Extreme values of non-Gaussian fields

In recent years the extremal behaviour of log-correlated spatial Gaussian processes has drawn a lot of attention. Among many other results, it is known for the lattice discrete Gaussian free field (DGFF) in d=2 as well as for general log-correlated Gaussian fields, that the limiting law of the centred maximum is a randomly shifted Gumbel distribution. While for Gaussian fields the picture is rather complete, many difficulties arise when the field of interest in non-Gaussian. In this thesis, we study the extreme values of the non-Gaussian sine-Gordon field and P(φ)₂ field on the unit torus in dimension d=2. Our analysis includes the φ⁴₂ field. To this end, we develop tools, which we subsequently use to establish results for their extreme values, which are analogous to the known results for the Gaussian free field in d=2. In particular, we prove that the centred global maximum of the non-Gaussian fields of interest converges in distribution to a randomly shifted Gumbel distribution, which confirms the conjectured behaviour of these fields. For the sine-Gordon field, we extend the scope of the extreme values to the local extremal process, which also includes information about the local extrema of the field. More precisely, we prove that this random measure converges to a certain Poisson point process with random intensity measure. For both the sine-Gordon field and the P(φ)₂ field, the main tool is a coupling result between the well-studied Gaussian free field and the field of interest. This allows to represent the non-Gaussian field as a sum of the Gaussian free field and a difference field for which further probabilistic regularity estimates are established using renormalisation group and stochastic control techniques, in particular the Polchinski renormalisation group approach and the Boué-Dupuis variational formula.