van Engelenburg, D. G. P., & Lis, M. (2025). On the duality between height functions and continuous spin models. Probability and Mathematical Physics (PMP), 6(4), 1291–1325. https://doi.org/10.2140/pmp.2025.6.1291
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 localization on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalization of the height function implies a BKT phase transition in planar models; and also delocalization itself for height functions on periodic “almost” planar graphs.
