# The Statistics of the Points Where Nodal Lines Intersect a Reference Curve - Nonlinear Sciences Chaotic Dynamics

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Abstract: We study the intersection points of a fixed planar curve $\Gamma$ with thenodal set of a translationally invariant and isotropic Gaussian random field$\Psi\bi{r}$ and the zeros of its normal derivative across the curve. Theintersection points form a discrete random process which is the object of thisstudy. The field probability distribution function is completely specified bythe correlation $G|\bi{r}-\bi{r}| = <\Psi\bi{r} \Psi\bi{r}>$.Given an arbitrary $G|\bi{r}-\bi{r}|$, we compute the two pointcorrelation function of the point process on the line, and derive otherstatistical measures repulsion, rigidity which characterize the short andlong range correlations of the intersection points. We use these statisticalmeasures to quantitatively characterize the complex patterns displayed byvarious kinds of nodal networks. We apply these statistics in particular tonodal patterns of random waves and of eigenfunctions of chaotic billiards. Ofspecial interest is the observation that for monochromatic random waves, thenumber 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 ofthat field.

Autor: Amit Aronovitch, Uzy Smilansky

Fuente: https://arxiv.org/