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Counting nodal components of boundary-adapted arithmetic random waves

Cann, James; (2019) Counting nodal components of boundary-adapted arithmetic random waves. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the two dimensional unit square have historically raised many questions, and continue to do so today. Prominent amongst them is the question of the number of ‘nodal components’ (connected components of the zero set) of a typical eigenfunction. In this thesis, we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundary-adapted arithmetic random waves’. We then study the number of nodal components -- now a random variable -- of this ensemble as the eigenvalue grows to infinity, and establish the existence of a limiting mean nodal intensity which is non-universal, in the sense that it depends (indeed relies) upon restriction to subsequences of eigenvalues with specific arithmetic properties. We further show that the number of nodal components concentrates exponentially in probability about this limiting mean intensity.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Counting nodal components of boundary-adapted arithmetic random waves
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2019. Original content in this thesis is licensed under the terms of the Creative Commons Attribution 4.0 International (CC BY 4.0) Licence (https://creativecommons.org/licenses/by/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Population Health Sciences
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Population Health Sciences > Institute of Health Informatics
UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Population Health Sciences > Institute of Health Informatics > Clinical Epidemiology
URI: https://discovery.ucl.ac.uk/id/eprint/10077034
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