  We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin that
exhibits a relation between the average local growth of a Laplace eigenfunction
on a closed surface and the global size of its nodal set. More precisely, we
provide a lower and an upper bound to the Hausdorff measure of the nodal set in
terms of the expected value of the growth exponent of an eigenfunction on disks
of wavelength like radius. Combined with Yau's conjecture, the result implies
that the average local growth of an eigenfunction on such disks is bounded by
constants in the semi-classical limit. We also obtain results that link the
size of the nodal set to the growth of solutions of planar Schr\"odinger
equations with small potential.
