Random Schr\"odinger operators and convolution on wreath products

Adam Arras

Random Schr\"odinger operators and convolution on wreath products

T0 review · 0 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The averaged spectrum of a random Schrödinger operator is the Plancherel measure of a deterministic convolution operator on a wreath product.

desk verdict Core conditional theorems are sound and cleanly proven; the abstract overstates scope, and a sign error in the Lifshitz tail proof needs fixing. read the letter →

arxiv 2505.22485 v1 pith:4MZEUGQT submitted 2025-05-28 math.PR math.GRmath.SP

Adam Arras This is my paper

classification math.PRmath.GRmath.SP MSC 60H2547B8020E2260B15

keywords randomSchrödingeroperatorswreathproductsconvolutionPlancherelmeasuredensityofstatesLifshitztailsdirectintegrallamplightergroups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a spectral bridge between random Schrödinger operators $H(\omega) = A + V$ on a countable group $\Gamma$, with $V$ an i.i.d. potential, and deterministic convolution operators on the wreath product $\Lambda \wr \Gamma$ built by attaching a lamp group $\Lambda$ to every site of $\Gamma$. The central identity (Theorem 1.1) says that the averaged spectral measure at the identity, i.e. the density of states of $H(\omega)$, equals the Plancherel measure of the convolution operator $M = L_{\hat m_\Gamma + \hat m_\Lambda}$ on $\Lambda \wr \Gamma$, provided the single-site potential is distributed according to the spectral measure of a convolution operator $B$ on $\Lambda$. When $\Lambda$ is finitely generated and Abelian, the correspondence is upgraded (Theorem 1.2) to a unitary equivalence between $M$ and the direct integral $\int_\Omega^\oplus H(\omega)\,P(d\omega)$. A sympathetic reader should care because the identity lets deterministic group-harmonic tools bear on random operators and, conversely, lets random-operator regularity tools bear on convolutions on wreath products; from it the paper derives an absolute-continuity criterion, Lifshitz tails with exponent $-d/2$ on any polynomial-growth base group, and an exact formula for the second moment of the Green function.

What carries the argument

The central object is the wreath product $\Lambda \wr \Gamma = (\bigoplus_\Gamma \Lambda) \rtimes \Gamma$, whose elements $(f,g)$ describe a lamplighter at position $g \in \Gamma$ with finitely many lamps $f: \Gamma \to \Lambda$ switched on. The engine of the proof is a moment expansion: powers of $H(\omega)$ enumerate closed walks on $\Gamma$ with potential weights, while powers of $M$ enumerate closed walks on $\Lambda \wr \Gamma$ where each visit of a potential is replaced by a lamp switch; identity (7) matches these sums term by term using the fact that the moments of the single-site potential are exactly the return moments of the lamp-group convolution operator. For Abelian lamps, the Fourier transform on $\bigoplus_\Gamma \Lambda$ turns the configuration space into the probability space $\Omega$ of the i.i.d. potential and yields the direct-integral unitary equivalence of Theorem 1.2.

What would settle it

For $\Gamma = \mathbb{Z}$, $\Lambda = \mathbb{Z}/2\mathbb{Z}$, with $m_\Gamma$ and $m_\Lambda$ each uniform on $\{\pm 1\}$, compute both sides of the moment identity (7) for $n = 1,2,3,4$: the left side is an explicit finite sum over closed walks on $\mathbb{Z}$ with Bernoulli weights, and the right side is an explicit finite sum over closed walks on the lamplighter group. A disagreement for any single $n$ would refute Theorem 1.1; agreement for all $n$ is a direct check of the proof's closed-walk bookkeeping.

