S}}}\log\|y-N\|\,dy.$$ It follows that there exists a rotated configuration $(\bar y_1,...,\bar y_n)$ such that $$\frac{1}{n}\sum_i \log\|\bar y_i-N\| = \dashint_{{{\mathcal S}}}\log\|y-N\|\,dy.$$ Transporting this equality back to ${\mathbb R}^2$ with $T^{-1}$, we obtain a minimizer $(\bar x_1,...,\bar x_n)$ of $\overline{w}_n$ such that $$\frac{1}{n}\sum_i\log(1+|\bar x_i|^2) =\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).$$

If $(x_1,\dots,x_n)$ is a minimizer of $w_n$ we use [@EquilSphere Theorem 15] about the optimal point separation which yields the existence of constants $C$ and $n_0$ such that for any $n\geq n_0$ and any minimizer $\{y_1,...,y_n\}\in\mathcal{S}^n$ of the logarithmic energy on the sphere, we have $$\min_{i\neq j}\|y_i-y_j\|>\frac{C}{\sqrt{n-1}}.$$ Letting $y_i = T(x_i)$ we have that $(N,y_1,\dots, y_n)$ is a minimizer of the logarithmic energy, hence for any $1\leq i\leq n$, $$\label{separation}
\|y_i-N\|>\frac{C}{\sqrt{n-1}}.$$

For $n\geq n_0$ and $\delta>0$ sufficiently small, we define, for any $0< r\leq \delta$, $$n(r):=\string #\left\{y_i \mid y_i\in B(N,r)\cap\mathcal{S}  \right\},$$ and $r_i=\|y_i-N\|$. From the separation property there exists a constant $C$ such that $n(r)\leq Cr^2n$ for any $r$. Hence we have, using integration by parts, $$\begin{aligned}
-\sum_{y_i\in B(N,\delta)}\log r_i &=-\int_{1/\sqrt{n-1}}^\delta \log r n'(r)dr\\
&=-n(\delta)\log \delta +\int_{1/\sqrt{n-1}}^\delta \frac{n(r)}{r}dr\\
&\leq -Cn \delta^2\log\delta +Cn\int_{1/\sqrt{n-1}}^\delta rdr\leq C\delta^2 n|\log\delta|.\end{aligned}$$ It follows that $$\begin{aligned}
\label{convlog}
\lim_{\delta\to 0}\lim_{n\to +\infty}\frac{1}{n}\int_{B(N,\delta)\cap\mathcal{S}}\log\|y-N\|dT\sharp\nu_n(y)&=   \lim_{\delta\to 0}\lim_{n\to +\infty}-\sum_{i=1}^{n_\delta}\frac{1}{n}\log\|y_i-N\|=0.\end{aligned}$$ By Lemma \[weakmeas\], $\displaystyle \frac{\nu_n}{n}$ goes weakly to the measure $\mu_V$ on $B_R$ for any $R$, hence we get $$\begin{aligned}
&\lim_{n\to +\infty}\frac{1}{n}\int_{{\mathbb R}^2}\log(1+|x|^2)d\nu_n(x)\\
&=\lim_{n\to +\infty}\frac{1}{n}\left(\int_{B_R}\log(1+|x|^2)d\nu_n(x)+\int_{B_R^c}\log(1+|x|^2)d\nu_n(x)\right)\\
&=\int_{B_R}\log(1+|x|^2)d\mu_V(x)+\lim_{n\to +\infty}\frac{1}{n}\int_{B_R^c}\log(1+|x|^2)d\nu_n(x).\end{aligned}$$ Therefore it follows from that $$\begin{aligned}
&\lim_{n\to +\infty}\frac{1}{n}\int_{{\mathbb R}^2}\log(1+|x|^2)d\nu_n(x)\\
&=\lim_{R\to+\infty}\left( \int_{B_R}\log(1+|x|^2)d\mu_V(x)+\lim_{n\to +\infty}\frac{1}{n}\int_{B_R^c}\log(1+|x|^2)d\nu_n(x)\right)\\
 &=\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).\end{aligned}$$ The convergence is proved.

