Session S33 - Spectral Geometry

Wednesday, July 21, 17:30 ~ 17:50 UTC-3

Filament structure in random plane waves

Melissa Tacy

The University of Auckland, New Zealand   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Numerical studies of random plane waves, functions \[u=\sum_{j}c_{j}e^{\frac{i}{h}\langle x,\xi_{j}\rangle}\] where the coefficients $c_{j}$ are chosen ``at random'', have detected an apparent filament structure. The waves appear enhanced along straight lines. There has been significant difference of opinion as to whether this structure is indeed a failure to equidistribute, numerical artefact or an illusion created by the human desire to see patterns. In this talk I will present some recent results that go some way to answering the question. We study the behaviour of a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ and show that $G(x,\xi)$ fails to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

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