Research output: Contribution to journal › Article › peer-review

**Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials.** / Beliaev, Dmitry; Muirhead, Stephen; Wigman, Igor.

Research output: Contribution to journal › Article › peer-review

Beliaev, D, Muirhead, S & Wigman, I 2020, 'Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials', *Annales de l’Institut Henri Poincaré, Probabilités et Statistiques*.

Beliaev, D., Muirhead, S., & Wigman, I. (Accepted/In press). Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials. *Annales de l’Institut Henri Poincaré, Probabilités et Statistiques*.

Beliaev D, Muirhead S, Wigman I. Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques. 2020 Dec 10.

@article{8b14f9212ee041f2ac5006b23b14d626,

title = "Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials",

abstract = "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.",

author = "Dmitry Beliaev and Stephen Muirhead and Igor Wigman",

year = "2020",

month = dec,

day = "10",

language = "English",

journal = "Annales de l{\textquoteright}Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials

AU - Beliaev, Dmitry

AU - Muirhead, Stephen

AU - Wigman, Igor

PY - 2020/12/10

Y1 - 2020/12/10

N2 - 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.

AB - 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.

M3 - Article

JO - Annales de l’Institut Henri Poincaré, Probabilités et Statistiques

JF - Annales de l’Institut Henri Poincaré, Probabilités et Statistiques

SN - 0246-0203

ER -

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