  We consider the classical XY model (or classical rotor model) on the
two-dimensional square lattice graph as well as its dual model, which is a
model of height functions. The XY model has a phase transition called the
Berezinskii-Kosterlitz-Thouless transition. There is a heuristic which predicts
that this phase transition should coincide with the localisation-delocalisation
transition for height functions: the primal and dual model share the same
partition function, and the phase transition of either model should coincide
with the unique non-analytic point of the partition function when expressed in
terms of the inverse temperature. We use probabilistic arguments to prove that
the correlation length (the reciprocal of the mass) of the XY model is exactly
twice the correlation length of the height function, which implies in
particular that the prediction of this heuristic is correct: namely, that the
BKT phase for the XY model coincides exactly with the delocalised phase of the
dual height function.
