Spectral geometry of flat tori with random impurities

We discuss new results on the geometry of eigenfunctions in disor- dered systems. More precisely, we study tori $R^d\mathrm{/}L{Z}^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\to \mathrm{\infty }$ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on $R^d$ . Finally, we discuss some results on the breakdown of localization in random displacement models above a certain energy threshold.