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