Counting open nodal lines of random waves on planar domains

TL;DR

This paper calculates the expected number of open nodal lines in random waves on smooth planar domains, showing it grows proportionally with the energy parameter, aligning with physics predictions.

Contribution

It provides the first asymptotic formulas for the expected count of open nodal lines in random waves on planar domains, confirming physics-based predictions.

Findings

Expected number of open nodal lines is proportional to mbda

Results hold for both long and short energy windows

Findings agree with previous physics literature predictions

Abstract

We compute the asymptotic expectation of the number of {\em open} nodal lines for random waves on smooth planar domains. We find that for both the long energy window $[0,\lambda]$ and the short one $[\lambda,\lambda+1]$ the expected number of open nodal lines is proportional to $\lambda$, asymptotically as $\lambda\to\infty$. Our results are consistent with the predictions in the physics literature made by Blum, Gnutzmann and Smilansky \cite{BGS}.