**Abstract:** FrÃ¶hlich and Spencer proved the existence of the Kosterlitz-Thouless phase transition in their landmark article in 1981. Amongst other things, they demonstrate that the solid-on-solid model on two-dimensional lattices delocalises at high temperature. In the first half of the talk, we present a new, simplified proof of this result in the infinite-volume setting, which is based on new insights regarding phase coexistence of planar percolation and a number of results from Sheffield’s Random Surfaces. This also proves that the model delocalises at any slope; a new result. In the second half of the talk, we discuss how this infinite-volume result relates to finite-volume measures. The bridge from infinite- to finite-volume is made through a new correlation inequality, which asserts that the absolute value of the solid-on-solid height function has the FKG lattice property. We shall present the proof of this fact, which relies on the presence of a “hidden” FK-Ising model. Finally, depending on time, we will discuss some additional (simple) properties of this FK-Ising model which might be useful to obtain quantitative versions of the result.

## Video