Watch

Extended reading notes

Core claim

On its own terms, the discovery is that randomness and wreath-product geometry are two faces of the same spectral object. Theorem 1.1 identifies the density of states of $H(\omega) = L_{m_\Gamma} + V$, where the i.i.d. potential follows $\mu^{e_\Lambda}_{L_{m_\Lambda}}$, with the Plancherel measure $\mu^{e_{\Lambda \wr \Gamma}}_{L_{\hat m_\Gamma + \hat m_\Lambda}}$ on the wreath product. Theorem 1.2 upgrades this to a unitary equivalence for finitely generated Abelian lamp groups: after Fourier transform on the lamp configurations, the deterministic operator $M$ on $\Lambda \wr \Gamma$ is exactly the direct integral of $H(\omega)$ over the probability space of potentials. The paper then uses the correspondence in both directions: a Wegner-estimate argument makes the density of states absolutely continuous with bounded density whenever the lamp-group spectral measure is, giving an absolute-continuity criterion for convolutions on wreath products; return-probability asymptotics on wreath products yield Lifshitz tails for the random operator on any polynomial-growth base group; and a reverse correspondence expresses matrix elements of $f(H(\omega))$ as Fourier series over lamp configurations, with the Parseval identity giving the second moment of the Green function.

Load-bearing premise

The results hold only when the single-site disorder distribution is exactly the spectral measure of an $\ell^1$ convolution operator on some lamp group, a class that does not cover every bounded i.i.d. potential, as the paper itself notes in Remark 3.3.

Editorial extensions

If this is right

The density of states of $H(\omega)$ becomes a deterministic spectral object, so Wegner-type regularity estimates for random operators translate into absolute-continuity results for convolution operators on wreath products; in particular, if the lamp-group spectral measure is absolutely continuous with bounded density, then the Plancherel measure of $M$ is purely absolutely continuous (Theorem 2.1
Lifshitz tails hold on any infinite polynomial-growth base group $\Gamma$ with the universal exponent $-d/2$, where $d$ is the growth degree of $\Gamma$, for every potential in the correspondence class (Theorem 2.3).
The reverse correspondence gives an exact formula for the second moment of the Green function, $E|\langle \delta_{e_\Gamma}, (H(\omega)-z)^{-1}\delta_{e_\Gamma}\rangle|^2$, as a sum of squared matrix elements of $(M-z)^{-1}$ over lamp configurations, linking delocalization of $H$ to decay of $M$'s eigenvectors along the lamp group (Proposition 2.5 and Remark 2.6).
The known lamplighter-to-Bernoulli correspondence is generalized from binary disorder to arbitrary spectral-measure disorder and from the lamplighter group to arbitrary base and lamp groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the paper leaves implicit is that any numerical method able to approximate Plancherel measures of convolution operators on finite truncations of wreath products becomes a numerical method for the density of states of the random operator, bypassing ensemble averaging entirely.
The restriction identified in Remark 3.3 suggests an inverse classification problem: characterize the probability measures on the real line that are spectral measures of $\ell^1$ convolution operators on countable Abelian groups; the paper's results would apply exactly on that class.
The Green-function second-moment formula rephrases the open problem of low-disorder delocalization on amenable groups as a deterministic question about decay of matrix coefficients of resolvents of wreath-product convolution operators along the lamp factor, a potentially more tractable target than direct random-operator analysis.
The correspondence is likely to extend to generalized wreath products with site-dependent lamp groups, as the paper itself notes in Remark 1.3, which would cover inhomogeneous or correlated disorder.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Core is solid; the packaging is too loud. Theorem 1.1 is a clean generalization of the lamplighter–percolation correspondence, and I believe the proof. The moment expansion on closed walks is the right way to see it, and the direct integral version for Abelian lamps (Theorem 1.2) is a genuine strengthening, not just a reinterpretation. The applications are real: the ac criterion via Wegner estimate works, the Green function second moment/delocalization dictionary is attractive, and using Erschler's return probabilities to get Lifshitz tails is a nice transfer from group theory to random operators.

The main caveat is structural: the correspondence only holds when the single-site disorder is exactly the spectral measure of an ell^1 convolution operator on the lamp group. That is a real modelling constraint, and Remark 3.3 concedes that not every distribution has this form. So the abstract's unqualified 'random Schrödinger operators' and the title of Theorem 2.3 overstate the scope. The result is still interesting for the correspondence class, but it is not a Lifshitz tail theorem for arbitrary bounded i.i.d. potentials on polynomial-growth groups.

