What does a typical (generalized) eigenfunction of the laplacian in the unbounded domain R², ie 2D free space, look like? Say we choose the eigenvalue 1, then the complex plane wave e^in.x, with n is a unit direction vector, is an eigenfunction, but is not typical. A typical sample from the eigenspace with eigenvalue 1 can be constructed by a superposition of such waves traveling in all directions, with random complex phases. In other words, it is the 2D Fourier transform of Gaussian white noise restricted to the unit circle S¹. Berry first proposed this as a model for quantum eigenfunctions of systems whose classical dynamics is chaotic, in the '70s.

This was one of the topics addressed at the recent AIM Workshop on Topological complexity of random sets (Aug 2009). Here is my contribution to the workshop, and some data generated there is below.

Here are some pictures of samples of random plane waves, showing the real part, chosen with mean-square value 1, computed with the code linked below. Click on each to magnify:

	color plot	squared	extreme value set (|f|>2.5)
25 wavelengths box
200 wavelengths box

One might study the local box-counting (Kolmogorov) fractal dimension of the extreme-value sets via the slope of the log-log plot of the number of boxes vs box size. Here is such a plot (different curves are for the different extreme value cut-offs), compared to the same for a spatially-uncorrelated model. They are similar but the stringiness of the level-set shows up as a slight slope in the region before the asymptotic slope of 2 is reached.

Some movies generated at the workshop:

• 200 wavelengths box with varying extreme value cut-off (6MB, black is above, white below, the value shown at bottom right).
• 2000 wavelength box with varying extreme value cut-off (3MB, black is below, white above, the value shown at top right), downsampled from 7000x7000 data below.
• color version (3MB) of the above.
• Worms movie (4MB) of slowly changing spectral measure from the whole disk, through an annulus, to a very thin annulus. Notice how the correlation length-scale grows.
• Mark Dennis' worms movie (4MB) of the same thing, instead using radially-sliding point-masses as spectral measure.

Here is Matlab function to produce the above pictures: rpw2dsample.m which needs codes circle_blobs.m and src.m. See the Matlab help for documentation and example usage. The code is spectrally accurate (with default parameter e=5 it has absolute errors of order 3e-4), and uses the nonuniform FFT by resampling onto a uniform square grid. With n=1024 it takes of order 5 sec on my laptop for all computations, using about 400MB RAM. Time scales like n^2 log n and memory use like n^2; both these are optimal.

For larger computations the windowed NUFFT becomes a RAM-hog and crude slow-FT summation wins out if you have CPU cycles to burn. Here's a code for that: naiverpw.m. The result is super-large plane-wave samples, such as the following extreme-value set downsampled from 7000x7000 pixels, 2000 wavelengths across (click to magnify):

extreme-value set |f|>3.0

mean Radon projection of (bottom-left quarter of) f2

Notice how the Radon projection seems to have isolated bright spots (or bowties), as you'd expect for intense lines in coordinate space. It is an open problem how to prove something statistical which quantifies the `stringiness' visible to the eye in the above pictures and movies. One question is why the Radon, or Gaussian beam, basis, leads to apparent sparsity of coefficients.

Some references (also see AIM contribution above):

• ``Properties of random superpositions of plane waves'', P. O'Connor, J. Gehlen, E. J. Heller, Phys. Rev. Lett. 58, 1296-99 (1987). This paper was how I first learned about this, showing that when the wavenumbers are allowed to vary in magnitude, the stringiness disappears.
• Terence Tao's notes on Tomas-Stein theorems on restrictions of the Fourier transform to the sphere. This seems not to be relevant since the functions do not live in L^p spaces.

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