Phase transition in the integrated density of states of the Anderson model arising from a supersymmetric sigma model

  • Margherita Disertori

    University of Bonn, Bonn, Germany
  • Valentin Rapenne

    Université Lyon 1, Villeurbanne, France
  • Constanza Rojas-Molina

    CY Cergy Paris Université, Cergy-Pontoise, France
  • Xiaolin Zeng

    Université de Strasbourg, Strasbourg, France
Phase transition in the integrated density of states of the Anderson model arising from a supersymmetric sigma model cover
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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