Abstract: The classical XY model is a ferromagnetic spin model taking values in the unit circle and in two dimensions, it is known to undergo a subtle type of phase transition: at high temperatures it is exponentially disordered, at low temperatures the correlations decay algebraically. This transition is called the Berezinskii-Kosterlitz-Thouless (BKT) transition. Particular to planar graphs is a duality of these spin systems with integer-valued height function, which played an important role in all proofs of the transition. These dual height functions also undergo a phase transition on the square lattice, but on a hypercubic lattice of dimensions larger than 3, they do not. So, what happens for planar graphs that are transient? If we require mild regularity, we are essentially forced to take a hyperbolic graph, but still what happens for the integer-valued height function? Will it stay the same as for d > 2, or undergo a transition as in d = 2? I will explain how a type of BKT transition does manifest itself, which is visible both for the height function and for the corresponding Coulomb gas. Based on joint work with Christophe Garban.