  Sandier and Serfaty studied the one-dimensional Log-gas model, in particular
they gave a crystallization result by showing that the one-dimensional lattice
$\mathbb{z}$ is a minimizer for the so-called renormalized energy which they
obtained as a limit of the $N$-particle Log-gas Hamiltonian for $N \to \infty$.
However, this minimizer is not unique among infinite point configurations (for
example small perturbations of $\mathbb{z}$ leave the renormalized energy
unchanged). In this paper, we establish that uniqueness holds at the level of
(stationary) point processes, the only minimizer being given by averaging
$\mathbb{z}$ over a choice of the origin in $[0,1]$. This is proved by showing
a quantitative estimate on the two-point correlation function of a process in
terms of its renormalized energy.
