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Anderson localization on the Bethe lattice

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Pith's one-line read The full probability distribution of the local density of states serves as the order parameter for the Anderson localization transition on the Bethe lattice.

desk verdict Lecture notes that review the standard cavity approach to Anderson localization on the Bethe lattice without adding new results or derivations. read the letter →

arxiv 2606.24671 v1 pith:P24WXXXH submitted 2026-06-23 cond-mat.dis-nn

Dafne Prado Bandeira , Angelica Vecchiarelli , Marco Tarzia This is my paper · ORCID
classification cond-mat.dis-nn
keywords AndersonlocalizationBethelatticeresolventformalismcavitymethodlocaldensityofstatesdisorder-driventransitiondirectedpolymers
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

The notes review how Anderson localization on the Bethe lattice becomes analytically tractable through the resolvent formalism and cavity method. They establish that the entire probability distribution of the local density of states, rather than its average value, distinguishes the localized and delocalized phases. This approach yields the critical disorder strength, the localization length on the insulating side, and the correlation volume on the delocalized side. A reader cares because the Bethe lattice supplies a solvable infinite-dimensional limit relevant to many-body localization problems.

What carries the argument

The cavity self-consistent equations for the diagonal resolvent, whose fixed-point solution is the probability distribution of the local density of states.

What would settle it

A direct numerical computation on large finite Bethe lattices showing that the full distribution of the local density of states fails to separate localized from delocalized regimes would falsify the central claim.

Watch

Extended reading notes

Core claim

On the Bethe lattice the Anderson transition is captured by the statistical properties of Green's functions. The cavity self-consistent equations for the diagonal resolvent are derived and solved both numerically by population dynamics and analytically in the large-connectivity limit; the resulting fixed-point distribution of the local density of states functions as the natural order parameter, giving access to critical disorder, localization length, and correlation volume while also linking the problem to directed polymers in random media.

Load-bearing premise

The Bethe lattice realizes the infinite-dimensional limit of the Anderson problem and serves as a useful mean-field description of localization in many-body configuration spaces.

Editorial extensions

If this is right

  • The critical disorder strength follows from the point at which the distribution of local density of states changes character.
  • The localization length in the insulating phase is obtained from the typical decay of the resolvent.
  • The correlation volume that controls the delocalized phase is extracted from the same distribution near criticality.
  • Rare resonant paths and associated freezing map the localization problem onto directed polymers in random media.

Reading between the lines

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

  • The same distribution-based order parameter could be monitored in numerical studies of finite-dimensional lattices or many-body systems to locate transitions without relying on averaged quantities.
  • Connections to directed polymers suggest that rare-region effects might dominate near the transition in real materials, offering a route to predict Griffiths-like singularities.
  • Analytical large-connectivity results could serve as benchmarks for approximate theories of localization on graphs with different topologies.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. This manuscript is a set of lecture notes that review Anderson localization on the Bethe lattice. It employs the resolvent formalism and cavity method to derive self-consistent equations whose solution is the full probability distribution of the local density of states (the order parameter). The notes cover numerical solution via population dynamics, analytic treatment in the large-connectivity limit, the critical disorder strength, the localization length on the insulating side, the correlation volume on the delocalized side, and the mapping to directed polymers in random media that interprets rare resonant paths and freezing.

Significance. If the derivations and numerical procedures are presented accurately, the notes would constitute a compact pedagogical resource for a technique that remains central to mean-field treatments of localization, both for the infinite-dimensional Anderson model and as an approximation for many-body configuration spaces. The explicit identification of the full distribution of the local density of states as the order parameter, together with the discussion of population dynamics and the large-connectivity limit, reproduces standard but non-trivial results from the literature in a single, accessible source.

minor comments (3)
  1. [Abstract] Abstract, final paragraph: the origin of the notes (Oropa Summer School, July 2025) is stated; a short sentence indicating which parts of the text are original derivations versus direct reproductions of earlier literature would help readers assess novelty.
  2. The manuscript repeatedly refers to 'the cavity self-consistent equations' without an early numbered equation that collects the full set of distributional equations; adding such an equation (e.g., after the first derivation) would improve readability.
  3. Section on the large-connectivity limit: the scaling of the critical disorder with coordination number is discussed, but the precise asymptotic expression is not written as a numbered equation; doing so would make the analytic result easier to cite.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our lecture notes and for recommending acceptance. The report contains no major comments requiring a point-by-point reply.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: pedagogical review of established results

full rationale

The paper is explicitly a review of prior derivations in the cavity-method literature for Anderson localization on the Bethe lattice. The central claim that the full distribution of the local density of states is the natural order parameter follows directly from the structure of the self-consistent resolvent equations (which close on the distribution rather than moments), a standard feature already present in the cited foundational works. No new parameters are fitted within the paper, no self-citation chain is load-bearing for a novel result, and no ansatz or uniqueness theorem is smuggled in. The derivation chain is therefore self-contained against external benchmarks.

