A Uniqueness Result for Minimizers of the 1D Log-gas Renormalized Energy

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.