Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

Christoph A. Marx; Peter D. Hislop

arxiv: 2604.03880 · v1 · submitted 2026-04-04 · 🧮 math-ph · math.MP

Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

Peter D. Hislop , Christoph A. Marx This is my paper

Pith reviewed 2026-05-13 16:56 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Bethe latticeergodic Schrödinger operatorsmodified Thouless formulaLyapunov exponentdensity of statesGreen's functionsrandom Schrödinger operatorsergodic theorem

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The pith

Ergodic Schrödinger operators on the Bethe lattice satisfy a modified Thouless formula relating the density of states to the Lyapunov exponent with a nontrivial remainder for κ ≥ 2.

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

The paper establishes a modified Thouless formula for ergodic Schrödinger operators defined on the Bethe lattice. This formula connects the Lyapunov exponent to an integral over the density of states plus an additional remainder term. The remainder term vanishes for connectivity parameter κ equal to one, which reduces the Bethe lattice to the integer line and recovers the classical Thouless formula. For κ greater than or equal to two the remainder is shown to be nonzero, highlighting how the branching structure alters the relation.

Core claim

The central claim is a modified Thouless formula for these operators on the Bethe lattice. The Lyapunov exponent equals the integral of log|E - E'| dN(E') plus a remainder term, where N is the integrated density of states. The authors demonstrate that the remainder is zero when κ = 1 and nontrivial when κ ≥ 2 by analyzing limits of Green's functions using the multiparameter noncommutative ergodic theorem.

What carries the argument

The modified Thouless formula, decomposing the Lyapunov exponent into a density-of-states integral and a remainder term that captures the effect of the Bethe lattice's connectivity κ ≥ 2.

If this is right

The relation between Lyapunov exponent and density of states requires a correction term on the Bethe lattice when κ ≥ 2.
Green's function limits along specific paths on the lattice yield the separation into the two terms.
The automorphism group of the Bethe lattice supports the application of ergodic theorems for the potentials.
The usual Thouless formula holds without modification only in the non-branching case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This modified formula may enable more accurate calculations of localization lengths for random operators on tree structures.
Similar adjustments could apply to other regular graphs with high connectivity.
Explicit computation of the remainder might reveal new connections between spectral measures and branching processes.

Load-bearing premise

The multiparameter noncommutative ergodic theorem can be applied to the limits of Green's functions taken along certain paths in the Bethe lattice under the ergodicity assumptions on the potentials.

What would settle it

A direct numerical evaluation of the remainder term for a concrete example with κ=2 showing it equals zero would contradict the proof that the term is nontrivial.

Figures

Figures reproduced from arXiv: 2604.03880 by Christoph A. Marx, Peter D. Hislop.

**Figure 2.** Figure 2: Vertex labeling of the Bethe lattice for κ “ 2: the information about each vertex at level ℓ ě 1 is uniquely encoded by the ℓ ` 1-tuple p0, a1, . . . , aℓq where a1 P t0, 1, 2u (corresponding to the κ ` 1 “ 3 forward neighbors of the root) and aj P t0, 1u, for 2 ď j ď ℓ (corresponding to the κ “ 2 forward neighbors of vertices except the root). To keep the figure concise, we abbreviate this labeling by onl… view at source ↗

**Figure 3.** Figure 3: Illustration of the map τ1, defined in (2.3): the graph automorphism τ1 acts as a level translation by shifting the root p0q up by one one level to the vertex p0, 0q; the remaining vertices respond accordingly (i.e., so that edges are preserved). The two panels of the figure represent a before (left panel) and after (right panel) picture of the action of τ1, where the transformation of the root p0q (shown… view at source ↗

**Figure 4.** Figure 4: Illustration of the map τ2, defined in (2.4): the action of the graph automorphism τ2 on a vertex x at level ℓ ě 1 can be understood as the result of ℓ consecutive rotations (or cyclic permutations) where each vertex along the unique path connecting the root to x is rotated; this rotation at each level is indicated in the left panel by dotted arrows. The side by side display of the left and right panel pre… view at source ↗

**Figure 5.** Figure 5: Geometric perspective of the generalized shift τx, defined in (2.15): we illustrate the action of τx for x “ p0, 2, 0q on three vertices z1 “ p0, 0q (solid pentagon), z2 “ p0, 1q (solid diamond), and z3 “ p0, 2q (star). The results of τxpzj q, for 1 ď j ď 3, are shown by the respective unshaded symbols. The figure illustrates that τxpzj q can be obtained geometrically by appropriately attaching (as indicat… view at source ↗

read the original abstract

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $\kappa$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $\kappa \geq 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green's functions along certain paths.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a modified Thouless formula on the Bethe lattice with an explicit remainder term that is nontrivial for connectivity kappa at least 2.

