  Given an ample real Hermitian holomorphic line bundle $L$ over a real
algebraic variety $X$, the space of real holomorphic sections of $L^{\otimes
d}$ inherits a natural Gaussian probability measure. We prove that the
probability that the zero locus of a real holomorphic section $s$ of
$L^{\otimes d}$ defines a maximal hypersurface tends to $0$ exponentially fast
as $d$ goes to infinity. This extends to any dimension a result of Gayet and
Welschinger valid for maximal real algebraic curves inside a real algebraic
surface.
  The starting point is a low degree approximation property which relates the
topology of the real vanishing locus of a real holomorphic section of
$L^{\otimes d}$ with the topology of the real vanishing locus a real
holomorphic section of $L^{\otimes d'}$ for a sufficiently smaller $d'<d$. Such
a statement is inspired by a recent work of Diatta and Lerario.
