Equidistribution of the conormal cycle of random nodal sets

TL;DR

This paper investigates the asymptotic behavior of the conormal cycle of random nodal sets on manifolds, showing equidistribution in odd dimensions and bounds in even dimensions, with implications for valuations like Euler characteristic.

Contribution

It provides new results on the asymptotic distribution of the conormal cycle of random nodal sets, including equidistribution in odd dimensions and bounds in even dimensions, extending understanding of geometric properties of random eigenfunctions.

Findings

In odd dimensions, the expectation of the conormal cycle equidistributes on cotangent bundle fibers.

In even dimensions, an upper bound of lower order on the expectation is established.

Results imply properties of the Euler characteristic and other valuations of random nodal sets.

Abstract

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidistributes on the fibers of the cotangent bundle. In the case where the dimension is even, we obtain an upper bound of lower order on the expectation. Using recent results of Alesker, we also deduce some properties on the asymptotic expectation of any smooth valuation including the Euler characteristic of random nodal sets.