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Filament structure in random plane waves

Banff International Research Station for Mathematical Innovation and Discovery

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. First we consider the behaviour of a random variable given by $F(x,\xi)=||u||_{L^{2}(\gamma_{(x,\xi)})}$ where $\gamma_{(x,\xi)}$ is a unit ray from the point $x$ in direction $\xi$. We will see that this random variable is uniformly equidistributed. That is, the probability that for any $(x,\xi)$, $F(x,\xi)$ differs from its equidistributed value is small (in fact exponentially small). This result rules out a strong scarring of random waves. However, when we look at the full phase space picture and study a random variable $G(x,\xi)=||P_{(x,\xi)}u||_{L^{2}}$ where $P_{(x,\xi)}$ is a semiclassical localiser at Planck scale around $(x,\xi)$ we do see a failure to equidistribute. This suggests that the observed filament structure is a configuration space reflection of the phase space concentrations.

Mathematics; Global Analysis; Analysis On Manifolds; Differential Geometry; Geometric Analysis

video/mp4

Author affiliation: University of Auckland

2023-10-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86340

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/