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.
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