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

TL;DR

This paper proves that, among stationary point processes, the only minimizer of the 1D Log-gas renormalized energy is the averaged lattice configuration, establishing a form of uniqueness at the process level.

Contribution

The paper demonstrates that the minimizer of the 1D Log-gas renormalized energy is unique among stationary processes, extending previous crystallization results.

Findings

Uniqueness of the minimizer among stationary point processes.

Quantitative estimate linking two-point correlation functions to renormalized energy.

The only minimizer is the averaged lattice configuration.

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 $\mathbb{z}$ 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.