We study the persistence event (i.e., the event that the field remains positive on a large ball) for stationary Gaussian fields on R^d or Z^d, with a spectral singularity at the origin. 

First, we give precise log-asymptotics for the persistence probability and connect it to a notion of capacity. Second, we establish that the field exhibits ``entropic repulsion”: conditioned on persistence on a ball of radius T, the field is propelled to height $C\sqrt{\log T}$ and fluctuates around a certain deterministic function (the “shape function”).

This generalises a classical result of Bolthausen, Deuschel and Zeitouni for the Gaussian free field (GFF) on Z^d for $d \ge 3$, to a wide class of Gaussian fields with long-range correlations.

Based on joint work with Ohad Feldheim and Stephen Muirhead.