an asymptotic formula still holds when the classical growth assumption is replaced with the weak growth assumption . However it is no longer obvious that the minimum of $w_n$ is achieved in this case, as the weak growth assumption could allows one point to go to infinity.

 \[thintro\] Let $V$ be an admissible potential [^1]. Then the following asymptotic expansion holds. $$\label{expansion}
\inf_{({\mathbb R}^2)^n} w_n=I_V(\mu_V)n^2-\frac{n}{2}\log n +\left(\frac{1}{\pi}\min_{\mathcal{A}_1}W-\frac{1}{2}\int_{{\mathbb R}^2}m_V(x)\log m_V(x)dx\right)n + o(n),$$ where $\mu_V = m_V(x)\,dx$ is the unique minimizer of $I_V$ (see Section \[RE\] for precise definitions of $W$ and $\mathcal{A}_1$.)

This result is proved using the methods in [@Sandier_Serfaty; @2DSandier] suitably adapted to equilibrium measures with possibly non-compact support together with the compactification approach in [@Hardy2; @Hardy; @Bloom:2014qy]. This compactification allows also to connect the discrete energy problem for log gases in the plane with the discrete logarithmic energy problem for finitely many points on the unit sphere $\S^2$ in the Euclidean space ${\mathbb R}^3$.\
The logarithmic energy of a configuration $(y_1,...,y_n)\in (\S^2)^n$ is given by $$\textnormal{E}_{\log}(y_1,...,y_n):=-\sum_{i\neq j}^n\log\|y_i-y_j\|,$$ where $\|\cdot\|$ is the Euclidean norm in ${\mathbb R}^3$. Finding a minimizer of such an energy functional is a problem with many links and ramifications as discussed in the fundamental paper 