On the duality between height functions and continuous spin models

TL;DR

This paper explores the duality between height functions and abelian spin models, deriving general results on variance bounds, phase transitions, and delocalisation phenomena in these models.

Contribution

It establishes a universal variance bound, links delocalisation to BKT phase transitions, and extends results to various graph classes, broadening understanding of height function behavior.

Findings

Universal upper bound on height function variance

Delocalisation implies BKT phase transition in planar models

Delocalisation established for height functions on periodic almost planar graphs

Abstract

We revisit the classical phenomenon of duality between random integer-valued height functions with positive definite potentials and abelian spin models with O(2) symmetry. We use it to derive new results in quite high generality including: a universal upper bound on the variance of the height function in terms of the Green's function (a GFF bound) which among others implies localisation on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalisation of the height function implies a BKT phase transition in planar models; and also delocalisation itself for height functions on periodic ``almost'' planar graphs.