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Title: Counting nodal components of boundary-adapted arithmetic random waves
Author: Cann, James
ISNI:       0000 0004 7965 1817
Awarding Body: UCL (University College London)
Current Institution: University College London (University of London)
Date of Award: 2019
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available