Filament structure of random waves

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.