The following result proves the existence of the constant $C$ in the Conjecture 1 of Rakhmanov, Saff and Zhou.

\[RSZ\] We have $$\mathcal{E}_{\log}(n)=\left( \frac{1}{2}-\log 2 \right)n^2-\frac{n}{2}\log n + \left( \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\log2\right)n +o(n), \quad \text{as } n\to +\infty.$$

As $\textnormal{E}_{\log}$ is invariant by translation of the $2$-sphere, we work on the sphere $\tilde{\S}^2$ of radius 1 and centred in $(0,0,1/2)$. Let $(y_1,...,y_n)\in\tilde{\S}^2$ be a minimizer of $\textnormal{E}_{\log}$. Without loss of generality, for any $n$, we can choose this configuration such that $y_i\neq N$ for any $1\leq i\leq n$. Hence there exists $(x_1,...,x_n)$ such that $\displaystyle \frac{y_i}{2}=T(x_i)$ for any $i$ and we get

$$\begin{aligned}
\textnormal{E}_{\log}(y_1,....,y_n)&=-\sum_{i\neq j}^n\log \|y_i-y_j\|\\
&=-\sum_{i\neq j}^n\log\|T(x_i)-T(x_j)\|-n(n-1)\log 2 \\
&=\overline{w}_n(x_1,...,x_n)-n(n-1)\log 2.\end{aligned}$$

By Lemma \[wminimizer\], $(y_1,...,y_n)$ is a minimizer of $\textnormal{E}_{\log}$ if and only if $(x_1,...,x_n)$ is a minimizer of $\overline{w}_n$. By the lower bound and the convergence of Lemma \[CVlog\], we have, for some minimizer $(\bar{x}_1,...,\bar{x}_n)$ of $\overline{w}_n$: $$\begin{aligned}
&\liminf_{n\to +\infty}\frac{1}{n}\left[ \overline{w}_n(\bar{x}_1,...,\bar{x}_n)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]\\
&=\liminf_{n\to +\infty}\frac{1}{n}\left[ w_n(\bar{x}_1,...,\bar{x}_n)-\sum_{i=1}^n\log(1+|\bar{x}_i|^2)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]\\
&\geq \alpha -\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).\end{aligned}$$ The upper bound and Lemma \[convlog\] yield, $(x_1,...,x_n)$ being a minimizer of $w_n$: $$\begin{aligned}
&\limsup_{n\to +\infty} \frac{1}{n}\left[ \overline{w}_n(\bar{x}_1,...,\bar{x}_n)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]\\
&\leq \limsup_{n\to +\infty} \frac{1}{n}\left[ \overline{w}_n(x_1,...,x_n)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]\\
&= \limsup_{n\to +\infty}\frac{1}{n}\left[ w_n(x_1,...,x_n)-\sum_{i=1}^n\log(1+|x_i|^2)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]\\
&=\alpha_V -\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).\end{aligned}$$ Thus, we get $$\lim_{n\to +\infty}\frac{1}{n}\left[ \overline{w}_n(\bar{x}_1,...,\bar{x}_n)-n^2 I_V(\mu_V)+\frac{n}{2}\log n  \right]=\alpha_V -\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x).$$ Therefore, we have the following asymptotic expansion, as $n\to +\infty$, when $(\bar{x}_1,...,\bar{x}_n)$ is a minimizer of $\overline{w}_n$: $$\begin{aligned}
&\overline{w}_n(\bar{x}_1,...,\bar{x}_n)\\
&=n^2I_V(\mu_V)-\frac{n}{2}\log n+\left(\frac{1}{\pi}\min_{\mathcal{A}_1}W-\frac{1}{2}\int_{{\mathbb R}^2}m_V(x)\log m_V(x)dx-\int_{{\mathbb R}^2}V(x)d\mu_V(x)\right)n + o(n). \end{aligned}$$ We know that $\displaystyle I_V(\mu_V)=\frac{1}{2}$ (see [@BrauchartSphere Eq. (2.26)]) and $$\begin{aligned}
\int_{{\mathbb R}^2}\log(1+|x|^2)d\mu_V(x)&=\frac{1}{\pi}\int_{{\mathbb R}^2}\frac{\log(1+|x|^2)}{(1+|x|^2)^2}\\
&=2\int_0^{+\infty}\frac{r\log(1+r^2)}{(1+r^2)^2}dr\\
&=-\left[\frac{\log(1+r^2)}{1+r^2}  \right]_0^{+\infty}+\int_{{\mathbb R}^2}\frac{2r}{(1+r^2)^2}dr\\
&=-\left[ \frac{1}{1+r^2} \right]_0^{+\infty}\\
&=1.\end{aligned}$$ Hence we obtain, as $n\to +\infty$, $$\begin{aligned}
\overline{w}_n(\bar{x}_1,...,\bar{x}_n)&=\frac{n^2}{2}-\frac{n}{2}\log n + \left(   \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{1}{2}\int \log(\pi(1+|x|^2)^2)d\mu_V(x)-1\right)n + o(n)\\
&=\frac{n^2}{2}-\frac{n}{2}\log n +\left(   \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\int \log(1+|x|^2)d\mu_V(x) -1\right)n +o(n) \\
&= \frac{n^2}{2}-\frac{n}{2}\log n +\left(   \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}\right)n +o(n),\end{aligned}$$ and the asymptotic expansion of $\mathcal{E}_{\log}(n)$ is $$\mathcal{E}_{\log}(n)=\left( \frac{1}{2}-\log 2 \right)n^2-\frac{n}{2}\log n + \left( \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\log 2\right)n +o(n).$$