There is also a concrete sign error in the lower bound of the Theorem 2.3 proof: the inequality should subtract e^{-\varepsilon t}, not e^{\varepsilon t}. As printed the bound is vacuous. It looks typographical—the next line already uses the correct e^{-\varepsilon t+···}—but it has to be fixed.

Citations look honest; Erschler, Pittet–Saloff-Coste and the Wegner estimate are used as inputs, not manufactured. No circularity. The paper is written carefully apart from the abstract/title overreach and a few typos.

Who is this for? People working at the intersection of random Schrödinger operators and geometric group theory, especially anyone interested in spectral measures of wreath products. I would send it to a serious referee; with the sign fixed and the scope qualified, it should be publishable.

Referee Report

0 major / 5 minor

Summary. The paper proves a spectral correspondence between random Schrödinger operators on a countable group Γ and deterministic convolution operators on the wreath product Λ≀Γ. Theorem 1.1 states that if H(ω)=A+V on ℓ²(Γ) has A=L_{mΓ} and i.i.d. single-site potential distributed as the spectral measure of B=L_{mΛ} at the identity, then the averaged spectral measure at eΓ equals the Plancherel measure of M=L_{m̂Γ+m̂Λ} on Λ≀Γ. Theorem 1.2 upgrades this, for finitely generated Abelian Λ, to unitary equivalence between M and the direct integral of H(ω) over the Pontryagin dual of the lamp configurations. Applications include an absolutely-continuous-spectrum criterion via Wegner estimates (Theorem 2.1), Lifshitz tail asymptotics for polynomial-growth base groups (Theorem 2.3), and a Fourier/Parseval formula for the second moment of the Green function (Proposition 2.5). Proofs are based on moment expansions and explicit Fourier transforms.

Significance. The central identity is exact and is proved by a clean moment computation, with the direct-integral version giving a constructive unitary. If correct, it provides a new bridge between random operator theory and group convolution, and the applications—especially the AC criterion and Lifshitz tails—are concrete and nontrivial. The proof of Theorem 1.1 is elementary and self-contained apart from standard spectral theory; Theorem 2.3 correctly uses Erschler's return-probability asymptotics as an external input. The main limitation is the modelling assumption that the disorder distribution coincides with the spectral measure of an ℓ¹ convolution operator on a lamp group, which is not satisfied by a generic bounded i.i.d. potential; this is acknowledged in Remark 3.3 but should be foregrounded. Overall the paper is a solid conditional contribution.

minor comments (5)

[§2.2, proof of Theorem 2.3, lower bound] The displayed lower bound '¯µ_H(I_ε) ≥ −e^{εt} + C e^{−ct^{d/(d+2)} log^α(t)}' should read '¯µ_H(I_ε) ≥ C e^{−ct^{d/(d+2)} log^α(t)} − e^{−εt}'. As printed the right-hand side is eventually negative and the bound is vacuous, while the following line already uses the corrected exponent e^{−εt}; this is a typographical error but it must be fixed because the lower bound is needed for the theorem.
[Abstract and Theorem 2.3] The abstract and the heading 'Lifshitz tail on any polynomial growth group' overstate the scope: the results apply only when the single-site potential distribution is exactly the spectral measure μ_B^{eΛ} of some convolution operator B=L_{mΛ} with mΛ∈ℓ¹, and Remark 3.3 concedes that not every distribution can be realized in this way. Please state this hypothesis explicitly in the abstract and temper the 'random Schrödinger operators' phrasing accordingly.
[Theorem 2.1] The final sentence 'M is purely absolutely continuous operator' appears to require that the support of m_{Λ≀Γ} generates the group so that δ_{e_{Λ≀Γ}} is cyclic; otherwise the bounded-density conclusion applies only to the cyclic subspace. This is satisfied in Example 2.2 because ˆS⊔ˆT is a generating set, but the theorem statement should include an explicit generating or cyclicity assumption.
[§2.2, upper-bound chain] In the upper-bound argument for Theorem 2.3, the log^α(t) factor is dropped when passing from k(t) to the bound '≤ 2Ce^{−ct^{d/(d+2)}+εt}'. The inequality is valid for large t since log^α(t)≥1, but the notation should be kept consistent so the reader can follow which constants absorb the logarithmic factor.
[Various] Please correct minor typos: the section heading 'Lifchitz tail' should be 'Lifshitz tail'; Theorem 1.1 contains 'on of the form'; the abstract repeats 'Lifschitz'; and Theorem 2.3's assumption 'containing a finite generating set' should specify that the measure is symmetric and that the generating set is meant in the group-theoretic sense.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Theorems 1.1 and 1.2 are proven self-containedly by moment expansions and direct integral computations; the only self-citation is a non-load-bearing pointer in a remark.

