DLR equations, number-rigidity and translation-invariance for infinite-volume limit points of the 2DOCP

TL;DR

This paper establishes that at any positive temperature, the infinite-volume limit points of the 2D one-component plasma satisfy DLR equations, exhibit number-rigidity, and are translation-invariant, extending known results from the Ginibre ensemble.

Contribution

It rigorously proves DLR equations, number-rigidity, and translation-invariance for the 2DOCP's infinite-volume limits, extending prior results to a broader setting.

Findings

Infinite-volume limit points satisfy DLR equations.

Limit points exhibit number-rigidity.

Limit points are translation-invariant.

Abstract

We prove that, at arbitrary positive temperature, every infinite-volume local limit point of the two-dimensional one-component plasma (2DOCP, also known as Coulomb or log-gas, or jellium) satisfies a system of Dobrushin-Lanford-Ruelle (DLR) equations - in particular, we explain how to rigorously make sense of those despite the long-range interaction. We also show number-rigidity and translation-invariance of the limiting processes. This extends results known for the infinite Ginibre ensemble. The proofs combine recent results on finite 2DOCP's and classical infinite-volume techniques.