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 