Expected volume and Euler characteristic of random submanifolds

In a closed manifold of positive dimension n, we estimate the expected volume and Euler characteristic for random submanifolds of codimension r∈{1,…,n} in two different settings. On one hand, we consider a closed Riemannian manifold and some positive λ. Then we take r independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than λ and consider the random submanifold defined as the common zero set of these r functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as λ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle L and a rank r holomorphic vector bundle E that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of E⊗Ld as d goes to infinity. The same techniques apply to both settings.