that it has compact support.

Recently, Hardy and Kuijlaars [@Hardy2] (see also [@Hardy]) proved that if one replaces by the so-called weak growth assumption $$\label{growthL}
\liminf_{|x|\to +\infty}\left\{V(x)-\log(1+|x|^2)  \right\}>-\infty,$$ then $I_V$ still admits a unique minimizer, which may no longer have compact support. Moreover Bloom, Levenberg and Wielonsky [@Bloom:2014qy] proved that the classical Frostman type inequalities still hold in this case.\
Coming back to the minimum of the discrete energy $w_n$, its relation to the minimum of $I_V$ is that as $n\to +\infty$, the minimum of $w_n$ is equivalent to $n^2 \min I_V$. The next term in the asymptotic expansion of $w_n$ was derived by Sandier and Serfaty [@2DSandier] in the classical case , it reads $$\min w_n =  n^2 \min I_V - \frac{n}{2} \log n + \alpha_V n +o(n),$$ where $\alpha_V$ is related to the minimum of a Coulombian renormalized energy studied in [@Sandier_Serfaty] which quantifies the discrete energy of infinitely many positive charges in the plane screened by a uniform negative background. Note that rather strict assumptions in addition to need to be made on $V$ for this expansion to hold, but they are satisfied in particular if $V$ is smooth and strictly convex.

Here, we show that such 