Classification:** Primary 52A40, 82B05 ; Secondary 41A60, 82B21, 31C20.\
**Keywords:** Coulomb gas ; Abrikosov lattices ; Triangular lattice ; Renormalized energy ; Crystallization ; Logarithmic energy ; Number theory ; Logarithmic potential theory ; Weak confinement ; Gamma-convergence ; Ginzburg-Landau ; Vortices.\

Introduction
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Let $(x_1,...,x_n)\in({\mathbb R}^2)^n$ be a configuration of $n$ points interacting through a logarithmic potential and confined by an external field $V$. The Hamiltonian of this system, also known as a Coulomb gas, is defined as $$w_n(x_1,...,x_n):=-\sum_{i\neq j}^n\log|x_i-x_j|+n\sum_{i=1}^nV(x_i)$$ where $|\cdot|$ is the Euclidean norm in ${\mathbb R}^2$. The minimization of $w_n$ is linked to the following classical problem of logarithmic potential theory : find a probability measure $\mu_V$ on ${\mathbb R}^2$ which minimizes $$\label{IV}
I_V(\mu):=\iint_{{\mathbb R}^2\times {\mathbb R}^2}\left(\frac{V(x)}{2}+\frac{V(y)}{2}-\log|x-y|\right)d\mu(x)d\mu(y)$$ amongst all probability measures $\mu$ on ${\mathbb R}^2$. This type of problem dates back to Gauss. More recent references are the thesis of Frostman [@Frost] and the monography of E.Saff and V.Totik [@SaffTotik]. The usual assumptions on $V:{\mathbb R}^2\to{\mathbb R}\cup\{+\infty\}$ are that it is lower semicontinuous, that it is finite on a set of nonzero capacity, and that it satisfies the growth assumption $$\label{classicgrowth}
\lim_{|x|\to +\infty}\left\{V(x)-2\log| x|\right\}=+\infty.$$ These assumptions ensure that a unique minimizer $\mu_V$ of $I_V$ exists and 