  Using the multiplicities of the Laplace eigenspace on the sphere (the space
of spherical harmonics) we endow the space with Gaussian probability measure.
This induces a notion of random Gaussian spherical harmonics of degree $n$
having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the
nodal lines of random spherical harmonics. It is known that the expected length
is of order $n$. It is natural to conjecture that the variance should be of
order $n$, due to the natural scaling. Our principal result is that, due to an
unexpected cancelation, the variance of the nodal length of random spherical
harmonics is of order $\log{n}$. This behaviour is consistent with the one
predicted by Berry for nodal lines on chaotic billiards (Random Wave Model). In
addition we find that a similar result is applicable for "generic" linear
statistics of the nodal lines.
