Uniform Lipschitz functions on the triangular lattice have logarithmic variations

TL;DR

This paper proves that uniform Lipschitz functions on the triangular lattice exhibit logarithmic variation growth, and constructs a unique infinite-volume Gibbs measure for the associated loop O(2) model, demonstrating scale-invariance and finite loops.

Contribution

It establishes the order of variation for Lipschitz functions on the triangular lattice and constructs a unique Gibbs measure for the loop O(2) model with finite loops and scale-invariance.

Findings

Variations grow as √log N for Lipschitz functions

Constructed a unique infinite-volume Gibbs measure for the loop O(2) model

Proved RSW-type estimates for the associated spin model

Abstract

Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop $O(2)$ model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.