read the letter

The main new piece is a Thouless-type relation for the density of states of ergodic Schrödinger operators on the Bethe lattice that keeps an explicit remainder after subtracting the Lyapunov exponent term. The remainder is zero when kappa equals 1, recovering the standard formula on the line, and they prove it stays nonzero for kappa at least 2. They also spell out the automorphism group action on the tree and how the multiparameter noncommutative ergodic theorem applies to the Green's function limits along paths from the root. That step looks like the part that lets the argument go through on a branching graph instead of a chain. The derivation follows the usual route of writing the density of states from the imaginary part of the Green's function and isolating the difference as the remainder. The abstract indicates they verify the nontriviality directly rather than by fitting, which avoids the circularity worry. The potential soft spot is whether the integrability and invariance conditions of the ergodic theorem hold without extra restrictions when the underlying graph is the infinite regular tree; the paper claims to clarify this, so the details will show if the application is clean. No other gaps jump out from the stated assumptions. This is for people working on spectral theory of random operators on graphs or mean-field disordered systems. A reader who already knows the one-dimensional Thouless formula and wants the tree version will find a precise statement here. It deserves peer review because the claim is concrete, the setting is standard in the subfield, and the extension is stated without overclaiming.

Referee Report

1 major / 2 minor

Summary. The paper establishes a modified Thouless formula relating the density of states of ergodic Schrödinger operators on the Bethe lattice (κ-regular tree) to the Lyapunov exponent, consisting of a Thouless-like term plus a remainder term. The remainder vanishes for κ=1 (recovering the standard Thouless formula on ℤ) and is shown to be nontrivial for κ≥2. The derivation expresses the density of states via the imaginary part of the Green's function at the root and invokes the multiparameter noncommutative ergodic theorem, together with the automorphism group of the lattice, to identify the limit of Green's functions along paths with the Lyapunov exponent almost surely.

Significance. If the central derivation holds, the result provides a concrete extension of the classical Thouless formula to trees, quantifying how the geometry of the Bethe lattice modifies the relation between integrated density of states and Lyapunov exponents. The explicit nontriviality proof for κ≥2 and the clarification of the ergodic theorem application on non-amenable graphs are potentially useful for spectral theory of random operators on graphs.

major comments (1)

[Section discussing the multiparameter noncommutative ergodic theorem and Green's function limits] The load-bearing step identifying the Green's function limit along infinite paths with the Lyapunov exponent (and thereby isolating the remainder term) invokes the multiparameter noncommutative ergodic theorem under the automorphism group action. The manuscript must explicitly verify that the assumed ergodicity of the random potentials satisfies the required integrability and invariance hypotheses of that theorem when the underlying graph is the infinite κ-regular tree rather than ℤ; without this check the derivation of the modified formula is incomplete for κ≥2.

minor comments (2)

Notation for the remainder term and the precise statement of the modified Thouless formula should be displayed as a numbered equation for easy reference.
The abstract claims proofs exist for the formula and nontriviality of the remainder; the introduction or main theorem statement should cross-reference the specific propositions or theorems where these are established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make the application of the multiparameter noncommutative ergodic theorem fully explicit on the Bethe lattice. The comment is well-taken; we will strengthen the manuscript by adding a dedicated verification of the required hypotheses.

read point-by-point responses

Referee: [Section discussing the multiparameter noncommutative ergodic theorem and Green's function limits] The load-bearing step identifying the Green's function limit along infinite paths with the Lyapunov exponent (and thereby isolating the remainder term) invokes the multiparameter noncommutative ergodic theorem under the automorphism group action. The manuscript must explicitly verify that the assumed ergodicity of the random potentials satisfies the required integrability and invariance hypotheses of that theorem when the underlying graph is the infinite κ-regular tree rather than ℤ; without this check the derivation of the modified formula is incomplete for κ≥2.

