On the expected Betti numbers of the nodal set of random fields

Wigman, Igor

doi:10.2140/apde.2021.14.1797

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On the expected Betti numbers of the nodal set of random fields

Igor Wigman

Vol. 14 (2021), No. 6, 1797–1816

DOI: 10.2140/apde.2021.14.1797

Abstract

This note concerns the asymptotics of the expected total Betti numbers of the nodal set for an important class of Gaussian ensembles of random fields on Riemannian manifolds. By working with the limit random field defined on the Euclidean space we were able to obtain a locally precise asymptotic result, though due to the possible positive contribution of large percolating components this does not allow us to infer a global result. As a by-product of our analysis, we refine the lower bound of Gayet and Welschinger for the important Kostlan ensemble of random polynomials and its generalisation to Kähler manifolds.

Keywords

Gaussian random fields, Betti numbers, nodal sets, scaling limit, ergodicity

Mathematical Subject Classification 2010

Primary: 57R99, 60G15

Milestones

Received: 30 April 2019

Revised: 18 November 2019

Accepted: 9 March 2020

Published: 7 September 2021

Authors

Igor Wigman
Department of Mathematics King’s College London London United Kingdom

Vol. 14, No. 6, 2021