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

As a review paper, the work relies on standard assumptions from the prior literature on Anderson localization and cavity methods without introducing new free parameters, axioms, or invented entities.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Anderson localization on the Bethe lattice." pith.science (2026). https://pith.science/paper/P24WXXXH

@misc{pith2026260624671,
  author       = {Pith},
  title        = {Pith review of: Anderson localization on the Bethe lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P24WXXXH}},
  note         = {Machine review of arXiv:2606.24671}
}
read the original abstract

Anderson localization is a paradigmatic disorder-driven quantum phase transition in which interference effects can completely suppress wave propagation. Its formulation on the Bethe lattice -- an infinite tree with fixed coordination number -- provides a unique setting in which the transition becomes analytically tractable. Owing to its exponential volume growth, the Bethe lattice realizes the infinite-dimensional limit of the Anderson problem and also serves as a useful mean-field-like description of localization in many-body configuration spaces. These lecture notes provide a pedagogical review of Anderson localization on the Bethe lattice, focusing on the resolvent (Green's function) formalism and the cavity approach. We discuss the main diagnostics of localization and show how they are encoded in the statistical properties of Green's functions, with the full probability distribution of the local density of states emerging as the natural order parameter of the transition. We derive the cavity self-consistent equations for the diagonal resolvent and review both their numerical solution by population dynamics and their analytical treatment in the large-connectivity limit. This framework gives access to the critical disorder and the critical behavior near the transition, including the localization length in the insulating phase and the correlation volume that controls the delocalized phase. We also discuss the connection with directed polymers in random media and its interpretation in terms of rare resonant paths and freezing phenomena. These notes originated from a series of lectures delivered by one of the authors at the Oropa Summer School on "Fundamental Problems in Statistical Physics XVI" (July 2025).

Figures

Figures reproduced from arXiv: 2606.24671 by the authors.

Figure 1
Figure 1. Schematic representation of the configuration space of the spin chain (Eq. (21)) for L = 3 in the basis of the simultaneous eigenstates of the {σ z i } operators. Each vertex corresponds to one of the 2 3 = 8 spin configurations. To each vertex, a random energy is assigned (indicated by color) containing linear combinations of the random fields hi , weighted by the spin values of that specific configuration. Edges c… view at source ↗
Figure 2
Figure 2. (Left) For a finite system, G(z) is analytic except at a discrete set of points on the real axis corresponding to the eigenvalues λn, where it exhibits simple poles. (Right) In the thermodynamic limit (N → ∞), these discrete poles may merge into a continuous branch cut along the real axis. The physical Green’s functions are recovered by approaching the real axis from the complex plane through the regularization z = … view at source ↗
Figure 3
Figure 3. Energy spectrum of the system in the extended phase. Each horizontal line represents an eigenstate at energy λn, with its oscillatory pattern indicating the spatial structure. The red marker shows that the amplitude of the wavefunction is uniformly of order O(1/N). may remain pure point (localized phase) or develop an absolutely continuous component (delocalized phase) in the thermodynamic limit. However, the averag… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Integrated local density of states Ri(E) in the extended phase. Both the step height (proportional to |ψn(i)| 2 ) and the level spacing are of order O(1/N). In the thermodynamic limit, these discrete steps merge into a smooth, continuous function. In the thermodynamic …
Figure 5
Figure 5. Figure 5: Energy spectrum in the localized regime. Each eigenstate is confined to a region around its localization center. At site i, the local density of states receives significant contributions only from eigenstates whose centers lie within ∼ ξ of i; distant states contribute…
Figure 6
Figure 6. Figure 6: Integrated local density of states Ri(E) in the localized phase. Most steps are exponentially small (|ψn(i)| 2 ∼ e −L/ξ), corresponding to eigenstates localized far (at a distance L) from site i. Rare large jumps of order one occur when an eigenstate is centered near i…
Figure 7
Figure 7. Figure 7: Cayley tree of degree k = 3 up to the third generation. Locally, it is identical to the neighborhood of the RRG and of the BL of same degree. Each concentric circle shows a successive generation of nodes, where the distance ℓ from the central root increases by one at e…
Figure 8
Figure 8. Figure 8: Schematic representation of the cavity method factorization. Thus, the matrix elements of the resolvent can be expressed as correlation functions of the Gaussian fields ϕi with the action S[ϕ] = − i 2 X N i=1 (z − εi)ϕ 2 i + i X ⟨i,j⟩ t ϕiϕj , (73) where i denotes the …
Figure 9
Figure 9. Figure 9: Branching structure entering the computation of the off-diagonal resolvent G0r(z) on the Bethe lattice. The final section of the unique path between nodes 0 and r is shown horizontally. The final node, r, has k neighbors to be integrated out that are not part of the pa…
Figure 10
Figure 10. Figure 10: Probability distribution of Im G in the localized phase for different values of the regulator η. The distributions are obtained using population dynamics (see Sec.7) for a population size Ω = 225, and fixed energy E = 0.0 and disorder strength W = 30.0, on a Bethe lat…
Figure 11
Figure 11. Figure 11: Probability distribution of Im G in the delocalized phase for different values of the disorder strength W. The distributions are obtained using population dynamics (see Sec.7) for a population size Ω = 225, and fixed energy E = 0.0 and no regulator η = 0.0, on a Bethe…
Figure 12
Figure 12. Figure 12: Schematic sketch of the spatial profile of the amplitude of typical delocalized eigenstates close to localization threshold, resulting of the hybridization of N/Λc resonant peaks, each one occupying a volume Λc. approaching the transition [38, 42, 31], see Eq. (127) b…
Figure 13
Figure 13. Figure 13: Schematic phase diagram of the Anderson model for the Bethe Lattice for k = 2 in the W-E plane [132, 133, 134]. The inner curve marks the mobility edge separating localized and extended states. The straight lines indicate the edges of the support of the density of sta…
Figure 14
Figure 14. Figure 14: Illustration of the largest eigenvalue of the integral operator (Eq. (106)) as a function of β ∈ [0, 1]. The green curve corresponds to W < Wc (unstable localized phase). The red curve represents the critical point W = Wc, and the other curves correspond to W > Wc (st…
Figure 15
Figure 15. Figure 15: Sketch of the critical behavior near the Anderson transition on the Bethe lattice [27, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. On the localized side, W ≳ Wc, the length ξ diverges as ξ ∼ (W − Wc) −1 (Eq. (124)), the tail exponent approaches its…
Figure 16
Figure 16. Figure 16: Distribution P(ImG) of the imaginary part of the cavity Green’s function for several values W ∈ [13, 17.3]. The lines correspond to fits using Eq. (155) to extract the correlation volume (see text). This large-deviation approach was applied in Ref. [139] to compute th…
Figure 17
Figure 17. Figure 17: Left: Probability distribution Q(0, mˆ )/ ⟨ρ⟩ obtained by conditioning on m = 0. Middle: Exponent β of the power-law tails as a function of W. The dashed curve shows the prediction of Eq. (125) near Wc from Ref. [31]. Right: Generalized inverse participation ratios Ip…

