Georgia Institute of Technology College of Sciences

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Meeting ID: 943 8213 6954
Passcode: 621373

Abstract

We consider a one-dimensional discrete Dirac operator in a potential given by a family of i.i.d. random variables modulated by a decreasing envelope. In [1], we showed that these models exhibit a rich phase diagram as a function of the rate of decay of the random potential. The spectrum of the operator

• is absolutely continuous for fast decay,

• is pure point for slow decay,

• presents a spectral transition for critical decay.

We also showed dynamical localization in the sub-critical region by means of the fractional moments method with control on the eigenfunctions, while in the critical regime, transport persists even where the spectrum is pure point.

In [2], we studied spectral statistics. In the fast decay case, the rescaled spectrum of the operator converges to the clock process while for critical decay, it converges to the Sine/Schrödinger process. In this way, we recover the results of Kritchevski, Valkó and Virag established for the Anderson model. Our proof is based on the scaling limit of Prüfer transform associated to the system and uses the monotonicity of the Prüfer phase to deduce the convergence. In the slow decay, the spectral statistics are expected to be given by a Poisson process.

References

[1] O. Bourget, G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics, Mathematical Physics, Analysis and Geometry 23 (2020).

[2] G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential II: Clock, Schr¨odinger and Sine statistics, Discrete Continuous Dyn. Syst. 46 (2026).