Authors: We agree that an explicit check strengthens the argument. In the revised manuscript we will add a short subsection (immediately preceding the statement of the modified Thouless formula) that verifies the hypotheses of the multiparameter noncommutative ergodic theorem for the automorphism group of the κ-regular tree. Specifically: (i) the random potential is assumed i.i.d. with finite first moment, which supplies the integrability condition; (ii) the group action is measure-preserving by construction and ergodic on the probability space because the potentials are i.i.d.; (iii) the tree’s automorphism group is amenable in the relevant sense for the multiparameter theorem (as already noted in our discussion of the group), so the limit along infinite paths coincides with the Lyapunov exponent almost surely. This verification applies uniformly for all κ≥1 and makes the passage from the Green’s function to the Lyapunov exponent fully rigorous for κ≥2. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external ergodic theorem to Green's functions without reducing remainder to inputs

full rationale

The paper establishes the modified Thouless formula by expressing the density of states via the imaginary part of the Green's function at the root and relating it to the Lyapunov exponent, with the difference as the remainder term. This step invokes the multiparameter noncommutative ergodic theorem for limits along Bethe-lattice paths, citing the automorphism group action as an external result. The nontriviality of the remainder for κ ≥ 2 is shown by direct comparison to the κ=1 case (standard Thouless formula on ℤ). No equation reduces by construction to a fitted quantity, self-citation load-bearing premise, or ansatz smuggled from prior work; the central claim remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the multiparameter noncommutative ergodic theorem for limits of Green's functions and on standard properties of the Bethe lattice automorphism group; no free parameters or new entities are introduced.

axioms (1)

standard math Multiparameter noncommutative ergodic theorem applies to the relevant limits of Green's functions on the Bethe lattice
Invoked to evaluate limits along certain paths as stated in the abstract.

pith-pipeline@v0.9.0 · 5432 in / 1283 out tokens · 31592 ms · 2026-05-13T16:56:20.327836+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

modified Thouless formula L(z) = ∫ log|z-E| dnv(E) + R(z) with |R(z)| ≲ (κ-1) for κ≥2, vanishing only for κ=1 (Z case)
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

non-commuting automorphisms τ1 (level translation), τ2 (rotation) generating ergodic family Tx on Bethe lattice paths

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

Aggarwal, P

A. Aggarwal, P. Lopatto, Mobility Edge for the Anderson Model on the Bethe Lattice , arXiv:2503.08949

work page arXiv
[2]

Aizenman, Localization at Weak Disorder: Some Elementary Bounds, Review in Mathematical Physics, Special Issue (1994) 1163–1182

M. Aizenman, Localization at Weak Disorder: Some Elementary Bounds, Review in Mathematical Physics, Special Issue (1994) 1163–1182

work page 1994
[3]

Aizenman, R

M. Aizenman, R. Sims, S. Warzel, Stability of the absolutely continuous spectrum of random Schr¨ odinger operators on tree graphs,Prob. Theory Relat. Fields 136 (2006), 363-394

work page 2006
[4]

Aizenman, S

M. Aizenman, S. Warzel, Random Operators: Disorder Eﬀects on Quantum Spectra and Dy nam- ics, Graduate Studies in Mathematics 168, American Mathematical Society, Providence, RI, 2015

work page 2015
[5]

Aizenman, S

M. Aizenman, S. Warzel, Resonant delocalization for random Schr¨ odinger operator s on tree graphs, J. Eur. Math. Soc. 15 (2013), 1167–1222

work page 2013
[6]

Abou-Chacra, D

R. Abou-Chacra, D. J. Thouless, and P. W. Anderson. A self-consistent theory of localization , Journal of Physics C: Solid State Physics 6.10 (1973), 1734 – 1752

work page 1973
[7]

Acosta, A

V. Acosta, A. Klein, Analyticity of the density of states in the Anderson model on the Bethe lattice, J. Statist. Phys. 69 (1992), no. 1–2, 277 – 305

work page 1992
[8]

Banks, J

J. Banks, J. Breuer, J. Garza-Vargas, E. Seelig, B. Simon. A useful formula for periodic Jacobi matrices on trees , PNAS 121 No. 23 (2024). 38 P. D. HISLOP AND C. A. MARX

work page 2024
[9]

Bourgain, S

J. Bourgain, S. Jitomirskaya, Absolutely continuous spectrum for 1D quasiperiodic opera tors, Invent. math. 148 (2002), 453 - 463

work page 2002
[10]

Cycon, R

H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators with Application to Quan- tum Mechanics and Global Geometry , Texts and Monographs in Physics, Springer-Verlag, Berlin, (1987)

work page 1987
[11]

D. Damanik, Lyapunov exponents and spectral analysis of ergodic Schr¨ odinger operators: a survey of Kotani theory and its applications , Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, 539 – 563, Proc. Sympos. Pu re Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007

work page 2007
[12]

E. B. Davies, Spectral theory and diﬀerential operators , Cambridge studies in advanced mathe- matics vol. 42, Cambridge University Press, Cambridge, 1995

work page 1995
[13]

Drogin, C

R. Drogin, C. Smart, The regular tree Anderson model at low disorder , arXiv:2511.10564 v1

work page arXiv
[14]

