Equidistribution of the conormal cycle of random nodal sets

Nguyen Viet Dang (1) , Gabriel Riviere (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Keywords: eigenfunctions of the Laplacian; Gaussian measures; nodal sets; conormal cycle

Domains: Spectral Theory [math.SP]; Differential Geometry [math.DG]; Probability [math.PR]; Mathematical Physics [math-ph]

Nguyen Viet Dang, Gabriel Riviere. Equidistribution of the conormal cycle of random nodal sets. Journal of the European Mathematical Society, 2018, ⟨10.4171/JEMS/828⟩. ⟨hal-01189071v2⟩