full rationale

The derivation chain is self-contained. Theorem 1.1 is proven by expressing both sides in terms of closed walks: the left side uses the independence of the potentials and the identity E[v^k] = <δ_{e_Λ}, B^k δ_{e_Λ}>, which holds because the potential distribution is defined as the spectral measure of B. This is a definitional input, not a fitted or predicted quantity; the equality of moments is then a combinatorial rearrangement, not a circular reduction. Theorem 1.2 and Theorem 3.4 are proven by constructing an explicit unitary F via Pontryagin duality and verifying (10) on basis elements. The applications rest on external inputs: Erschler's asymptotic return probability for wreath products [13] and Wegner estimates from [3], both used as hypotheses rather than as consequences of the paper's results. The only self-citation, [5, Lemma 20] in Remark 2.6, is a pointer for a possible future approach to proving absolutely continuous spectrum and is not used in any proof in this paper. Remark 3.3 honestly states the modeling limitation that not every potential distribution is realizable (the Fourier representation need not be ℓ^1); this is a restrictiveness caveat, not circularity. The lower bound in the proof of Theorem 2.3 appears to contain a sign error (the first term is printed as -e^{εt} instead of e^{-εt}, making the bound vacuously true as written); this is a correctness issue in a proof step, not a circular self-reference. No step reduces a prediction to its own input, and no load-bearing self-citation was found.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters and no ad hoc physical or mathematical entities. It relies on standard spectral theory and on external theorems from random operator theory (Wegner estimates) and random walks on groups (Erschler's return probability asymptotics), all cited. The only modelling restriction is the use of potential distributions that are spectral measures of convolution operators on the lamp group, which the author explicitly acknowledges in Remark 3.3.

assumptions (9)

standard math Spectral theorem and functional calculus for self-adjoint operators on Hilbert space.
Used throughout to define spectral measures and functional calculus; explicitly invoked in Section 1 notation and in the direct integral arguments of Theorem 1.2 (Ref. [33]).
standard math Ergodicity of random operators: almost sure spectrum and Lebesgue decomposition of spectral measures.
Assumed to define the density of states measure and the almost sure spectrum; cited from [3, Theorem 3.10] in Section 1.
domain assumption Erschler's return probability asymptotics for wreath products (Ref. [13, Theorem 2]).
Used in Theorem 2.3 to transfer return probability bounds to Lifshitz tail bounds; the asymptotics m_{2n} ~ exp(-n^{d/(d+2)} log^α n) is cited, not proved.
domain assumption Wegner estimate / rank-one perturbation bound for the density of the averaged spectral measure (Ref. [3, Chapter 4]).
Used in Theorem 2.1 to deduce absolute continuity of the DOS with bounded density from the single-site density bound.
standard math Kesten criterion: spectral radius of a symmetric generating probability measure on an amenable group equals 1.
Used in Theorem 2.3 to identify σ = max Σ(M) = 1.
domain assumption Stability of return probability asymptotics for symmetric measures with finite second moment that generate the group (Ref. [32]).
Used in Theorem 2.3 to relate moments of the spectral measure to return probabilities and heat kernel estimates.
standard math Pontryagin duality and Fourier transform on countable discrete Abelian groups, extended to product spaces.
Used in Theorem 1.2 and Lemma 3.2 to construct the probability space Ω and the unitary transform F via characters on the lamp configuration group.
standard math Gromov's polynomial growth theorem (Ref. [21]).
Used to define the growth degree d of the group, which appears in the Lifshitz exponent of Theorem 2.3.
domain assumption Absolute continuity with bounded density of the spectral measure of the adjacency operator on the free group F_d and on Z^d for d ≥ 3.
Used in Example 2.2 to verify the hypothesis of Theorem 2.1 for wreath products with those lamp groups.

