Non UBC
DSpace
Wigman, Igor
2019-03-13T05:05:50Z
2018-07-19T09:01
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
https://circle.library.ubc.ca/rest/handle/2429/68679?expand=metadata
58.0
video/mp4
Author affiliation: King's College London
Banff (Alta.)
10.14288/1.0376845
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Faculty
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Quantum theory
Global analysis, analysis on manifolds
Mathematical physics
Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials
Moving Image
http://hdl.handle.net/2429/68679