Code for illustrations and pictures.
See
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Phase portrait of a random polynomial of degree 2001 with +/-1 random coefficients. By Ryll-Nardzewski's theorem, the limiting random analytic function has a natural border at the border of the disk of radius 1, its domain of convergence, see JP Kahane's book Some random series of functions, chapter 4.
Phase portrait of the Gaussien Entire Function on [-8,8]x[-8,8]. I only kept the Taylor polynomial up to degree 200. The zero set is stationary, see the GEF Book by Hough et al.
Phase portrait of the complex gaussian random wave, ie a random solution of Helmotz's equation in the complex plane (real part + complex part). See this paper by Nourdin Peccati and Rossi and especially the representation in equation 1.5.
As you see I get some kind of pixelized effect, I don't know why.
The solutions of the Laplace equations on the torus have energy 4pin where n can be written as the sum of two squares and the dimension of the associated eigenspace is the number of ways to write n as a sum of two squares. Here are the nodal domains of one random (gaussian) eigenfunction associated with n=85.
See also the Krishnapur et al. paper.
Picture is in 4k so might take time to load. More will follow.
