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 