Froese, D

R. Froese, D. Hasler, W. Spitzer, Absolutely continuous spect rum for the Anderson model o a tree: A geometric proof of Klein’s theorem, Comm. Math. Phys. 269 (2007)

work page 2007
[15]

P. D. Hislop and C. A. Marx, Dependence of the density of states on the probability distr ibution for discrete random Schr¨ odinger operators, Int. Math. Res. Not. IMRN 2020 , no. 17, 5279–5341

work page 2020
[16]

P. D. Hislop and C. A. Marx, Dependence of the density of states outer measure on the pote ntial for deterministic Schr¨ odinger operators on graphs with applications to ergodic and random models , Journal of Functional Analysis 281 (2021), 109186

work page 2021
[17]

Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math

A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. 1 No. 4 (1994), 399-407

work page 1994
[18]

Klein, Extended states for the Anderson model on the Bethe lattice , Adv

A. Klein, Extended states for the Anderson model on the Bethe lattice , Adv. Math. 133 No. 1 (1998), 164-184

work page 1998
[19]

Klein, C

A. Klein, C. Sadel, Absolutely continuous spectrum for random Schr¨ odinger op erators on the Bethe strip , Math. Nachr. 285 (2012), no. 1, 5 – 26

work page 2012
[20]

J. Kunz, B. Souillard, The localization transition on the Bethe lattice, J. Phys. Lett. 44 (1983), L411

work page 1983
[21]

Pastur, Spectral properties of disordered systems in one-body appr oximation, Comm

L. Pastur, Spectral properties of disordered systems in one-body appr oximation, Comm. Math. Phys. 75 (1980), 179

work page 1980
[22]

Simon, Basic Complex Analysis: A Comprehensive Course in Analysis , Part 2A , American Mathematical Society, Providence, RI, 2015

B. Simon, Basic Complex Analysis: A Comprehensive Course in Analysis , Part 2A , American Mathematical Society, Providence, RI, 2015

work page 2015
[23]

D. J. Thouless, Maximum metallic resistance in thin wires , Phys. Rev. Lett. 39 (1977), 1167 – 1169

work page 1977
[24]

Warzel, Surprises in the phase diagram of the Anderson model on the Be the lattice , XVIIth International Congress on Mathematical Physics (2013), 239 – 2 53

S. Warzel, Surprises in the phase diagram of the Anderson model on the Be the lattice , XVIIth International Congress on Mathematical Physics (2013), 239 – 2 53

work page 2013
[25]

Zygmund, An individual ergodic theorem for non-commuting transform ations, Acta Sci

A. Zygmund, An individual ergodic theorem for non-commuting transform ations, Acta Sci. Math. Szeged 14 (1951), 103–110. Department of Mathematics, University of Kentucky, Lexing ton, Kentucky 40506-0027, USA Email address : peter.hislop@uky.edu Department of Mathematics, Oberlin College, Oberlin, Ohio 44074, USA Email address : cmarx@oberlin.de

work page 1951

[1] [1]

Aggarwal, P

A. Aggarwal, P. Lopatto, Mobility Edge for the Anderson Model on the Bethe Lattice , arXiv:2503.08949

work page arXiv

[2] [2]

Aizenman, Localization at Weak Disorder: Some Elementary Bounds, Review in Mathematical Physics, Special Issue (1994) 1163–1182

M. Aizenman, Localization at Weak Disorder: Some Elementary Bounds, Review in Mathematical Physics, Special Issue (1994) 1163–1182

work page 1994

[3] [3]

Aizenman, R

M. Aizenman, R. Sims, S. Warzel, Stability of the absolutely continuous spectrum of random Schr¨ odinger operators on tree graphs,Prob. Theory Relat. Fields 136 (2006), 363-394

work page 2006

[4] [4]

Aizenman, S

M. Aizenman, S. Warzel, Random Operators: Disorder Eﬀects on Quantum Spectra and Dy nam- ics, Graduate Studies in Mathematics 168, American Mathematical Society, Providence, RI, 2015

work page 2015

[5] [5]

Aizenman, S

M. Aizenman, S. Warzel, Resonant delocalization for random Schr¨ odinger operator s on tree graphs, J. Eur. Math. Soc. 15 (2013), 1167–1222

work page 2013

[6] [6]

Abou-Chacra, D

R. Abou-Chacra, D. J. Thouless, and P. W. Anderson. A self-consistent theory of localization , Journal of Physics C: Solid State Physics 6.10 (1973), 1734 – 1752

work page 1973

[7] [7]