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Cite this review

Pith. "Pith review of Random Schr\"odinger operators and convolution on wreath products." pith.science (2026). https://pith.science/paper/4MZEUGQT

@misc{pith2026250522485,
  author       = {Pith},
  title        = {Pith review of: Random Schr\"odinger operators and convolution on wreath products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4MZEUGQT}},
  note         = {Machine review of arXiv:2505.22485}
}

read the original abstract

We establish a spectral correspondence between random Schr\"odinger operators and deterministic convolution operators on wreath products, generalizing previous results that relate Lamplighter groups to Schr\"odinger operators with Bernoulli potentials. Using this correspondence in both directions, we obtain an elementary criterion for the absolute continuity of convolutions on wreath products, Lifschitz tail estimates for Schr\"odinger operators on Cayley graphs of polynomial growth, and an exact formula for the second moment of the Green function, expressed in terms of the wreath product with an Abelian group of lamps.

Figures

Figures reproduced from arXiv: 2505.22485 by the authors.

**Figure 1.** The tree TΓ,Λ An automorphism is a permutation of the vertices that preserves the adjacency relation. In particular, it must fix the root. Given an element h = (f, g) ∈ Λ≀Γ, one associates the automorphism ϕh sending s ∈ Γ to ϕh(s) = gs and sl ∈ Γ × Λ to ϕh(sl) = gsf(g)l. An important example is the wreath product (Z/2Z) ≀ Z, often referred to as the lamplighter group. It can be visualized as follows: consider a la… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A random matrix approach to lamplighter groups
math.PR 2026-07 accept novelty 7.0 of 10

Random permutation-plus-diagonal matrices reproduce the natural-generator spectral measure of lamplighter groups, with a CLT for Γ=Z.

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Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper

[1]

Localization at large dis order and at extreme energies: An elementary derivations

Michael Aizenman and Stanislav Molchanov. Localization at large dis order and at extreme energies: An elementary derivations. Communications in Mathematical Physics, 157:245–278, 1993

work page 1993
[2]

Stability of the absolutely continuous spectrum of random schr¨ odinger operators on tree graphs

Michael Aizenman, Robert Sims, and Simone Warzel. Stability of the absolutely continuous spectrum of random schr¨ odinger operators on tree graphs. Probability theory and related ﬁelds , 136:363–394, 2006

work page 2006
[3]

Random operators, volume 168

Michael Aizenman and Simone Warzel. Random operators, volume 168. American Mathematical Soc., 2015

work page 2015
[4]

Absence of diﬀusion in certain random lattices

Philip W Anderson. Absence of diﬀusion in certain random lattices. Physical review, 109(5):1492, 1958

work page 1958
[5]

Existence of absolutely con tinuous spectrum for galton–watson random trees

Adam Arras and Charles Bordenave. Existence of absolutely con tinuous spectrum for galton–watson random trees. Communications in Mathematical Physics , 403(1):495– 527, 2023

work page 2023
[6]

Spectrum of random graphs

Charles Bordenave. Spectrum of random graphs. Advanced topics in random matri- ces, 53:91–150, 2016

work page 2016
[7]

Random lattice schr¨ odinger operators with decayin g potential: some higher dimensional phenomena

J Bourgain. Random lattice schr¨ odinger operators with decayin g potential: some higher dimensional phenomena. In Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002 , pages 70–98. Springer, 2003

work page 2001
[8]

On random schr¨ odinger operators on Z2

Jean Bourgain. On random schr¨ odinger operators on Z2. Discrete and Continuous Dynamical Systems, 8(1):1–15, 2001

work page 2001

Show all 36 references

[9]

Spectra of inﬁnite cayley graphs, examples wit h pure band spectra

Pierre de la Harpe. Spectra of inﬁnite cayley graphs, examples wit h pure band spectra. arXiv preprint arXiv:2402.06279 , 2024

2024 arXiv
[10]

Les c [*]-alg´ ebres et leurs repr´ esentations

Jacques Dixmier. Les c [*]-alg´ ebres et leurs repr´ esentations. (No Title) , 1969

