Boundedness of the nodal domains of additive Gaussian fields

2021/11/17 by Stephen Muirhead, Muirhead, Stephen
Mathematics · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Geometry and complex manifolds #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2111.09059

Abstract

We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets \f ≤ ℓ\ of additive planar Gaussian fields are bounded almost surely at the critical level ℓ_c = 0. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension d ≥ 3 the excursion sets have unbounded components at all levels.

Related

• Existence of an unbounded nodal hypersurface for smooth Gaussian fields\n in dimension d \≥ 3
• A covariance formula for the number of excursion set components of Gaussian fields and applications
• A central limit theorem for the number of excursion set components of Gaussian fields
• Three central limit theorems for the unbounded excursion component of a Gaussian field
• Quasi-independence for nodal lines