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

Banff International Research Station for Mathematical Innovation and Discovery

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. This is a joint with with D. Beliaev and S. Muirhead

Mathematics; Quantum theory; Global analysis, analysis on manifolds; Mathematical physics

video/mp4

Author affiliation: King's College London

2019-03-13

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68679

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/