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.
