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

## TL;DR

This paper investigates the asymptotic behavior of the expected Betti numbers of the nodal sets of Gaussian random fields on manifolds, providing local asymptotics and refining bounds for related polynomial ensembles.

## Contribution

It introduces a local asymptotic analysis of Betti numbers for Gaussian fields and improves existing bounds for Kostlan ensembles on Kähler manifolds.

## Key findings

- Derived locally precise asymptotics for Betti numbers
- Refined lower bounds for Kostlan polynomial ensembles
- Highlighted limitations in inferring global results due to large components

## 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 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.