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The regular tree Anderson model at low disorder

Reuben Drogin, Charles K Smart

TL;DR

The paper proves delocalization for the Anderson model on an infinite regular tree at low disorder by exploiting a tree-specific self-consistent equation for the Green's function and a hyperbolic-geometry analysis of Möbius dynamics. It establishes a Green's-function concentration near the free Green's function, analyzes elliptic, parabolic, and hyperbolic dynamical regimes to obtain a unique fixed point and robust control of fluctuations, and shows the Lyapunov exponent is continuous as disorder vanishes. Using the Aizenman–Warzel criterion, the authors deduce almost-sure absolute continuity of the spectrum in a central energy window and dynamical delocalization when the potential is bounded, thereby filling gaps left by prior work. The approach extends to disorder with small fourth moment and regular density, providing a unified framework for localization-delocalization on trees via self-consistent Green's functions and Möbius dynamics on the upper half-plane.

Abstract

We prove delocalization for the Anderson model on an infinite regular tree (or Cayley graph or Bethe lattice) at low disorder. This extends earlier results of Klein and Aizenman--Warzel by filling in the previously missing parts of the spectrum. Our argument generalizes to any disorder with small fourth moment and sufficiently regular density. We prove continuity of the Lyapunov exponent as the disorder vanishes.

The regular tree Anderson model at low disorder

TL;DR

The paper proves delocalization for the Anderson model on an infinite regular tree at low disorder by exploiting a tree-specific self-consistent equation for the Green's function and a hyperbolic-geometry analysis of Möbius dynamics. It establishes a Green's-function concentration near the free Green's function, analyzes elliptic, parabolic, and hyperbolic dynamical regimes to obtain a unique fixed point and robust control of fluctuations, and shows the Lyapunov exponent is continuous as disorder vanishes. Using the Aizenman–Warzel criterion, the authors deduce almost-sure absolute continuity of the spectrum in a central energy window and dynamical delocalization when the potential is bounded, thereby filling gaps left by prior work. The approach extends to disorder with small fourth moment and regular density, providing a unified framework for localization-delocalization on trees via self-consistent Green's functions and Möbius dynamics on the upper half-plane.

Abstract

We prove delocalization for the Anderson model on an infinite regular tree (or Cayley graph or Bethe lattice) at low disorder. This extends earlier results of Klein and Aizenman--Warzel by filling in the previously missing parts of the spectrum. Our argument generalizes to any disorder with small fourth moment and sufficiently regular density. We prove continuity of the Lyapunov exponent as the disorder vanishes.

Paper Structure

This paper contains 12 sections, 23 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Delocalization results
  3. The Aizenman--Warzel criterion
  4. Green's function estimates
  5. The self-consistent equation
  6. Additional hypotheses
  7. Acknowledgements
  8. Hyperbolic distance
  9. Fixed point iteration
  10. The elliptic and parabolic cases
  11. The hyperbolic case
  12. Delocalization

Key Result

Theorem 1.3

For every $\varepsilon > 0$, there is a $\delta > 0$ depending only on $(K,L,\varepsilon)$ such that, if $\beta < \delta$, then, almost surely, the spectral measure of $H$ is absolutely continuous in $\{ E \in \mathbb{R} : |E| < K + 1 - \varepsilon \}$ and pure point in $\{ E \in \mathbb{R} : |E| >

Theorems & Definitions (45)

  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.7: Aizenman--Warzel MR3055759
  • Theorem 1.9
  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.6
  • proof
  • ...and 35 more