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Works this paper leans on

159 extracted references · 79 canonical work pages

  1. [1]

    5 for the Anderson model on the Bethe lattice must be interpreted as self-consistent integral equations for the probability CONTENTS52 distribution of the cavity Green’s functions

    Population Dynamics algorithm(s) As discussed above, the cavity equations derived in Sec. 5 for the Anderson model on the Bethe lattice must be interpreted as self-consistent integral equations for the probability CONTENTS52 distribution of the cavity Green’s functions. These equations are given by Eqs. (87) and (88). The local density of states, the inve...

  2. [2]

    Connection with Derrida’s directed polymers in random media Before concluding this review, we would like to discuss the close relationship between Anderson localization on the Bethe lattice and a paradigmatic problem of classical disordered systems, namely the freezing glass transition of directed polymers in random media (DPRM), first introduced and stud...

  3. [3]

    Plugging the expression ofm ⋆ into the replicated free energy, one recovers the expression (167). It is easy to check that for the particular case in whichγ(ω)is a Gaussian distribution with zero mean and varianceσ2 ω one recovers the results (165) and (166)— withβ ⋆ = √ 2 lnk/σ ω—corresponding to the approximation in which one neglects the correlations b...

  4. [4]

    Fundamental Problems in Statistical Physics XVI

    Conclusions and perspectives 9.1. Summary More than six decades after Philip W. Anderson’s seminal discovery [1], the problem of quantum localization driven by disorder continues to reveal new frontiers, challenges, and subtleties. This pedagogical review has been dedicated to exploring this phenomenon within a unique and highly singular framework: the in...

    2025

  5. [5]

    Rev.109(5) 1492–1505 URLhttps://link.aps.org/doi/10.1103/ PhysRev.109.1492

    Anderson P W 1958Phys. Rev.109(5) 1492–1505 URLhttps://link.aps.org/doi/10.1103/ PhysRev.109.1492

  6. [6]

    Bloch F 1928Zeitschrift für Physik52555–600

  7. [7]

    Drude P 1900Annalen der Physik306566–613 (Preprinthttps://onlinelibrary.wiley.com/ doi/pdf/10.1002/andp.19003060312) URLhttps://onlinelibrary.wiley.com/doi/abs/ 10.1002/andp.19003060312

    work page doi:10.1002/andp.19003060312

  8. [8]

    Sommerfeld A 1927Naturwissenschaften15825–832

Show all 159 references
  1. [9]

    Lee P A and Ramakrishnan T V 1985Rev. Mod. Phys.57(2) 287–337 URLhttps://link.aps. org/doi/10.1103/RevModPhys.57.287

    doi

  2. [10]

    Evers F and Mirlin A D 2008Rev. Mod. Phys.80(4) 1355–1417 URLhttps://link.aps.org/ doi/10.1103/RevModPhys.80.1355 CONTENTS69

    doi

  3. [11]

    Lagendijk A, Tiggelen B v and Wiersma D S 2009Physics today6224–29 URLhttps: //physicstoday.aip.org/features/fifty-years-of-anderson-localization

  4. [12]

    Scardicchio A and Thiery T 2017 Perturbation theory approaches to anderson and many-body localization: some lecture notes (Preprint1710.01234) URLhttps://arxiv.org/abs/1710. 01234

    2017 arXiv

  5. [13]

    Pascazio S, Scardicchio A and Tarzia M 2023 Anderson localization on the bethe latticeSpin Glass Theory and Far Beyond: Replica Symmetry Breaking After 40 Years(World Scientific) pp 335–352

    2023

  6. [14]

    Mott N 1967Advances in Physics1649–144 (Preprinthttps://doi.org/10.1080/ 00018736700101265) URLhttps://doi.org/10.1080/00018736700101265

    doi

  7. [15]

