Extremes of Gaussian fields with a product term in the variance

TL;DR

This paper analyzes the probability of high excursions in a Gaussian field with a specific variance structure, revealing new localization regimes and asymptotics not covered by traditional assumptions.

Contribution

It introduces a novel analysis of Gaussian fields with product-form variance loss, identifying new regimes and asymptotics beyond locally additive models.

Findings

Identifies side-attached localization regimes for certain parameter ranges.

Derives high-level asymptotics including logarithmic and side-dominated regimes.

Extends understanding of Gaussian fields beyond classical locally additive assumptions.

Abstract

We study the high excursion probability of a centered Gaussian field on a square. Writing \(\sigma\) and \(r\) for its standard deviation and correlation function, we assume that \(\sigma\) has a unique maximum at the corner \(\boldsymbol{0}=(0,0)\) and \[ 1-\sigma(\boldsymbol{t}) \sim t_1^\beta+t_2^\beta+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0} \] in \(\mathbb R_+^2\). The local correlation is assumed to satisfy \[ 1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^\alpha+|t_2-s_2|^\alpha, \qquad 0<\alpha<\beta . \] This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range \(a<\beta/2\), the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case.