1969
[11]

On the analytic zero divisor conjecture of linnell

G´ abor Elek. On the analytic zero divisor conjecture of linnell. Bulletin of the London Mathematical Society, 35(2):236–238, 2003

2003
[12]

Lˆ 2-spectral invariants and convergent sequences of ﬁnite graphs

G´ abor Elek. Lˆ 2-spectral invariants and convergent sequences of ﬁnite graphs. arXiv preprint arXiv:0709.1261 , 2007

2007 arXiv
[13]

Isoperimetry for wreath products of markov chains and multiplicity of selﬁntersections of random walks

Anna Erschler. Isoperimetry for wreath products of markov chains and multiplicity of selﬁntersections of random walks. Probability Theory and related ﬁelds , 136:560–586, 2006

2006
[14]

A course in abstract harmonic analysis

Gerald B Folland. A course in abstract harmonic analysis . CRC press, 2016

2016
[15]

Absolutely continuous spec- trum for the anderson model on a tree: a geometric proof of klein’s theorem

Richard Froese, David Hasler, and Wolfgang Spitzer. Absolutely continuous spec- trum for the anderson model on a tree: a geometric proof of klein’s theorem. Com- munications in mathematical physics , 269:239–257, 2007

2007
[16]

Absence of diﬀusion in the anderson tight binding model for large disorder or low energy

J¨ urg Fr¨ ohlich and Thomas Spencer. Absence of diﬀusion in the anderson tight binding model for large disorder or low energy. Communications in Mathematical Physics, 88(2):151–184, 1983. 13

1983
[17]

Anderson localization and the space-time char acteristic of con- tinuum states

Gian Michele Graf. Anderson localization and the space-time char acteristic of con- tinuum states. Journal of statistical physics , 75:337–346, 1994

1994
[18]

Laplace and schr \” odinger op- erators without eigenvalues on homogeneous amenable graphs

Rostislav Grigorchuk and Christophe Pittet. Laplace and schr \” odinger op- erators without eigenvalues on homogeneous amenable graphs. arXiv preprint arXiv:2102.13542, 2021

2021 arXiv
[19]

The lamplighter group as a group generated by a 2-state automaton, and its spectrum

Rostislav I Grigorchuk and Andrzej ˙Zuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geometriae Dedicata, 87(1):209–244, 2001

2001
[20]

The ihara zeta functio n of inﬁnite graphs, the kns spectral measure and integrable maps

Rostislav I Grigorchuk and Andrzej Zuk. The ihara zeta functio n of inﬁnite graphs, the kns spectral measure and integrable maps. Random walks and geometry , pages 141–180, 2004

2004
[21]

Groups of polynomial growth and expanding ma ps (with an ap- pendix by jacques tits)

Michael Gromov. Groups of polynomial growth and expanding ma ps (with an ap- pendix by jacques tits). Publications Math´ ematiques de l’IH´ES, 53:53–78, 1981

1981
[22]

Quantum theory for mathematicians , volume 267

Brian C Hall. Quantum theory for mathematicians , volume 267. Springer Science & Business Media, 2013

2013
[23]

The a nderson model on the bethe lattice: Lifshitz tails

Francisco Hoecker-Escuti and Christoph Schumacher. The a nderson model on the bethe lattice: Lifshitz tails. 2014

2014
[24]

Absolutely cont inuous spectrum for random operators on trees of ﬁnite cone type

Matthias Keller, Daniel Lenz, and Simone Warzel. Absolutely cont inuous spectrum for random operators on trees of ﬁnite cone type. Journal d’Analyse Math´ ematique, 118(1):363–396, 2012

2012
[25]

Symmetric random walks on groups

Harry Kesten. Symmetric random walks on groups. Transactions of the American Mathematical Society, 92(2):336–354, 1959

1959
[26]

Extended states in the anderson model on the beth e lattice

Abel Klein. Extended states in the anderson model on the beth e lattice. Advances in Mathematics , 133(1):163–184, 1998

1998
[27]

