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arXiv · 2302.01156

No smooth Phase Transition for the Nodal Length of Band-limited Spherical Random Fields

Anna Paola Todino
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Abstract

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with respect to the frequency, while it is logarithmic when a single eigenfunction is considered. Then, it is natural to conjecture that there exists a smooth transition with respect to the number of eigenfunctions in the frequency window; however, we show here that the asymptotic variance is logarithmic whenever this number grows sublinearly, so that the window "shrinks". The result is achieved by exploiting the Christoffel-Darboux formula to establish the covariance function of the field and its first and second derivatives. This allows us to compute the two-point correlation function at high frequency and then to derive the asymptotic behaviour of the variance.

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Anna Paola Todino. 2023-02-02. No smooth Phase Transition for the Nodal Length of Band-limited Spherical Random Fields. https://arxiv.org/abs/2302.01156

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