Russo–Seymour–Welsh estimates for the Kostlan ensemble of random polynomials

D. Beliaev, S. Muirhead, I. Wigman

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Beginning with the predictions of Bogomolny–Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo–Seymour–Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations. The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a ‘typical’ real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert’s 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara–Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus.

Original languageEnglish
Pages (from-to)2189-2218
Number of pages30
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number4
Publication statusPublished - Nov 2021
Externally publishedYes


  • Gaussian field
  • Kostlan ensemble
  • Nodal set
  • Percolation
  • Russo–Seymour–Welsh estimates


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