On th e spectrum of lamp- lighter groups and percolation clusters

Franz Lehner, Markus Neuhauser, and Wolfgang Woess. On th e spectrum of lamp- lighter groups and percolation clusters. Mathematische Annalen, 342(1):69–89, 2008

2008
[28]

Energy spectrum structure and quan tum states of disor- dered condensed systems

Il’ya Mikhailovich Lifshitz. Energy spectrum structure and quan tum states of disor- dered condensed systems. Soviet Physics Uspekhi , 7(4):549, 1965

1965
[29]

Analytic versions of the zero divisor conjecture

Peter A Linnell. Analytic versions of the zero divisor conjecture . London Mathemat- ical Society Lecture Note Series , pages 209–248, 1998

1998
[30]

Spectral analysis for convo lution operators on locally compact groups

M M˘ antoiu and R Tiedra De Aldecoa. Spectral analysis for convo lution operators on locally compact groups. Journal of Functional Analysis , 253(2):675–691, 2007

2007
[31]

A survey on spectra of inﬁnit e graphs

Bojan Mohar and Wolfgang Woess. A survey on spectra of inﬁnit e graphs. Bulletin of the London Mathematical Society , 21(3):209–234, 1989

1989
[32]

On the stability of the beha vior of random walks on groups

Ch Pittet and Laurent Saloﬀ-Coste. On the stability of the beha vior of random walks on groups. Journal of Geometric Analysis , 10(4), 2000

2000
[33]

Methods of modern mathematical physics, volume 1

Michael Reed, Barry Simon, Barry Simon, and Barry Simon. Methods of modern mathematical physics, volume 1. Elsevier, 1972. 14

1972
[34]

Lifschitz tails for the anderson model

Barry Simon. Lifschitz tails for the anderson model. Journal of statistical physics , 38:65–76, 1985

1985
[35]

An introduction to the mathematics of anderso n localization

G¨ unter Stolz. An introduction to the mathematics of anderso n localization. Entropy and the quantum II. Contemp. Math , 552:71–108, 2011

2011
[36]

Springer, 2008

Ivan Veseli´ c.Existence and regularity properties of the integrated dens ity of states of random Schr¨ odinger operators, volume 1917. Springer, 2008. Adam Arras, HUN-REN Alfr´ ed R´ enyi Institute of Mathematics, Budapest, Hungary, arras@renyi.hu 15

1917

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Localization at large dis order and at extreme energies: An elementary derivations

Michael Aizenman and Stanislav Molchanov. Localization at large dis order and at extreme energies: An elementary derivations. Communications in Mathematical Physics, 157:245–278, 1993

work page 1993

[2] [2]

Stability of the absolutely continuous spectrum of random schr¨ odinger operators on tree graphs

Michael Aizenman, Robert Sims, and Simone Warzel. Stability of the absolutely continuous spectrum of random schr¨ odinger operators on tree graphs. Probability theory and related ﬁelds , 136:363–394, 2006

work page 2006

[3] [3]

Random operators, volume 168

Michael Aizenman and Simone Warzel. Random operators, volume 168. American Mathematical Soc., 2015

work page 2015

[4] [4]

Absence of diﬀusion in certain random lattices

Philip W Anderson. Absence of diﬀusion in certain random lattices. Physical review, 109(5):1492, 1958

work page 1958

[5] [5]

Existence of absolutely con tinuous spectrum for galton–watson random trees

Adam Arras and Charles Bordenave. Existence of absolutely con tinuous spectrum for galton–watson random trees. Communications in Mathematical Physics , 403(1):495– 527, 2023

work page 2023

[6] [6]

Spectrum of random graphs

Charles Bordenave. Spectrum of random graphs. Advanced topics in random matri- ces, 53:91–150, 2016

work page 2016

[7] [7]

Random lattice schr¨ odinger operators with decayin g potential: some higher dimensional phenomena

J Bourgain. Random lattice schr¨ odinger operators with decayin g potential: some higher dimensional phenomena. In Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002 , pages 70–98. Springer, 2003

work page 2001

[8] [8]

On random schr¨ odinger operators on Z2

Jean Bourgain. On random schr¨ odinger operators on Z2. Discrete and Continuous Dynamical Systems, 8(1):1–15, 2001

work page 2001