  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 to infer a global result. As a by-product of our
analysis, we refine the lower bound of Gayet-Welschinger for the important
Kostlan ensemble of random polynomials and its generalisation to K\"{a}hler
manifolds.