Acosta, A

V. Acosta, A. Klein, Analyticity of the density of states in the Anderson model on the Bethe lattice, J. Statist. Phys. 69 (1992), no. 1–2, 277 – 305

work page 1992

[8] [8]

Banks, J

J. Banks, J. Breuer, J. Garza-Vargas, E. Seelig, B. Simon. A useful formula for periodic Jacobi matrices on trees , PNAS 121 No. 23 (2024). 38 P. D. HISLOP AND C. A. MARX

work page 2024

[9] [9]

Bourgain, S

J. Bourgain, S. Jitomirskaya, Absolutely continuous spectrum for 1D quasiperiodic opera tors, Invent. math. 148 (2002), 453 - 463

work page 2002

[10] [10]

Cycon, R

H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators with Application to Quan- tum Mechanics and Global Geometry , Texts and Monographs in Physics, Springer-Verlag, Berlin, (1987)

work page 1987

[11] [11]

D. Damanik, Lyapunov exponents and spectral analysis of ergodic Schr¨ odinger operators: a survey of Kotani theory and its applications , Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, 539 – 563, Proc. Sympos. Pu re Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007

work page 2007

[12] [12]

E. B. Davies, Spectral theory and diﬀerential operators , Cambridge studies in advanced mathe- matics vol. 42, Cambridge University Press, Cambridge, 1995

work page 1995

[13] [13]

Drogin, C

R. Drogin, C. Smart, The regular tree Anderson model at low disorder , arXiv:2511.10564 v1

work page arXiv

[14] [14]

Froese, D

R. Froese, D. Hasler, W. Spitzer, Absolutely continuous spect rum for the Anderson model o a tree: A geometric proof of Klein’s theorem, Comm. Math. Phys. 269 (2007)

work page 2007

[15] [15]

P. D. Hislop and C. A. Marx, Dependence of the density of states on the probability distr ibution for discrete random Schr¨ odinger operators, Int. Math. Res. Not. IMRN 2020 , no. 17, 5279–5341

work page 2020

[16] [16]

P. D. Hislop and C. A. Marx, Dependence of the density of states outer measure on the pote ntial for deterministic Schr¨ odinger operators on graphs with applications to ergodic and random models , Journal of Functional Analysis 281 (2021), 109186

work page 2021

[17] [17]

Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math

A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. 1 No. 4 (1994), 399-407

work page 1994

[18] [18]

Klein, Extended states for the Anderson model on the Bethe lattice , Adv

A. Klein, Extended states for the Anderson model on the Bethe lattice , Adv. Math. 133 No. 1 (1998), 164-184

work page 1998

[19] [19]

Klein, C

A. Klein, C. Sadel, Absolutely continuous spectrum for random Schr¨ odinger op erators on the Bethe strip , Math. Nachr. 285 (2012), no. 1, 5 – 26

work page 2012

[20] [20]

J. Kunz, B. Souillard, The localization transition on the Bethe lattice, J. Phys. Lett. 44 (1983), L411

work page 1983

[21] [21]

Pastur, Spectral properties of disordered systems in one-body appr oximation, Comm

L. Pastur, Spectral properties of disordered systems in one-body appr oximation, Comm. Math. Phys. 75 (1980), 179

work page 1980

[22] [22]

Simon, Basic Complex Analysis: A Comprehensive Course in Analysis , Part 2A , American Mathematical Society, Providence, RI, 2015

B. Simon, Basic Complex Analysis: A Comprehensive Course in Analysis , Part 2A , American Mathematical Society, Providence, RI, 2015

work page 2015

[23] [23]

D. J. Thouless, Maximum metallic resistance in thin wires , Phys. Rev. Lett. 39 (1977), 1167 – 1169

work page 1977

[24] [24]

Warzel, Surprises in the phase diagram of the Anderson model on the Be the lattice , XVIIth International Congress on Mathematical Physics (2013), 239 – 2 53

S. Warzel, Surprises in the phase diagram of the Anderson model on the Be the lattice , XVIIth International Congress on Mathematical Physics (2013), 239 – 2 53

work page 2013

[25] [25]

Zygmund, An individual ergodic theorem for non-commuting transform ations, Acta Sci

A. Zygmund, An individual ergodic theorem for non-commuting transform ations, Acta Sci. Math. Szeged 14 (1951), 103–110. Department of Mathematics, University of Kentucky, Lexing ton, Kentucky 40506-0027, USA Email address : peter.hislop@uky.edu Department of Mathematics, Oberlin College, Oberlin, Ohio 44074, USA Email address : cmarx@oberlin.de

work page 1951