Mean conservation of nodal volume and connectivity measures for Gaussian ensembles

Dmitry Beliaev; Stephen Muirhead; Igor Wigman

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arXiv · 1901.09000

Mean conservation of nodal volume and connectivity measures for Gaussian ensembles

Dmitry Beliaev·Stephen Muirhead·Igor Wigman

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Abstract

We study in depth the nesting graph and volume distribution of the nodal domains of a Gaussian field, which have been shown in previous works to exhibit asymptotic laws. A striking link is established between the asymptotic mean connectivity of a nodal domain (i.e.\ the vertex degree in its nesting graph) and the positivity of the percolation probability of the field, along with a direct dependence of the average nodal volume on the percolation probability. Our results support the prevailing ansatz that the mean connectivity and volume of a nodal domain is conserved for generic random fields in dimension $d=2$ but not in $d \ge 3$, and are applied to a number of concrete motivating examples.

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Dmitry Beliaev, Stephen Muirhead, Igor Wigman. 2019-01-25. Mean conservation of nodal volume and connectivity measures for Gaussian ensembles. https://arxiv.org/abs/1901.09000

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