January 2023 Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d3
Hugo Duminil-Copin, Alejandro Rivera, Pierre-François Rodriguez, Hugo Vanneuville
Author Affiliations +
Ann. Probab. 51(1): 228-276 (January 2023). DOI: 10.1214/22-AOP1594


For the Bargmann–Fock field on Rd with d3, we prove that the critical level c(d) of the percolation model formed by the excursion sets {f} is strictly positive. This implies that for every sufficiently close to 0 (in particular for the nodal hypersurfaces corresponding to the case =0), {f=} 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 d3 is very different from the behavior of nodal lines of their 2-dimensional analogues.

Funding Statement

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No. 757296 and No. 851565). The authors acknowledge funding from the SwissMap funded by the Swiss FNS.


The problem was suggested to us by Peter Sarnak. We thank him as well as Vincent Beffara and Damien Gayet for inspiring discussions. We also thank Matthis Lehmkühler very much for providing the proofs of the FKG inequalities from Section A to us. Moreover, we thank Damien Gayet who provided the proof of Lemma D.1 to us and Thomas Letendre for help with such topological questions. An important part of this work was done while AR and HV were visiting HD-C and P-FR in IHES that we thank for hospitality. Finally, we wish to thank an anonymous referee for helpful comments and their careful reading of our paper.


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Hugo Duminil-Copin. Alejandro Rivera. Pierre-François Rodriguez. Hugo Vanneuville. "Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d3." Ann. Probab. 51 (1) 228 - 276, January 2023. https://doi.org/10.1214/22-AOP1594


Received: 1 September 2021; Revised: 1 June 2022; Published: January 2023
First available in Project Euclid: 22 November 2022

MathSciNet: MR4515694
zbMATH: 1515.60153
Digital Object Identifier: 10.1214/22-AOP1594

Primary: 60G60 , 60K35

Keywords: Gaussian fields , percolation

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 1 • January 2023
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