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}
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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 