It follows from lower bound proved by Rakhmanov, Saff and Zhou [@2400418 Theorem 3.1], that $$\begin{aligned}
 \frac{1}{\pi}\min_{\mathcal{A}_1}W +\frac{\log\pi}{2}+\log 2&=\lim_{n\to +\infty}\frac{1}{n}\left[\textnormal{E}_{\log}(y_1,...,y_n)-\left(\frac{1}{2}-\log2\right)n^2+\frac{n}{2}\log n  \right]\\
 &\geq -\frac{1}{2}\log\left[\frac{\pi}{2}(1-e^{-a})^b  \right],\end{aligned}$$ where $\displaystyle a:=\frac{2\sqrt{2\pi}}{\sqrt{27}}\left(\sqrt{2\pi+\sqrt{27}} +\sqrt{2\pi} \right)$ and $\displaystyle b:=\frac{\sqrt{2\pi+\sqrt{27}}-\sqrt{2\pi}}{\sqrt{2\pi+\sqrt{27}}+\sqrt{2\pi}}$, and we get $$\min_{\mathcal{A}_1}W\geq -\frac{\pi}{2}\log\left[ 2\pi^2(1-e^{-a})^b \right]\approx - 4.6842707.$$

Computation of renormalized energy for the triangular lattice and upper bound for the term of order $n$ {#NTh}
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Sandier and Serfaty proved in [@Sandier_Serfaty Lemma 3.3] that $$W(\Lambda_{1/2\pi})=-\frac{1}{2}\log\left(\sqrt{2\pi b}|\eta(\tau)|^2\right),$$ where $\Lambda_{1/2\pi}$ is the triangular lattice corresponding to the density $m=1/2\pi$, $\displaystyle \tau=a+ib=1/2+i\frac{\sqrt{3}}{2}$ and $\eta$ is the Dedekind eta function defined, with $q=e^{2i\pi\tau}$, by $$\eta(\tau)=q^{1/24}\prod_{n\geq 1}(1-q^n).$$ We recall Chowla-Selberg formula (see [@SelbergChowla] or [@Cohen Proposition 10.5.11] for details): $$\prod_{m=1}^{|D|}\Gamma\left(\frac{m}{|D|}  \right)^{\frac{w}{2}\left( \frac{D}{m} \right)}=4\pi\sqrt{-D}b|\eta(\tau)|^4,$$ for $\tau$ a root of the integral quadratic equation $\alpha z^2+\beta z+\gamma=0$ where $D=\beta^2-4\alpha\gamma<0$, $\displaystyle \left(\frac{D}{m}  \right)$ is the Kronecker symbol, $w$ the number of roots of unity in ${\mathbb Q}(i\sqrt{-D})$ and when the class number of ${\mathbb Q}(i\sqrt{-D})$ is equal to 1. In our case $b=\sqrt{3}/2$, $w=6$, $\alpha=\beta=\gamma=1$ because $\tau$ is a root of unity, hence $D=-3$, $\displaystyle\left(  \frac{-3}{1}\right)=1$ and $\displaystyle \left(  \frac{-3}{2}\right)=-1$ by the Gauss Lemma. Therefore we obtain $$\Gamma(1/3)^3\Gamma(2/3)^{-3}=4\pi\sqrt{3}\frac{\sqrt{3}}{2}|\eta(\tau)|^4,$$ and by Euler’s reflection formula $\displaystyle \Gamma(1-1/3)\Gamma(1/3)=\frac{\pi}{\sin(\pi/3)}$, we get $$\frac{\Gamma(1/3)^6 3\sqrt{3}}{8\pi^3}=\frac{4\pi\sqrt{3}\times\sqrt{3}|\eta(\tau)|^4}{2}.$$ Finally we obtain $$|\eta(\tau)|^4=\frac{\Gamma(1/3)^6\sqrt{3}}{16\pi^4}.$$ Now it is possible to find the exact value of the renormalized energy of the triangular lattice $\Lambda_1$ of density $m=1$: $$\begin{aligned}
