# Large Deviation theorems for Dirichlet determinants of analytic quasi-periodic Jacobi operators with Brjuno-R\"ussmann frequency

## TL;DR

This paper establishes large deviation theorems for Dirichlet determinants of quasi-periodic Jacobi operators with Brjuno-Rüssmann frequencies, linking eigenvalue distribution to the irrationality of the frequency through the smallest deviation.

## Contribution

It introduces large deviation theorems for these operators using a new approach based on the strong Birkhoff Ergodic Theorem for subharmonic functions with Brjuno-Rüssmann shifts.

## Key findings

- Large deviation estimates depend on the smallest deviation related to the frequency's irrationality.
- Eigenvalue distribution of Jacobi operators is characterized by the smallest deviation.
- The Brjuno-Rüssmann function plays a key role in the deviation theorems.

## Abstract

In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-R\"ussmann shift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-R\"ussmann function, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.