  The paper deals with asymptotic nodal geometry for the Laplace-Beltrami
operator on closed surfaces. Given an eigenfunction f corresponding to a large
eigenvalue, we study local asymmetry of the distribution of sign(f) with
respect to the surface area. It is measured as follows: take any disc centered
at the nodal line {f=0}, and pick at random a point in this disc. What is the
probability that the function assumes a positive value at the chosen point? We
show that this quantity may decay logarithmically as the eigenvalue goes to
infinity, but never faster than that. In other words, only a mild local
asymmetry may appear. The proof combines methods due to Donnelly-Fefferman and
Nadirashvili with a new result on harmonic functions in the unit disc.
