Mean conservation of nodal volume and connectivity measures for Gaussian ensembles

Dmitry Beliaev, Stephen Muirhead, Igor Wigman

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number107521
JournalAdvances in Mathematics
Publication statusPublished - 12 Feb 2021
Externally publishedYes


  • Connectivity measures
  • Gaussian random fields
  • Mean connectivity
  • Nodal sets
  • Percolating fields


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