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Mean conservation of nodal volume and connectivity measures for Gaussian ensembles

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Dmitry Beliaev, Stephen Muirhead, Igor Wigman

Original languageEnglish
Article number107521
Published12 Feb 2021

Bibliographical note

Funding Information: Let {?L}L?L be a Gaussian ensemble on [Formula presented], C3?-smooth and non-degenerate, possessing a translation invariant local limit K as L?? that is independent of x. Assume further that the spectral measure ? of the Gaussian field F corresponding to the limit covariance K satisfies (?1)?(?4), and also that {?L}L?L satisfies the nodal lower concentration property in Definition 1.8. Then> Acknowledgments. The research leading to these results has received funding from the Engineering & Physical Sciences Research Council (EPSRC) Fellowship EP/M002896/1 held by Dmitry Beliaev (D.B. & S.M.), the EPSRC Grant EP/N009436/1 held by Yan Fyodorov (S.M.), the ARC Discovery Early Career Researcher Award DE200101467 held by Stephen Muirhead (S.M.), and the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013), ERC grant agreement no 335141 (I.W.) We are grateful to P. Sarnak and M. Sodin for the very inspiring and fruitful conversations concerning subjects relevant to this manuscript. Finally, we are grateful to the anonymous referee for pointing out the reference [7] to us. Publisher Copyright: © 2020 The Authors Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

King's Authors


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≥3, and are applied to a number of concrete motivating examples.

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