In recent years there has been significant progress in the study of extreme values of log-correlated Gaussian fields, thanks to the work of Bramson, Ding, Roy, Zeitouni and Biskup, Louidor. For instance, it has been shown that for the discrete Gaussian free field (DGFF) in d=2 and for log-correlated Gaussian fields the limiting law of the centred maximum is a randomly shifted Gumbel distribution.
In this talk I will present analogous results for non-Gaussian fields such as the sine-Gordon field and the \Phi^4 field in d = 2. The main tool is a coupling at all scales between the field of interest and the DGFF which emerges from the Polchinski renormalisation group approach as well as the Boue-Dupuis variational formula. The talk is based on joint works with Roland Bauerschmidt and Trishen Gunaratnam, Nikolay Barashkov.