A uniqueness result for minimizers of the 1D Log-gas renormalized energy

In [21] Sandier and Serfaty studied the one-dimensional Log-gas model, in particular they gave a crystallization result by showing that the one-dimensional lattice 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→∞. However, this minimizer is not unique among infinite point configurations (for example local perturbations of 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 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.