Phase transition in the integrated density of states of the Anderson model arising from a supersymmetric sigma model
Margherita Disertori
University of Bonn, Bonn, GermanyValentin Rapenne
Université Lyon 1, Villeurbanne, FranceConstanza Rojas-Molina
CY Cergy Paris Université, Cergy-Pontoise, FranceXiaolin Zeng
Université de Strasbourg, Strasbourg, France

Abstract
We study the integrated density of states (IDS) of the random Schrödinger 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.
Cite this article
Margherita Disertori, Valentin Rapenne, Constanza Rojas-Molina, Xiaolin Zeng, Phase transition in the integrated density of states of the Anderson model arising from a supersymmetric sigma model. J. Spectr. Theory (2025), published online first
DOI 10.4171/JST/545