On a Riemann surface $\Sigma$, consider the space $H_\lambda$ spanned by eiigenfunctions corresponding to eigenvalues $\leq \lambda$.   By Weyl's asymptotic formula we know that

$$ \dim H_\lambda  \sim const \lambda $$

as $\lambda \to \infty$.  Denote  by $S_\lambda$ the unit sphere in $H_\lambda$ with respect to the $L^2$-norm.  Equip with with the unique rotationally invaraint measure of total volume $1$ so now  you can think of $S_\lambda$ as a probability space. Thus, for any $f\in S_\lambda$, the number $N_f$ of zonal regions of $f$ is a  random   variable.  We denote by $N_\lambda$ its expectation, i.e., the average number of  zonal domains  of a function $f\in S_\lambda$.  One can show  that there exists a constant  $C>0$ such that

$$ N_\lambda \leq C\lambda $$

for $\lambda \gg 0$.     For a proof see [this preprint][1].


I actually believe that 

$$ N_\lambda \sim C\lambda  $$

as $\lambda \to \infty$, but I have no promising idea how to approach this.


  [1]: http://www3.nd.edu/~lnicolae/CritSetStat.pdf