Dyson's spike for random Schroedinger operators and Novikov-Shubin invariants of groups

TL;DR

This paper investigates the spectral properties of random Schrödinger operators, revealing a Dyson spike near zero and computing Novikov-Shubin invariants for specific groups, advancing understanding of spectral measures in random and geometric group contexts.

Contribution

It demonstrates the Dyson spike phenomenon without independence assumptions and calculates Novikov-Shubin invariants for lamplighter groups and Sol lattices.

Findings

Spectral measure near zero behaves as 1/|log ε|^2

Limiting eigenvalue distribution differs from Poisson and random matrix models

Computed Novikov-Shubin invariants for various groups

Abstract

We study Schroedinger operators with random edge weights and their expected spectral measures $\mu_H$ near zero. We prove that the measure exhibits a spike of the form $\mu_H(-\varepsilon,\varepsilon) \sim \frac{C}{| \log\varepsilon|^2}$ (first observed by Dyson), without assuming independence or any regularity of edge weights. We also identify the limiting local eigenvalue distribution, which is different from Poisson and the usual random matrix statistics. We then use the result to compute Novikov-Shubin invariants for various groups, including lamplighter groups and lattices in the Lie group Sol.