    Abrahams E, Anderson P W, Licciardello D C and Ramakrishnan T V 1979Physical Review Letters42673

  8. [16]

    Slevin K and Ohtsuki T 1999Physical Review Letters82382–385 ISSN 1079-7114 URL http://dx.doi.org/10.1103/PhysRevLett.82.382

    doi

  9. [17]

    Slevin K and Ohtsuki T 2014New Journal of Physics16015012 ISSN 1367-2630 URLhttp: //dx.doi.org/10.1088/1367-2630/16/1/015012

    doi

  10. [18]

    Tarquini E, Biroli G and Tarzia M 2017Physical Review B95094204

  11. [19]

    Hikami S 1992Progress of Theoretical Physics Supplement107213–227

  12. [20]

    Wegner F 1979Zeitschrift für Physik B Condensed Matter35207–210

  13. [21]

    Schäfer L and Wegner F 1980Zeitschrift für Physik B Condensed Matter38113–126

  14. [22]

    John S 1984Phys. Rev. Lett.53(22) 2169–2172 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.53.2169

  15. [23]

    1080/13642818508240619) URLhttps://doi.org/10.1080/13642818508240619

    Anderson P W 1985Philosophical Magazine B52505–509 (Preprinthttps://doi.org/10. 1080/13642818508240619) URLhttps://doi.org/10.1080/13642818508240619

    doi

  16. [24]

    Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clément D, Sanchez-Palencia L, Bouyer P and Aspect A 2008Nature453891–894

  17. [25]

    Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M and Inguscio M 2008Nature453895–898

  18. [26]

    Kondov S S, McGehee W R, Zirbel J J and DeMarco B 2011Science33466–68 ISSN 1095-9203 URLhttp://dx.doi.org/10.1126/science.1209019

    doi

  19. [27]

    Jendrzejewski F, Bernard A, Müller K, Cheinet P, Josse V, Piraud M, Pezzé L, Sanchez- Palencia L, Aspect A and Bouyer P 2012Nature Physics8398–403 ISSN 1745-2481 URL http://dx.doi.org/10.1038/nphys2256

    doi

  20. [28]

    Semeghini G, Landini M, Castilho P, Roy S, Spagnolli G, Trenkwalder A, Fattori M, Inguscio M and Modugno G 2015Nature Physics11554–559 ISSN 1745-2481 URLhttp://dx.doi.org/ 10.1038/nphys3339

    doi

  21. [29]

    Chabé J, Lemarié G, Grémaud B, Delande D, Szriftgiser P and Garreau J C 2008Physical Review Letters101ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.101.255702

    doi

  22. [30]

    Hu H, Strybulevych A, Page J H, Skipetrov S E and van Tiggelen B A 2008Nature Physics4 945–948 ISSN 1745-2481 URLhttp://dx.doi.org/10.1038/nphys1101

    doi

  23. [31]

    Abou-Chacra R, Thouless D J and Anderson P W 1973Journal of Physics C: Solid State Physics 61734 URLhttps://doi.org/10.1088/0022-3719/6/10/009

    doi

  24. [33]

    Rizzo T and Tarzia M 2024Phys. Rev. B110(18) 184210 URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.110.184210

    doi

  25. [34]

    Tikhonov K S and Mirlin A D 2019Phys. Rev. B99(2) 024202 URLhttps://link.aps.org/ doi/10.1103/PhysRevB.99.024202

    doi

  26. [35]

    Tikhonov K S and Mirlin A D 2019Phys. Rev. B99(21) 214202 URLhttps://link.aps.org/ doi/10.1103/PhysRevB.99.214202

    doi

  27. [36]

    Efetov K 1985Sov. Phys. JETP61606–617 URLhttps://jetp.ras.ru/cgi-bin/dn/e_061_ CONTENTS70 03_0606.pdf

  28. [37]

    Efetov K 1987Sov. Phys. JETP65360–370 URLhttps://jetp.ras.ru/cgi-bin/dn/e_065_ 02_0360.pdf

  29. [38]

    Efetov K 1987Sov. Phys. JETP66634–642 URLhttps://jetp.ras.ru/cgi-bin/dn/e_066_ 03_0634.pdf

  30. [39]

    Zirnbauer M R 1986Phys. Rev. B34(9) 6394–6408 URLhttps://link.aps.org/doi/10.1103/ PhysRevB.34.6394

  31. [42]

    sciencedirect.com/science/article/pii/055032139190028V

    Mirlin A D and Fyodorov Y V 1991Nuclear Physics B366507–532 URLhttps://www. sciencedirect.com/science/article/pii/055032139190028V

  32. [43]

    Mirlin A and Fyodorov Y V 1991Journal of Physics A: Mathematical and General242273 URL https://doi.org/10.1088/0305-4470/24/10/016

    doi

  33. [44]

    Fyodorov Y V and Mirlin A D 1991Phys. Rev. Lett.67(15) 2049–2052 URLhttps://link.aps. org/doi/10.1103/PhysRevLett.67.2049

    doi

  34. [45]

    Fyodorov Y V, Mirlin A D and Sommers H J 1992J. Phys. I France21571–1605 ISSN 1286-4862 URLhttp://dx.doi.org/10.1051/jp1:1992229

    doi

  35. [46]

