H\"older continuity of the integrated density of states for quasi-periodic Jacobi block matrices

TL;DR

This paper establishes H"older continuity of the integrated density of states for quasi-periodic Jacobi block matrices with Diophantine frequencies, extending and strengthening previous results in the Schr"odinger operator setting.

Contribution

It introduces a new scheme to derive local zero counts from global ones, enabling broader H"older continuity results for these matrices.

Findings

H"older exponent can be any 5 with 0<5<1/(2kk^d)

Generalizes previous results to more Diophantine frequencies

Provides a new method for analyzing zero counts of characteristic polynomials

Abstract

In this paper, we prove H\"older continuity of the integrated density of states for discrete quasiperiodic Jacobi $d\times d$ block matrices with Diophantine frequencies. The H\"older exponent is shown to be any $\beta$ such that $0<\beta<1/(2\kappa^d)$, where $\kappa^d$ is the acceleration, i.e., the slope of the sum of the top $d$ Lyapunov exponents in the imaginary direction of the phase. This generalizes the H\"older continuity results in the Schr\"odinger operator setting in \cites{GS2,HS1}, and also strengthens them in that setting by covering more Diophantine frequencies. The proof is built on a new scheme for obtaining a local zero count for finite-volume characteristic polynomials from a global one.