Probabilistic averages of Jacobi operators

TL;DR

This paper investigates the Lyapunov exponent and density of states for Jacobi operators, reducing complex questions to ergodic cases and applying results to specific non-ergodic models, including a probabilistic version of a key spectral theorem.

Contribution

It introduces a reduction technique for spectral questions of Jacobi operators to ergodic cases and extends the Denisov–Rakhmanov–Remling theorem to probabilistic settings.

Findings

Reduction of spectral questions to ergodic Jacobi operators

Application to non-ergodic models with specific coefficient functions

Development of a probabilistic version of a classical spectral theorem

Abstract

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these, can be reduced to questions about ergodic Jacobi operators. Then, I apply this to $a(n) = 1$ and $b(n) = f(n^\rho \pmod{1})$ for $\rho > 0$ not an integer, and to obtain a probabilistic version of the Denisov--Rakhmanov--Remling Theorem.