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.
Highlights
Height functions occupy a central role in statistical mechanics models on lattices
The predicted conformal invariance of these models is tightly linked to the convergence of their associated height functions to the Gaussian Free Field (GFF) or variations of it
In this paper we study integer-valued height functions defined on the vertices of the two-dimensional triangular lattice T, or equivalently on the faces of the hexagonal lattice H
Summary
Height functions occupy a central role in statistical mechanics models on lattices. The predicted conformal invariance of these models is tightly linked to the convergence of their associated height functions to the Gaussian Free Field (GFF) or variations of it. Both statements were proved only in a handful of cases, and remain fascinating conjectures in general. In this paper we study integer-valued height functions defined on the vertices of the two-dimensional triangular lattice T, or equivalently on the faces of the hexagonal lattice H. For any finite domain D of H, we will consider a uniformly chosen Lipschitz function among those with values 0 outside of D. The question of interest is the behaviour of such a function, especially as the domain D increases towards H
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