Filament structure of random waves

TL;DR

This paper studies the small-scale filament structures of random waves in high-dimensional spaces, revealing their equidistribution properties and phase space fluctuations through theoretical and numerical analysis.

Contribution

It provides new insights into the filament structure of random waves, connecting phase space concentrations with configuration space patterns.

Findings

X-ray along any line segment is uniformly equidistributed

Logarithmic fluctuations observed at Planck scale in phase space

Filament structure may reflect phase space concentrations

Abstract

We investigate the small scale equidistribution properties of random waves in $\mathbb{R}^{n}$. Numerical evidence suggests that such objects display a fine scale filament structure. We show that the X-ray along any line segment is uniformly equidistributed so any limiting behaviour must be weaker than $L^{2}$ scaring. On the other hand, we show that at Planck scale in phase space there are (with high probability) logarithmic fluctuations above what would be expected given equidistribution. Taken together these results suggest that the filament structure may be a configuration space echo of the phase space concentrations.