  In a closed manifold of positive dimension $n$, we estimate the expected
volume and Euler characteristic for random submanifolds of codimension $r\in
\{1,...,n\}$ in two different settings. On one hand, we consider a closed
Riemannian manifold and some positive $\lambda$. Then we take $r$ independent
random functions in the direct sum of the eigenspaces of the Laplace-Beltrami
operator associated to eigenvalues less than $\lambda$ 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 $\lambda$ goes to infinity. On the other hand, we consider a
complex projective manifold defined over the reals, equipped with an ample line
bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$
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 $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to
infinity. The same techniques apply to both settings.
