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Mobility Edge for the Anderson Model on Random Regular Graphs

Suhan Liu, Patrick Lopatto

Abstract

We determine the phase diagram of the Anderson tight-binding model on regular random graphs with Gaussian disorder distribution and sufficiently large degree. In particular, we prove that if the degree is fixed and the number of vertices goes to infinity, the spectrum asymptotically consists of a finite delocalized interval surrounded by two unbounded localized components. Our argument uses a recent description of the spectrum of the tight-binding model on the Bethe lattice (Aggarwal--Lopatto, 2025). By viewing the Bethe lattice as the local limit of a random regular graph, and proving several preliminary estimates, we transfer this characterization of the spectrum of the limiting model to its finite approximants.

Mobility Edge for the Anderson Model on Random Regular Graphs

Abstract

We determine the phase diagram of the Anderson tight-binding model on regular random graphs with Gaussian disorder distribution and sufficiently large degree. In particular, we prove that if the degree is fixed and the number of vertices goes to infinity, the spectrum asymptotically consists of a finite delocalized interval surrounded by two unbounded localized components. Our argument uses a recent description of the spectrum of the tight-binding model on the Bethe lattice (Aggarwal--Lopatto, 2025). By viewing the Bethe lattice as the local limit of a random regular graph, and proving several preliminary estimates, we transfer this characterization of the spectrum of the limiting model to its finite approximants.
Paper Structure (37 sections, 45 theorems, 272 equations)

This paper contains 37 sections, 45 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Proof Ideas
  5. Acknowledgments
  6. Proof Outline
  7. Notations and Conventions
  8. Mobility Edge on the Bethe Lattice
  9. Resolvent Criteria for Localization and Delocalization
  10. Proof of Main Result
  11. Localization and Delocalization Criteria
  12. Resolvent Estimates
  13. Preliminary Lemmas
  14. Localization
  15. Delocalization
  16. ...and 22 more sections

Key Result

Theorem 1.3

For every real number $\mathfrak L >0$, there exists a constant $d_0(\mathfrak L) > 0$ such that the following holds for all $d \ge d_0$. Fix $g \in \mathbb{R}$ such that and let $t= g ( d \ln d )^{-1}$. Then there exists $\mathfrak E >0$ such that the set $\{E : \rho(E) = 1/ (4g)\}$ equals $\{ -\mathfrak E, \mathfrak E \}$. Further, there exists $\mathfrak M(d) > 0$ such that $|\mathfrak M - \m

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof : Proof of \ref{['t:main']}
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 77 more