A note on the reduction principle for the nodal length of planar random waves

Inspired by Marinucci et al. (2020), we prove that the nodal length of a planar random wave BE, i.e. the length of its zero set BE−1(0), is asymptotically equivalent, in the L2-sense and in the high-frequency limit E→∞, to the integral of H4(BE(x)), H4 being the fourth Hermite polynomial. As straightforward consequences, we obtain Moderate Deviation estimates and a central limit theorem in Wasserstein distance. This complements recent findings by Nourdin et al. (2019) and Peccati and Vidotto (2020).