The ‘nodal sets’ (zero sets) of Dirichlet Laplace eigenfunctions for the two-dimensional unit square have raised many questions over the past century. The nodal domain theorems of Courant (1924), Stern (1926) and Pleijel (1956), give deterministic (upper, liminf, and limsup resp.) bounds for the number of nodal domains which can be exhibited by an n-th eigenfunction, assuming the eigenvalues are ordered increasingly and with multiplicities listed. Prominent amongst contemporary research is the closely related question of the number of ‘nodal components’ (connected components of the zero set) of a ‘typical’ eigenfunction.

The spectral degeneracy for the square means that the nodal count of an n-th eigenfunction could take a wide range of values, and to understand the ‘typical’ behaviour of this number we attribute Gaussian random coefficients to a standard basis of eigenfunctions for each eigenspace, to form the ensemble of ‘boundary-adapted arithmetic random waves’. The number of nodal components —now a random variable— can then be studied, and this talk will draw together tools from various areas of mathematics (random fields, integral geometry, number theory, mathematical physics, …) in order to say a few things about its asymptotic properties.

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Last updated: 18 Sep 2024