Nodal lines, ergodicity and complex numbers

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is,the zeros of the analytic continuation of the eigenfunctions tothe complexification of the underlying space. Such acomplexification of the problem is analogous to studying thecomplex zeros of polynomials with real coefficients. The firstresult determines the limit distribution of complex zeros of 'ergodic eigenfunctions' such as eigenfunctions of classicallychaotic systems. The second result determines the expecteddistribution of complex zeros for complexifications of Gaussianrandom waves adapted to the Riemannian manifold. The resultingdistribution is the same in both cases. It is singular along theset of real points.