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Absolutely continuous spectrum for multi-type Galton Watson trees

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Matthias Keller

Mathematical Physics · Submitted 9 Sep 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.

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arXiv ID1109.1959 (version 2)
Pith IDOGPJGI25, Pith's permanent record of this paper and its review
CitationMatthias Keller. "Absolutely continuous spectrum for multi-type Galton Watson trees." arXiv:1109.1959 (2011). https://pith.science/paper/OGPJGI25
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,866 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.24 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.66 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. contribution 0.54 0.32 0.03 asserted 0.90
s1 Moreover, we impose that each vertex has at least one forward neighbor. method 0.50 0.43 0.02 asserted 0.95
s2 Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type. result 1.00 0.96 0.95 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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  • The fundamental difference to the situation in this work is that [ref] deal with small random perturbations appearing everywhere while here we deal with very large perturbations (removal and addition of edges) that occur with small probability.

    On the spectral theory of trees with finite cone type · Outline of the proof · 1.00

  • While for small disorder induced by a random potential large parts of the absolutely continuous spectrum are preserved, [ref] , the spectrum is almost surely pure point whenever the disorder is induced by an arbitrary small radial symmetric random potential, [ref] .

    Resonant delocalization for random Schr\"odinger operators on tree graphs · Introduction and main result · 0.82

Cites as background 4 papers

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