Three central limit theorems for the unbounded excursion component of a Gaussian field

Abstract

For a smooth, stationary Gaussian field $f$ on Euclidean space with fast correlation decay, there is a critical level ${\ell }_c$ such that the excursion set $\{f\ge \ell \}$ contains a (unique) unbounded component if and only if $\ell <{\ell }_c$. We prove central limit theorems for the volume, surface area and Euler characteristic of this unbounded component restricted to a growing box. For planar fields, the results hold at all supercritical levels (i.e. all $\ell <{\ell }_c$). In higher dimensions the results hold at all sufficiently low levels (all $\ell <-{\ell }_c<{\ell }_c$) but could be extended to all supercritical levels by proving the decay of truncated connection probabilities. Our proof is based on the martingale central limit theorem.

References in corpus (12)

• Nodal portraits of quantum billiards: Domains, lines, and statistics
• On Nonlinear Functionals of Random Spherical Eigenfunctions
• The sharp phase transition for level set percolation of smooth planar Gaussian fields
• On the distribution of the critical values of random spherical harmonics
• On the Excursion Sets of Spherical Gaussian Eigenfunctions
• Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics
• Universality class for bootstrap percolation with $m=3$ on the cubic lattice
• Fluctuations of the number of excursion sets of planar Gaussian fields
• On Bogomolny-Schmit conjecture
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• A central limit theorem for the number of excursion set components of Gaussian fields
• A covariance formula for the number of excursion set components of Gaussian fields and applications