Work

Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3

Year: 2022

Type: article

Abstract: For the Bargmann–Fock field on Rd with d≥3, we prove that the critical level ℓc(d) of the percolation model formed by the excursion sets {f≥ℓ} is strictly positive. This implies that for every ℓ sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case ℓ=0), {f=ℓ} contains an unbounded connected component that visits “most” of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behavior of nodal hypersurfaces of these Gaussian fields in Rd for d≥3 is very different from the behavior of nodal lines of their 2-dimensional analogues.

Source: The Annals of Probability

DOI: 10.1214/22-aop1594

Authors: Hugo Duminil-Copin, Alejandro Rivera, Pierre-François Rodriguez, Hugo Vanneuville

Institutions: University of Geneva, École Polytechnique Fédérale de Lausanne, Imperial College London, Institut Fourier, Université Grenoble Alpes

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: National Centres of Competence in Research SwissMAP, European Commission, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award: G5923453602