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Resonant delocalization on the Bethe strip
Mathematical Physics · Submitted 16 Apr 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble.
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The abstract, sentence by sentence
Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). | background 1.00 | 0.03 | 0.09 | asserted 1.00 |
| s1 | We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. | contribution 1.00 | 0.94 | 0.11 | asserted 0.99 |
| s2 | These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops. | interpretation 0.35 | 0.29 | 0.07 | asserted 0.94 |
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The paper
Original abstract
Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schroedinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 10 sentences without a confident label
Builds on 1 paper
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The goal of this present work is to extend the result of [ref] to the case $W > 1$ of the Bethe strip.
Resonant delocalization for random Schr\"odinger operators on tree graphs · Introduction · 1.00
Uses the method or tool of 1 paper
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We apply [ref] .
Resonant delocalization for random Schr\"odinger operators on tree graphs · Proof of Theorem [ref] · 0.95
Cites as background 2 papers
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The interval $[K-1,K+1]$ is exactly the $ℓ^1$ spectrum of the unperturbed operator; the importance of the $ℓ^1$ spectrum is further discussed in [ref] and in the survey [ref] by Warzel.
Resonant delocalization for random Schr\"odinger operators on tree graphs · Introduction · 0.86
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The interval $[K-1,K+1]$ is exactly the $ℓ^1$ spectrum of the unperturbed operator; the importance of the $ℓ^1$ spectrum is further discussed in [ref] and in the survey [ref] by Warzel.
Surprises in the phase diagram of the Anderson model on the Bethe lattice · Introduction · 0.86
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