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
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\[THM\] 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 