Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin

TL;DR

This paper analyzes the persistence probabilities and entropic repulsion of a broad class of stationary Gaussian fields with spectral singularities, providing explicit asymptotics and revealing their universality depending only on key parameters.

Contribution

It generalizes previous results on Gaussian free fields to Gaussian fields with spectral singularities, deriving explicit asymptotics based on capacity and equilibrium potential.

Findings

Derived exact log-asymptotics of persistence probability.

Determined entropic repulsion profile conditioned on persistence.

Showed universality depending only on spectral singularity order and dimension.

Abstract

We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $\alpha \in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $\alpha$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $\alpha$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.