In recent years the extremal behaviour of log-correlated spatial Gaussian processes has drawn a lot of attention. For the lattice discrete Gaussian free field (DGFF) in d=2 as well as for general log-correlated Gaussian fields, the limiting law of the centred maximum has been identified as a randomly shifted Gumbel distribution.
In this talk I will explain how an analogous result is obtained for the non-Gaussian sine-Gordon field. I will present a strong coupling at all scales of the sine-Gordon field with the Gaussian free field and demonstrate how this can be used to extend existing methods for the maximum of the DGFF . The talk is based on a joint work with R. Bauerschmidt.