    Mirlin A D and Fyodorov Y V 1994Journal de Physique I4655–673

  36. [47]

    Mirlin A D and Fyodorov Y V 1994Phys. Rev. Lett.72(4) 526–529 URLhttps://link.aps. org/doi/10.1103/PhysRevLett.72.526

    doi

  37. [48]

    Gornyi I V, Mirlin A D and Polyakov D G 2005Phys. Rev. Lett.95(20) 206603 URLhttps: //link.aps.org/doi/10.1103/PhysRevLett.95.206603

    doi

  38. [49]

    Basko D, Aleiner I and Altshuler B 2006Ann. Phys. (N. Y.)3211126–1205 ISSN 0003-4916 URLhttp://dx.doi.org/10.1016/j.aop.2005.11.014

    2005 doi

  39. [50]

    Nandkishore R and Huse D A 2015Annu. Rev. Condens. Mat. Phys.615–38 ISSN 1947-5462 URLhttps://doi.org/10.1146/annurev-conmatphys-031214-014726

    1947 doi

  40. [51]

    Abanin D A, Altman E, Bloch I and Serbyn M 2019Rev. Mod. Phys.91(2) 021001 URL https://link.aps.org/doi/10.1103/RevModPhys.91.021001

    doi

  41. [52]

    Physique19498–525 ISSN 1878-1535 URL http://dx.doi.org/10.1016/j.crhy.2018.03.003

    Alet F and Laflorencie N 2018Comptes Rendus. Physique19498–525 ISSN 1878-1535 URL http://dx.doi.org/10.1016/j.crhy.2018.03.003

    2018 doi

  42. [53]

    Sierant P, Lewenstein M, Scardicchio A, Vidmar L and Zakrzewski J 2025Reports on Progress in Physics88026502 ISSN 1361-6633 URLhttp://dx.doi.org/10.1088/1361-6633/ad9756

    doi

  43. [54]

    Huse D A, Nandkishore R and Oganesyan V 2014Phys. Rev. B90ISSN 1550-235X URL http://dx.doi.org/10.1103/PhysRevB.90.174202

    doi

  44. [55]

    Altshuler B L, Gefen Y, Kamenev A and Levitov L S 1997Physical Review Letters782803–2806 ISSN 1079-7114 URLhttp://dx.doi.org/10.1103/PhysRevLett.78.2803

    doi

  45. [56]

    Tikhonov K and Mirlin A 2021Annals of Physics435168525 ISSN 0003-4916 URLhttp: //dx.doi.org/10.1016/j.aop.2021.168525

    2021 doi

  46. [57]

    Tikhonov K S and Mirlin A D 2021Phys. Rev. B103(6) 064204 URLhttps://link.aps.org/ doi/10.1103/PhysRevB.103.064204

    doi

  47. [58]

    Tarzia M 2020Physical Review B102ISSN 2469-9969 URLhttp://dx.doi.org/10.1103/ PhysRevB.102.014208

  48. [59]

    De Luca A and Scardicchio A 2013EPL (Europhysics Letters)10137003 ISSN 1286-4854 URL http://dx.doi.org/10.1209/0295-5075/101/37003

    doi

  49. [60]

    Biroli G and Tarzia M 2017Phys. Rev. B96(20) 201114(R) URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.96.201114

    doi

  50. [61]

    Biroli G and Tarzia M 2020Phys. Rev. B102(6) 064211 URLhttps://link.aps.org/doi/10. 1103/PhysRevB.102.064211

  51. [62]

    Logan D E and Welsh S 2019Physical Review B99ISSN 2469-9969 URLhttp://dx.doi.org/ CONTENTS71 10.1103/PhysRevB.99.045131

    doi

  52. [63]

    García-Mata I, Martin J, Giraud O, Georgeot B, Dubertrand R and Lemarié G 2022Phys. Rev. B106(21) 214202 URLhttps://link.aps.org/doi/10.1103/PhysRevB.106.214202

    doi

  53. [64]

    Herre J N, Karcher J F, Tikhonov K S and Mirlin A D 2023Physical Review B108ISSN 2469-9969 URLhttp://dx.doi.org/10.1103/PhysRevB.108.014203

    doi

  54. [65]

    org/abs/1211.7334

    Biroli G, Ribeiro-Teixeira A C and Tarzia M 2012 Difference between level statistics, ergodicity and localization transitions on the bethe lattice (Preprint1211.7334) URLhttps://arxiv. org/abs/1211.7334

    2012 arXiv

  55. [66]

    Kravtsov V, Altshuler B and Ioffe L 2018Annals of Physics389148–191 ISSN 0003-4916 URL http://dx.doi.org/10.1016/j.aop.2017.12.009

    2017 doi

  56. [67]

    Pino M 2020Phys. Rev. Res.2(4) 042031 URLhttps://link.aps.org/doi/10.1103/ PhysRevResearch.2.042031

  57. [68]

    Bera S, De Tomasi G, Khaymovich I M and Scardicchio A 2018Phys. Rev. B98(13) 134205 URLhttps://link.aps.org/doi/10.1103/PhysRevB.98.134205

    doi

  58. [69]

    De Tomasi G, Bera S, Scardicchio A and Khaymovich I M 2020Phys. Rev. B101(10) 100201 URLhttps://link.aps.org/doi/10.1103/PhysRevB.101.100201

    doi

  59. [70]

