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 , 