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

Abstract

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 $N\to \infty$. However, this minimizer is not unique among infinite point configurations (for example small 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.

23 pages

References in corpus (2)

• Next Order Asymptotics and Renormalized Energy for Riesz Interactions
• Face-centered cubic crystallization of atomistic configurations