W(\Lambda_1)&=2\pi W(\Lambda_{1/2\pi})-\pi\frac{\log(2\pi)}{2}\\
&=-\pi \log\left(\sqrt{2\pi b}|\eta(\tau)|^2\right)-\pi\frac{\log(2\pi)}{2}\\
&= \pi\log\pi -\frac{\pi}{2}\log3-3\pi\log(\Gamma(1/3))+\frac{3}{2}\pi\log2\\
&=\pi\log\left( \frac{2\sqrt{2}\pi}{\sqrt{3}\Gamma(1/3)^3} \right)\\
&\approx  -4.1504128.\end{aligned}$$ Thus, we get $$\begin{aligned}
&\frac{1}{\pi}W(\Lambda_1)+\frac{\log\pi}{2}+\log 2\\
&=\frac{1}{\pi}\left( \pi\log\pi -\frac{\pi}{2}\log3-3\pi\log(\Gamma(1/3))+\frac{3}{2}\pi\log2  \right)+\frac{\log\pi}{2}+\log 2\\
&=2\log2 +\frac{1}{2}\log\frac{2}{3}+3\log\frac{\sqrt{\pi}}{\Gamma(1/3)}=C_{BHS}\approx  -0.0556053,\end{aligned}$$ and we find exactly the value $C_{BHS}$ conjectured by Brauchart, Hardin and Saff in [@Brauchart Conjecture 4]. Therefore Conjecture 2 is true if and only if the triangular lattice $\Lambda_1$ is a global minimizer of $W$ among vector-fields in $\mathcal{A}_1$, i.e. $$\min_{\mathcal{A}_1}W=W(\Lambda_1)=\pi\log\left( \frac{2\sqrt{2}\pi}{\sqrt{3}\Gamma(1/3)^3} \right).$$ Thus we obtain the following result

\[equiconj\] We have:

1.  It holds $$\lim_{n\to +\infty}\frac{1}{n}\left[\mathcal{E}_{\log}(n)-\left(\frac{1}{2}-\log2\right)n^2+\frac{n}{2}\log n  \right]\leq 2\log2 +\frac{1}{2}\log\frac{2}{3}+3\log\frac{\sqrt{\pi}}{\Gamma(1/3)}.$$

2.  Conjectures 2 and 3 are equivalent, i.e. $\displaystyle \min_{\mathcal{A}_1} W= W(\Lambda_1) \iff C=C_{BHS}$.

LAURENT BETERMIN\
Institut für Angewandte Mathematik,\
Universität Heidelberg,\
Im Neuenheimer Feld 294, 69120 Heidelberg. Deutschland\
`betermin@uni-heidelberg.de`\
ETIENNE SANDIER\
Université Paris-Est,\
LAMA - CNRS UMR 8050,\
61, Avenue du Général de Gaulle, 94010 Créteil. France\
& Institut Universitaire de France\
`sandier@u-pec.fr`

[^1]: See Section \[ad\] for the precise definition

[^2]: A Bravais lattice of ${\mathbb R}^2$, also called “simple lattice" is $L={\mathbb Z}\vec{u}\oplus {\mathbb Z}\vec{v}$ where $(\vec{u},\vec{v})$ is a basis of ${\mathbb R}^2$.

[^3]: See Section \[NTh\]

[^4]: We note that $\mu_V$ is an equilibrium measure is sufficient to obtain , as explained by Mizuta in [@Mizuta Theorem 6.1, Chapter 2] or by Bloom, Levenberg and Wielonsky in [@Bloom:2014qy Lemma 3.2]
