On the Real Eigenvalues of the Non-Hermitian Anderson Model

TL;DR

This paper analyzes the sensitivity of eigenvalues in the non-Hermitian Anderson model on a ring, providing exact decay rates and conditions for eigenvalues to remain real under perturbation.

Contribution

It offers an exact rate of eigenvalue sensitivity to non-Hermiticity and confirms the eigenvalues stay real within a specific parameter range, extending previous results.

Findings

Eigenvalues remain real for $0 < g < \gamma(\lambda_{0})$

Decay rate of eigenvalue sensitivity is $\sim \gamma(\lambda_{0}) - g$

Provides an alternative proof for eigenvalue reality under perturbation

Abstract

We study the non-Hermitian Anderson model on the ring. We provide the exact rate of decay of the sensitivity of the eigenvalues to the non-Hermiticity parameter $g$, on the logarithmic scale, as the Lyapunov exponent minus the non-Hermiticity parameter. Namely, for $0 < g < \gamma(\lambda_{0})$ we show that $-\frac{1}{n}\log|\lambda_{g}-\lambda_{0}|\sim \gamma(\lambda_{0})-g$ and that the eigenvalue remains real for all such $g$. This provides an alternative proof to that of Goldsheid and Sodin that the perturbed eigenvalue remains real and specifies the exact rate at which the eigenvalue is exponentially close to the unperturbed eigenvalue.