    Wormald N Cet al.1999London mathematical society lecture note series239–298

  60. [71]

    Tikhonov K S, Mirlin A D and Skvortsov M A 2016Phys. Rev. B94(22) 220203 URL https://link.aps.org/doi/10.1103/PhysRevB.94.220203

    doi

  61. [72]

    García-Mata I, Giraud O, Georgeot B, Martin J, Dubertrand R and Lemarié G 2017Phys. Rev. Lett.118(16) 166801 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.118.166801

    doi

  62. [73]

    Tarquini E, Biroli G and Tarzia M 2016Phys. Rev. Lett.116(1) 010601 URLhttps://link. aps.org/doi/10.1103/PhysRevLett.116.010601

    doi

  63. [74]

    Biroli G and Tarzia M 2018 Delocalization and ergodicity of the anderson model on bethe lattices (Preprint1810.07545) URLhttps://arxiv.org/abs/1810.07545

    2018 arXiv

  64. [75]

    Baroni M, Lorenzana G G, Rizzo T and Tarzia M 2024Phys. Rev. B109(17) 174216 URL https://link.aps.org/doi/10.1103/PhysRevB.109.174216

    doi

  65. [77]

    Efetov K 1983Advances in Physics3253–127 (Preprinthttps://doi.org/10.1080/ 00018738300101531) URLhttps://doi.org/10.1080/00018738300101531

    doi

  66. [78]

    Rep.326259–382 ISSN 0370-1573 URLhttps://www.sciencedirect

    Mirlin A D 2000Phys. Rep.326259–382 ISSN 0370-1573 URLhttps://www.sciencedirect. com/science/article/pii/S0370157399000915

  67. [79]

    Sierant P, Lewenstein M and Scardicchio A 2023SciPost Physics15045

  68. [80]

    García-Mata I, Martin J, Dubertrand R, Giraud O, Georgeot B and Lemarié G 2020Phys. Rev. Res.2(1) 012020(R) URLhttps://link.aps.org/doi/10.1103/PhysRevResearch.2. 012020

    doi

  69. [81]

    Vanoni C, Altshuler B L, Kravtsov V E and Scardicchio A 2024Proceedings of the National Academy of Sciences121e2401955121

  70. [82]

    Sonner M, Tikhonov K and Mirlin A 2017Physical Review B96214204

  71. [83]

    Tikhonov K S and Mirlin A D 2016Physical Review B94184203

  72. [84]

    Derrida B and Spohn H 1988Journal of Statistical Physics51817–840

  73. [85]

    Ashcroft N W and Mermin N D 1976Solid statevol 1 (Harcourt College Publishers)

  74. [86]

    Pietracaprina F, Ros V and Scardicchio A 2016Physical Review B93ISSN 2469-9969 URL http://dx.doi.org/10.1103/PhysRevB.93.054201

    doi

  75. [87]

    Mörters P, Moser R, Penrose M, Schwetlick H and Zimmer J (eds) 2008Analysis and Stochastics of Growth Processes and Interface Models(Oxford, UK: Oxford University Press) ISBN 978- 0-19-923925-2

  76. [88]

    Rodriguez A, Vasquez L J, Slevin K and Römer R A 2011Phys. Rev. B84(13) 134209 URL https://link.aps.org/doi/10.1103/PhysRevB.84.134209

    doi

  77. [89]

    Potters M and Bouchaud J P 2020A first course in random matrix theory: for physicists, CONTENTS72 engineers and data scientists(Cambridge University Press) chap. 4

  78. [90]

    Livan G, Novaes M and Vivo P 2018Introduction to Random Matrices(Springer International Publishing) ISBN 9783319708850 URLhttp://dx.doi.org/10.1007/978-3-319-70885-0

    doi

  79. [91]

    Wigner E P 1958Annals of Mathematics67325–327 ISSN 0003486X, 19398980 URLhttp: //www.jstor.org/stable/1970008

  80. [92]

    Mott N F and Twose W D 1961Advances in Physics10107–163

  81. [93]

    Series A

    Borland R E 1963Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences274529–545

  82. [94]

    Ishii K 1973Progress of Theoretical Physics Supplement5377–138

  83. [95]

    Aspect A and Inguscio M 2009Physics Today6230–35

  84. [96]

    1088/0022-3719/5/1/010

    Thouless D J 1972Journal of Physics C: Solid State Physics577 URLhttps://doi.org/10. 1088/0022-3719/5/1/010

  85. [97]

    Comtet A, Luck J M, Texier C and Tourigny Y 2013Journal of Statistical Physics15013–65

  86. [98]

    Comtet A, Texier C and Tourigny Y 2013Journal of Physics A: Mathematical and Theoretical 46254003

  87. [99]

    Kramer B and MacKinnon A 1993Reports on Progress in Physics561469–1564

  88. [100]

    Abrahams E and Ramakrishnan T V 1980Philosophical Magazine42827–838

  89. [101]

    Foster M S, Ryu S and Ludwig A W W 2009Phys. Rev. B80(7) 075101 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.80.075101

    doi

  90. [102]

