The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

Abstract

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field $\Psi(\bi{r})$ and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation $G(|\bi{r}-\bi{r}'|) = <\Psi(\bi{r}) \Psi(\bi{r}')>$. Given an arbitrary $G(|\bi{r}-\bi{r}'|)$, we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.