Vol. 14, No. 6, 2021

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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

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.

Gaussian random fields, Betti numbers, nodal sets, scaling limit, ergodicity
Mathematical Subject Classification 2010
Primary: 57R99, 60G15
Received: 30 April 2019
Revised: 18 November 2019
Accepted: 9 March 2020
Published: 7 September 2021
Igor Wigman
Department of Mathematics
King’s College London
United Kingdom