One-dimensional Discrete Dirac Operators in a Decaying Random Potential II: Clock, Schr\"odinger and Sine statistics

TL;DR

This paper studies spectral statistics of one-dimensional discrete Dirac operators in decaying random potentials, showing convergence to clock and Sine processes depending on decay rate, and discusses implications for eigenfunctions.

Contribution

It establishes the spectral statistics and scaling limits of Dirac models with decaying randomness, connecting to known random matrix theory processes and providing new analytical methods.

Findings

Rescaled spectrum converges to clock process for fast decay.

Rescaled spectrum converges to Sine process at critical decay.

Provides alternative proof techniques using Pr"ufer phase analysis.

Abstract

We consider one-dimensional discrete Dirac models in vanishing random environments. In a previous work [6], we showed that these models exhibit a rich phase diagram in terms of their spectrum as a function of the rate of decay of the random potential. This article is devoted to their spectral statistics. We show that the rescaled spectrum converges to the clock process for fast decay and to the Schr\"odinger/Sine processes from random matrix theory for critical decay. This way, we recover all the regimes previously identified for the Anderson model in a similar context [25]. Poisson statistics, which should appear in the model with slow decay, are left as an open problem. The core of the proof consists in a suitable scaling limit for the Pr\"ufer phase and monotonicity arguments, yielding an alternative to the approach of [25]. For one of the models, we also obtain the scaling limit of the Pr\"ufer radii and discuss the consequences for the limiting shape of the eigenfunctions.