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

Margherita Disertori, Valentin Rapenne, Constanza Rojas-Molina, Xiaolin Zeng

TL;DR

The paper analyzes the Integrated Density of States (IDS) for the random Schrödinger operator $H_{eta}$ tied to the supersymmetric $H^{2|2}$ sigma-model, revealing a dimension- and disorder-dependent phase transition in the low-energy behavior of the IDS and showing the absence of Lifshitz tails in the strong-disorder regime. It develops a multifaceted proof strategy combining finite-volume Dirichlet bounds, a careful comparison with simple boundary conditions, and a detailed analysis of conditional measures whose densities drive the lower bounds, complemented by a Wegner-type estimate that yields Hölder regularity of the IDS. The results establish a dichotomy in the asymptotics: in dimension $d=1$ the IDS behaves like $N(E) oughly c\sqrt{E}$ for all $W$, while in higher dimensions strong disorder preserves the $\sqrt{E}$-type behavior at low energies, and weak disorder yields faster decay, up to a linear bound in $E$ in $d\ge 3$. The work also connects the spectral phase transition to phase transitions in the associated reinforced random processes and the $H^{2|2}$ sigma-model, offering a physically motivated example where Lifshitz tails break down despite localization phenomena, with implications for Anderson-type transitions and localization theory.

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.

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

TL;DR

The paper analyzes the Integrated Density of States (IDS) for the random Schrödinger operator tied to the supersymmetric sigma-model, revealing a dimension- and disorder-dependent phase transition in the low-energy behavior of the IDS and showing the absence of Lifshitz tails in the strong-disorder regime. It develops a multifaceted proof strategy combining finite-volume Dirichlet bounds, a careful comparison with simple boundary conditions, and a detailed analysis of conditional measures whose densities drive the lower bounds, complemented by a Wegner-type estimate that yields Hölder regularity of the IDS. The results establish a dichotomy in the asymptotics: in dimension the IDS behaves like for all , while in higher dimensions strong disorder preserves the -type behavior at low energies, and weak disorder yields faster decay, up to a linear bound in in . The work also connects the spectral phase transition to phase transitions in the associated reinforced random processes and the sigma-model, offering a physically motivated example where Lifshitz tails break down despite localization phenomena, with implications for Anderson-type transitions and localization theory.

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.
Paper Structure (24 sections, 28 theorems, 196 equations)

This paper contains 24 sections, 28 theorems, 196 equations.

Table of Contents

  1. Introduction and main results
  2. Discussion on the results and open questions.
  3. Behavior of the IDS near $\mathbf{E=0}$.
  4. Critical value of the disorder strength $W$.
  5. The Anderson transition.
  6. Theorem \ref{['thm-main-theorem']} : strategy of the proof.
  7. Theorem \ref{['wegner-type-theorem']}: strategy of the proof.
  8. Theorem \ref{['corol-upper-bound-on-IDS']}: strategy of the proof.
  9. Organization of this paper.
  10. Acknowledgements.
  11. Some previous results on the $\mathcal{H}_\beta$ operator
  12. Remarks on various marginals.
  13. Connection with the $H^{2|2}$ model.
  14. How the results above follow from Disertori2010
  15. Comparing $W_{c}$ and $W_{c}'$.
  16. ...and 9 more sections

Key Result

Theorem 1

We define where $W_{c}>0$ (resp. $W_{c}'>0$) is the (dimensional dependent) parameter introduced in Theorem thm-localization-of-the-ground-state-green-function (resp. Theorem thm-localization-fractional-moment). In particular $W_{cr}=\infty$ for $d=1.$ Then, for each $0<W<W_{cr}$ there exist constants $c=c(W,

Theorems & Definitions (28)

  • Theorem 1: lower bound on the IDS
  • Theorem 2: Wegner type estimate
  • Theorem 3: upper bound and regularity for the IDS
  • Theorem 4: The multivariate inverse Gaussian distribution
  • Lemma 5: connection to $H^{2|2}$
  • Theorem 6: decay of the ground state Green's function (1)
  • Theorem 7: decay with wired bc (1)
  • Theorem 8: decay of the ground state Green's function (2)
  • Corollary 9: decay with wired b.c. (2)
  • Lemma 10
  • ...and 18 more