    García-García A M and Cuevas E 2007Physical Review B75ISSN 1550-235X URLhttp: //dx.doi.org/10.1103/PhysRevB.75.174203

    doi

  91. [103]

    Castellani C, Castro C D and Peliti L 1986Journal of Physics A: Mathematical and General19 L1099–L1103 ISSN 1361-6447 URLhttp://dx.doi.org/10.1088/0305-4470/19/17/009

    doi

  92. [104]

    Minami N 1996Communications in Mathematical Physics177709–725

  93. [105]

    Elgart A 2009Duke Mathematical Journal146331–360

  94. [106]

    Klein A 1994Mathematical Research Letters1399–407 URLhttps://www.intlpress.com/ site/pub/files/_fulltext/journals/mrl/1994/0001/0004/MRL-1994-0001-0004-a001. pdf

    1994

  95. [107]

    Aizenman M, Sims R and Warzel S 2006Communications in Mathematical Physics264371–389 URLhttps://arxiv.org/abs/math-ph/0504039

    arXiv

  96. [108]

    Froese R, Hasler D and Spitzer W 2006Journal of Functional Analysis230184–221

  97. [109]

    Aggarwal A and Lopatto P 2025 Mobility edge for the anderson model on the bethe lattice (Preprint2503.08949) URLhttps://arxiv.org/abs/2503.08949

    2025

  98. [110]

    Liu S and Lopatto P 2026 Mobility edge for the anderson model on random regular graphs (Preprint2603.14230) URLhttps://arxiv.org/abs/2603.14230

    2026

  99. [111]

    Imbrie J Z 2016Phys. Rev. Lett.117(2) 027201 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.117.027201

  100. [112]

    Prado Bandeira D and Tarzia M 2026Physical Review B113ISSN 2469-9969 URLhttp: //dx.doi.org/10.1103/q3bz-97v6

    doi

  101. [113]

    Economou E 2006Green’s Functions in Quantum PhysicsSpringer Series in Solid-State Sciences (Springer Berlin Heidelberg) ISBN 9783540288411 URLhttps://books.google.fr/books? id=HdJDAAAAQBAJ

  102. [114]

    Craig W and Simon B 1983Communications in Mathematical Physics90207–218

  103. [115]

    Ising E 1925Zeitschrift für Physik31253–258

  104. [116]

    Erdős P and Rényi A 1959Publicationes Mathematicae Debrecen6290–297

  105. [117]

    Erdos P L and Rényi A 1984Transactions of the American Mathematical Society286257–257

  106. [118]

    akjournals.com/view/journals/10473/12/1-2/article-p261.xml

    Erdős P and Rényi A 1961Acta Mathematica Hungarica12261–267 URLhttps://www. akjournals.com/view/journals/10473/12/1-2/article-p261.xml

  107. [119]

    Bogacz L, Burda Z and Wacław B 2006Physica A: Statistical Mechanics and its Applications 366587–607 ISSN 0378-4371 URLhttp://dx.doi.org/10.1016/j.physa.2005.10.024

    2005 doi

  108. [120]

    Molloy M and Reed B 1995Random Structures & Algorithms6161–180 (Preprint CONTENTS73 https://onlinelibrary.wiley.com/doi/pdf/10.1002/rsa.3240060204) URLhttps:// onlinelibrary.wiley.com/doi/abs/10.1002/rsa.3240060204

    doi

  109. [121]

    Newman M 2018Networks(Oxford university press)

  110. [122]

    Dorogovtsev S N and Mendes J F 2022The nature of complex networks(Oxford University Press)

  111. [123]

    Marinari E and Monasson R 2004Journal of Statistical Mechanics: Theory and Experiment2004 P09004

  112. [124]

    Marinari E, Monasson R and Semerjian G 2006Europhysics Letters738–14

  113. [125]

    Bollobás B 2001Random graphs(Cambridge University Press)

  114. [126]

    Braunstein A and Semerjian G 2023 The cavity method: from exact solutions to algorithmsSpin Glass Theory and Far Beyond: Replica Symmetry Breaking After 40 Years(World Scientific) pp 375–387 URLhttps://arxiv.org/abs/2209.11499

    2023

  115. [127]

    Krzakala F 2011 Belief propagation for the (physicist) layman URLhttp://www.lps.ens.fr/ ~krzakala/BP.pdf

    2011

  116. [128]

    Mezard M and Montanari A 2009Information, physics, and computation(Oxford University Press) chap. 14, Sec. 14.6.4

  117. [129]

    Susca V A R, Vivo P and Kühn R 2021SciPost Physics Lecture Notes331–43

  118. [130]

    Bordenave C and Lelarge M 2010Random Structures & Algorithms37332–352

  119. [131]

    Arous G B and Guionnet A 2008Commun. Math. Phys.278715–751 URLhttps://doi.org/ 10.1007/s00220-007-0389-x

    doi

  120. [132]

    Schur I 1917Jahresbericht der Deutschen Mathematiker-Vereinigung25205–232

  121. [133]

    Parisi G, Pascazio S, Pietracaprina F, Ros V and Scardicchio A 2019Journal of Physics A: Mathematical and Theoretical53014003 ISSN 1751-8121 URLhttp://dx.doi.org/10.1088/ 1751-8121/ab56e8

  122. [134]

