be admissible. Then, for every configuration $(x_1,...x_n)\in ({\mathbb R}^2)^n$, $n\geq 2$, we have $$w_n(x_1,...,x_n)=n^2I_V(\mu_V)-\frac{n}{2}\log n +\frac{1}{\pi}W(\nabla H'_n,\indic_{{\mathbb R}^2})+2n\sum_{i=1}^n\zeta(x_i). \label{splitting}$$

We may proceed as in the proof of [@2DSandier Lemma 3.1] and make use of the Frostman type inequalities and and Lemma \[convHn\]. The important point is that, as shown in the proof of the previous lemma, we have $H_n(x)=O(|x|^{-1})$ and $\nabla H_n(x)=O(|x|^{-2})$ as $|x|\to +\infty$ which implies, exactly like in the compact support case, that $$\lim_{R\to +\infty} \int_{\partial B_R} H_n(x)\nabla H_n(x).\vec{\nu}(x)dx=0$$ where $\vec{\nu}(x)$ is the outer unit normal vector at $x\in \partial B_R$.

Lower bound {#UsLem}
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Here we follow the strategy of [@2DSandier], pointing out the required modifications in the noncompact case.

Mass spreading result and modified density $g$
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We have the following result from [@2DSandier Proposition 3.4]:

Let $V$ be admissible and assume $(\nu,E)$ are such that $\displaystyle \nu=\sum_{p\in\Lambda}\delta_p$ for some finite subset $\Lambda \subset {\mathbb R}^2$ and ${\mathop{\rm div}\nolimits}E=2\pi(\nu -m_V)$, ${\mathop{\rm curl}\nolimits}E=0$ in ${\mathbb R}^2$. Then, given any $\rho>0$ there exists a signed measure $g$ supported on ${\mathbb R}^2$ and such that:

-   there exists a family $\mathcal{B}_\rho$ of disjoint closed balls covering Supp$(\nu)$, with the sum of the radii of the balls in $\mathcal{B}_\rho$ intersecting 