Phase transition in the Integrated Density of States of the Anderson model arising from a supersymmetric sigma model

TL;DR

This paper investigates the phase transition in the Integrated Density of States of the Anderson model linked to a supersymmetric sigma-model, revealing a transition between disorder regimes and contrasting behavior with other disordered systems.

Contribution

It establishes a phase transition in the IDS of the Anderson model via supersymmetric sigma-model analysis, including bounds and regularity results, and confirms the absence of Lifshitz tails in strong disorder.

Findings

Identifies a phase transition in the IDS between weak and strong disorder regimes in dimensions ≥ 3.

Shows the IDS lacks Lifshitz tails in the strong disorder regime, confirming a recent conjecture.

Provides a Wegner type estimate indicating the regularity of the IDS.

Abstract

We study the Integrated Density of States (IDS) of the random Schr\"odinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model. We rely on previous results on the supersymmetric sigma-model to obtain lower and upper bounds on the asymptotic behavior of the IDS near the bottom of the spectrum in all dimension. We show a phase transition for the IDS between weak and strong disorder regime in dimension larger or equal to three, that follows from a phase transition in the corresponding random process and supersymmetric sigma-model. In particular, we show that the IDS does not exhibit Lifshitz tails in the strong disorder regime, confirming a recent conjecture. This is in stark contrast with other disordered systems, like the Anderson model. A Wegner type estimate is also derived, giving an upper bound on the IDS and showing the regularity of the function.