Year: 2021
Type: article
Abstract: Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order $$\sqrt{\log N}$$ 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.
Authors: Alexander Glazman, Ioan Manolescu
Institutions: University of Vienna, University of Fribourg
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,690
OpenAlex ID: W2895862273