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Mobility Edge for the Anderson Model on the Bethe Lattice

Amol Aggarwal, Patrick Lopatto

TL;DR

This work proves the existence of mobility edges for the Anderson model on the Bethe lattice with unbounded on-site disorder, showing a finite set of energy thresholds separating localized (pure-point) and delocalized (absolutely continuous) spectral regions for large degree $K$. The authors reduce the spectral question to a transfer-operator problem and establish a key monotonicity property of the leading eigenvalue of this operator, $oldsymbol{bbambda}_{s,E}$, near energies where the disorder density meets a critical value. The analysis hinges on a Krein–Rutman framework for the transfer operator, careful control of self-energy densities, and a bootstrap scheme for localized regions, combined with precise Lipschitz bounds and asymptotics of the transfer densities. The result confirms the Abou-Chacra–Thouless–Anderson mobility-edge prediction in a rigorous setting and provides a detailed mechanism linking the disorder distribution to phase transitions in the spectrum, with potential implications for related random-graph and random-matrix models. The techniques offer a robust blueprint for proving phase transitions in tree-like disordered systems through spectral-radius criteria and monotonicity arguments.

Abstract

We pinpoint the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of our proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model's off-diagonal resolvent entries.

Mobility Edge for the Anderson Model on the Bethe Lattice

TL;DR

This work proves the existence of mobility edges for the Anderson model on the Bethe lattice with unbounded on-site disorder, showing a finite set of energy thresholds separating localized (pure-point) and delocalized (absolutely continuous) spectral regions for large degree . The authors reduce the spectral question to a transfer-operator problem and establish a key monotonicity property of the leading eigenvalue of this operator, , near energies where the disorder density meets a critical value. The analysis hinges on a Krein–Rutman framework for the transfer operator, careful control of self-energy densities, and a bootstrap scheme for localized regions, combined with precise Lipschitz bounds and asymptotics of the transfer densities. The result confirms the Abou-Chacra–Thouless–Anderson mobility-edge prediction in a rigorous setting and provides a detailed mechanism linking the disorder distribution to phase transitions in the spectrum, with potential implications for related random-graph and random-matrix models. The techniques offer a robust blueprint for proving phase transitions in tree-like disordered systems through spectral-radius criteria and monotonicity arguments.

Abstract

We pinpoint the spectral decomposition for the Anderson tight-binding model with an unbounded random potential on the Bethe lattice of sufficiently large degree. We prove that there exist a finite number of mobility edges separating intervals of pure-point spectrum from intervals of absolutely continuous spectrum, confirming a prediction of Abou-Chacra, Thouless, and Anderson. A central component of our proof is a monotonicity result for the leading eigenvalue of a certain transfer operator, which governs the decay rate of fractional moments for the tight-binding model's off-diagonal resolvent entries.

Paper Structure

This paper contains 73 sections, 72 theorems, 512 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Related Works
  5. Proof Ideas
  6. Reduction to a Transfer Operator
  7. Local Monotonicity for $\lambda_{s, E}$
  8. Proof Outline
  9. Notation and Conventions
  10. Resolvent
  11. Localization and Delocalization Criteria
  12. Transfer Operator
  13. Bootstrap for the Localized Regime
  14. Proof of Main Result
  15. Miscellaneous Preliminaries
  16. ...and 58 more sections

Key Result

Theorem 1.3

For any real number $\mathfrak L> 1$, there exists a constant $K_0 (\mathfrak L) > 1$ such that the following holds for all $K \ge K_0$. Fix $g \in \mathbb{R}$ and an $\mathfrak L$-regular probability density $\rho$ such that Suppose further that for any point $E$ such that $\rho(E) = 1/ (4g)$, either $\rho'(E') \ge 1/ \mathfrak L$ for all $E' \in [ E - 1/\mathfrak L, E + 1/\mathfrak L]$, or $\rh

Figures (1)

  • Figure 1: Illustration of \ref{['t:main']}. Shaded regions on the horizontal axis indicate regions of absolutely continuous spectrum, and unshaded regions indicate pure-point spectrum.

Theorems & Definitions (139)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5: aizenman2013resonant
  • Definition 2.6
  • ...and 129 more