# Pseudo-gaps for random hopping models

## TL;DR

This paper develops a perturbation theory for the rotation number in one-dimensional random Schrödinger operators at hyperbolic critical energies, demonstrating the existence of pseudo-gaps through H"older continuity and renewal theory, supported by numerical evidence.

## Contribution

It introduces a controlled perturbation approach for rotation numbers at hyperbolic energies, establishing pseudo-gaps in random hopping models.

## Key findings

- H"older continuity of the rotation number at critical energies
- Existence of pseudo-gaps in the density of states
- Numerical illustrations confirming theoretical results

## Abstract

For one-dimensional random Schr\"odinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Pr\"ufer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy, at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models. The main result is a H\"older continuity of the rotation number at the critical energy that, under certain conditions on the randomness, implies the existence of a pseudo-gap. The proof uses renewal theory. The result is illustrated by numerics.