CS23: Stochastic Processes Under Constraints

Organizer: Dominic T. Schickentanz (Paderborn University)

Persistence of Strongly Correlated Stationary Gaussian fields: From Universal Probability Decay to Entropic Repulsion

Ohad Noy Feldheim

We study the persistence event (i.e., the event that the field remains positive on a large ball) for stationary Gaussian fields on \(\mathbb{R}^d\) or \(\mathbb{Z}^d\), whose correlation function is non-summable and asymptotically positively correlated (a spectral singularity at the origin).

Firstly, we give precise log-asymptotics for the persistence probability in terms of capacity, expressed in terms of the order of the singularity at the origin. We then establish that the field conditioned to persist is propelled to height \(C\sqrt{\log T}\) and takes the shape of a deterministic function (the “equilibrium potential”).

This generalises a classical result of Bolthausen, Deuschel and Zeitouni for the Gaussian free field (GFF) on \(\mathbb{Z}^d\), \(d \ge 3\), to a wide class of Gaussian fields with spectral singularity, showing that both the leading order of persistence probability decay and entropic repulsion are, in a sense, universal, depending only on the order of the spectral singularity.