    Bapst V 2014Journal of Mathematical Physics55ISSN 1089-7658 URLhttp://dx.doi.org/ 10.1063/1.4894055

    doi

  123. [135]

    archives-ouvertes.fr/tel-01727220

    Tarquini E 2016Anderson Localization in High Dimensional LatticesPhd thesis Université Paris-Saclay (COmUE) nNT: 2016SACLS540 (Preprinttel-01727220) URLhttps://tel. archives-ouvertes.fr/tel-01727220

  124. [136]

    BiroliG,SemerjianGandTarziaM2010Progress of Theoretical Physics Supplement184187–199 ISSN 0375-9687 URLhttp://dx.doi.org/10.1143/PTPS.184.187

    doi

  125. [137]

    Tonetti L, Cugliandolo L F and Tarzia M 2025 Testing the localization landscape theory on the bethe lattice (Preprint2512.04037) URLhttps://arxiv.org/abs/2512.04037

    2025

  126. [138]

    Tonetti L, Cugliandolo L F and Tarzia M 2026 Geometry and localization: Probing localization landscape theory on the bethe lattice (Preprint2605.29745) URLhttps://arxiv.org/abs/ 2605.29745

    2026 arXiv

  127. [139]

    Warzel S 2012 Surprises in the phase diagram of the anderson model on the bethe latticeXVIIth International Congress on Mathematical Physics(Singapore: World Scientific) pp 239–253 URLhttps://www.worldscientific.com/doi/abs/10.1142/9789814449243_0014

    2012 doi

  128. [140]

    Aizenman M and Molchanov S 1993Communications in Mathematical Physics157245–278

  129. [141]

    Aizenman M 1994Reviews in mathematical physics61163–1182

  130. [142]

    Aizenman M and Warzel S 2011Physical review letters106136804

  131. [143]

    Biroli G, Hartmann A K and Tarzia M 2022Phys. Rev. B105(9) 094202 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.105.094202

    doi

  132. [144]

    Mézard M and Parisi G 2001The European Physical Journal B20217–233 ISSN 1434-6028 URL http://dx.doi.org/10.1007/PL00011099

    doi

  133. [145]

    Lemarié G 2019Physical review letters122030401

  134. [146]

    Monthus C and Garel T 2008Journal of Physics A: Mathematical and Theoretical42075002

  135. [147]

    Monthus C and Garel T 2011Journal of Physics A: Mathematical and Theoretical44145001

  136. [148]

    Chakrabarti A, Martins C, Laflorencie N, Georgeot B, Brunet É and Lemarié G 2022arXiv preprint arXiv:2212.13593

  137. [149]

    Swain N, Adam S and Lemarié G 2026Physical Review Research8013038 CONTENTS74

  138. [150]

    Biroli G, Hartmann A K and Tarzia M 2024Phys. Rev. B110ISSN 2469-9950, 2469-9969 URL https://link.aps.org/doi/10.1103/PhysRevB.110.014205

    doi

  139. [151]

    MirandaGA,AletF,BiroliG,CugliandoloLF,LaflorencieNandTarziaM2025Largedeviations in the many-body localization transition: The case of the random-field xxz chain (Preprint 2510.18545) URLhttps://arxiv.org/abs/2510.18545

  140. [152]

    Derrida B 1980Physical Review Letters4579

  141. [153]

    Mézard M, Parisi G and Virasoro M A 1987Spin glass theory and beyond: An Introduction to the Replica Method and Its Applicationsvol 9 (World Scientific Publishing Company)

  142. [154]

    Kolmogorov A, Petrovsky I and Piskounov B 1937Moscow Univ. Bull. Math.11

  143. [155]

    Roy S and Logan D E 2020Phys. Rev. B101(13) 134202 URLhttps://link.aps.org/doi/ 10.1103/PhysRevB.101.134202

    doi

  144. [156]

    Roy S and Logan D E 2020Phys. Rev. Lett.125(25) 250402 URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.125.250402

    doi

  145. [157]

    Scoquart T, Gornyi I V and Mirlin A D 2024Phys. Rev. B109(21) 214203 URLhttps: //link.aps.org/doi/10.1103/PhysRevB.109.214203

    doi

  146. [158]

    Logan D E and Roy S 2025Phys. Rev. B112(18) 184202 URLhttps://link.aps.org/doi/ 10.1103/5vqd-v4b8

    doi

  147. [159]

    Angelini M C, Palazzi S, Rizzo T and Tarzia M 2025SciPost Physics18030

  148. [160]

    Tarzia M 2022Phys. Rev. B105(17) 174201 URLhttps://link.aps.org/doi/10.1103/ PhysRevB.105.174201

  149. [161]

    Cugliandolo L F, Schehr G, Tarzia M and Venturelli D 2024Physical Review B110174202

  150. [162]

    Lett.14441001 ISSN 1286-4854 URLhttp://dx.doi

    Tapias D and Sollich P 2023Europhys. Lett.14441001 ISSN 1286-4854 URLhttp://dx.doi. org/10.1209/0295-5075/ad1001

    doi

  151. [163]

    da Silva J D, Tapias D, Sollich P and Metz F L 2025SciPost Phys.18047 URLhttps: //scipost.org/10.21468/SciPostPhys.18.2.047

    doi

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Reviewed June 25, 2026 · model on record in the stance chip above.