Spectral geometry of flat tori with random impurities

Ueberschär, Henrik

Journées équations aux dérivées partielles (2015), article no. 11, 7 p.

We discuss new results on the geometry of eigenfunctions in disordered systems. More precisely, we study tori ${ℝ}^d/L{\mathbb{Z}}^d$, $d=2,3$, 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 $L\to \infty$ 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.

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