Random Block Operators

TL;DR

This paper investigates the spectral properties of random block operators relevant to disordered mesoscopic systems, revealing how randomness influences spectral gaps, density of states, and Lifshits tails.

Contribution

It provides new insights into the spectral behavior of random block operators, including ergodic properties, spectrum location, and effects of randomness on density of states and Lifshits tails.

Findings

Existence of a robust spectral gap due to block structure

Randomness smears out singularities in the density of states

Established a Wegner estimate valid at all energies

Abstract

We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.