Skip to main content
SpringerLink
Log in

Existence of Absolutely Continuous Spectrum for Galton–Watson Random Trees

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We establish a quantitative criterion for an operator defined on a Galton–Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance below a threshold. As a by-product, we prove that the adjacency operator of a supercritical Poisson Galton–Watson tree has a non-trivial absolutely continuous part if the average degree is large enough. We also prove that its Karp and Sipser core has purely absolutely spectrum on an interval if the average degree is large enough. We finally illustrate our criterion on the Anderson model on a d-regular infinite tree with \(d\ge 3\) and give a quantitative version of Klein’s Theorem on the existence of absolutely continuous spectrum at disorder smaller that \(C \sqrt{d}\) for some absolute constant C.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abou-Chacra, R., Anderson, P., Thouless, D.: Absence of diffusion in certain random lattices. J. Phys. C 6, 1734–1752 (1973)

    ADS  Google Scholar 

  2. Abou-Chacra, R., Thouless, D.: Self-consistent theory of localization. II. Localization near the band edges. J. Phys. C Solid State Phys. 7, 65 (2001)

    Article  ADS  Google Scholar 

  3. Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157(2), 245–278 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Aizenman, M., Sims, R., Warzel, S.: Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs. Probab. Theory Relat. Fields 136(3), 363–394 (2006)

    Article  MATH  Google Scholar 

  5. Aizenman, M., Warzel, S.: Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. (JEMS) 15(4), 1167–1222 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Aizenman, M., Warzel, S.: Random Operators. Disorder Effects on Quantum Spectra and Dynamics, vol. 168. American Mathematical Society (AMS), Providence (2015)

    MATH  Google Scholar 

  7. Anantharaman, N., Sabri, M.: Quantum ergodicity on graphs: from spectral to spatial delocalization. Ann. Math. (2) 189(3), 753–835 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Anantharaman, N., Sabri, M.: Recent results of quantum ergodicity on graphs and further investigation. Ann. Fac. Sci. Toulouse Math. (6) 28(3), 559–592 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Anderson, P.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)

    Article  ADS  Google Scholar 

  10. Aronson, J., Frieze, A., Pittel, B.G.: Maximum matchings in sparse random graphs: Karp–Sipser revisited. Rand. Struct. Algorithms 12(2), 111–177 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York (1972). Die Grundlehren der mathematischen Wissenschaften, Band 196

  12. Bapst, V.: The large connectivity limit of the Anderson model on tree graphs. J. Math. Phys. 55(9), 092101 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Bauer, M., Golinelli, O.: Random incidence matrices: moments of the spectral density. J. Stat. Phys. 103, 08 (2000)

    MathSciNet  Google Scholar 

  14. Bauer, M., Golinelli, O.: Exactly solvable model with two conductor-insulator transitions driven by impurities. Phys. Rev. Lett. 86, 2621–2624 (2001)

    Article  ADS  Google Scholar 

  15. Bauerschmidt, R., Huang, J., Yau, H.-T.: Local Kesten–McKay law for random regular graphs. Commun. Math. Phys. 369(2), 523–636 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Bordenave, C.: On quantum percolation in finite regular graphs. Ann. Henri Poincaré 16(11), 2465–2497 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Bordenave, C.: Spectral measures of random graphs. In: Benaych-Georges, F., Chafai, D., Péché, S., de Tilière B. (eds) Advanced Topics in Random Matrices, Volume 53 of Panoramas et synthèses, p. xii+190. Société Mathématiques de France (2018)

  18. Bordenave, C., Lelarge, M., Salez, J.: The rank of diluted random graphs. Ann. Probab. 39(3), 1097–1121 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bordenave, C., Sen, A., Virág, B.: Mean quantum percolation. J. Eur. Math. Soc. (JEMS) 19(12), 3679–3707 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Brooks, S., Lindenstrauss, E.: Non-localization of eigenfunctions on large regular graphs. Isr. J. Math. 193, 1–14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coste, S., Salez, J.: Emergence of extended states at zero in the spectrum of sparse random graphs. Ann. Probab. (to appear)

  22. de Gennes, P., Lafore, P., Millot, J.: Sur un exemple de propagation dans un milieu désordonné. In: Simple Views on Condensed Matter, pp. 3–12

  23. de Gennes, P., Lafore, P., Millot, J.: Amas accidentels dans les solutions solides désordonnées. J. Phys. Chem. Solids 11(1), 105–110 (1959)

    Article  Google Scholar 

  24. Dumitriu, I., Pal, S.: Sparse regular random graphs: spectral density and eigenvectors. Ann. Probab. 40(5), 2197–2235 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Durrett, R.: Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 20. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  26. Evangelou, S.N., Economou, E.N.: Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble. Phys. Rev. Lett. 68, 361–364 (1992)

    Article  ADS  Google Scholar 

  27. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  28. Froese, R., Hasler, D., Spitzer, W.: Absolutely continuous spectrum for the Anderson model on a tree: a geometric proof of Klein’s theorem. Commun. Math. Phys. 269(1), 239–257 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Harris, A.B.: \(\frac{1}{\sigma }\) expansion for quantum percolation. Phys. Rev. B 29, 2519–2530 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  30. Karp, R.M., Sipser, M.: Maximum matching in sparse random graphs. In: 22nd Annual Symposium on Foundations of Computer Science (SFCS 1981), pp. 364–375 (1981)

  31. Keller, M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincaré 13(8), 1745–1766 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Keller, M., Lenz, D., Warzel, S.: Absolutely continuous spectrum for random operators on trees of finite cone type. Journal d’Analyse Mathématique 118(1), 363–396 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1), 163–184 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Müller, P., Stollmann, P.: Percolation Hamiltonians (2010)

  35. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich], New York (1978)

  36. Salez, J.: Every totally real algebraic integer is a tree eigenvalue. J. Combin. Theory Ser. B 111, 249–256 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Salez, J.: Spectral atoms of unimodular random trees. J. Eur. Math. Soc. (JEMS) 22(2), 345–363 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Simon, B.: \({L^p}\) norms of the Borel transform and the decomposition of measures. Proc. Am. Math. Soc. 123(12), 3749–3755 (1995)

    MATH  Google Scholar 

  39. Simon, B.: Trace Ideals and Their Applications, Volume 120 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (2005)

  40. van der Hofstad, R.: Random Graphs and Complex Networks, vols. 1, 43. Cambridge University Press, Cambridge (2017)

  41. Zdeborová, L., Mézard, M.: The number of matchings in random graphs. J. Stat. Mech. Theory Exp. 2006(05), P05003 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Nalini Anantharaman and Mostafa Sabri for their explanations on their work [7]. CB is supported by French ANR Grant ANR-16-CE40-0024-01.

Author information

Authors and Affiliations

  1. Aix-Marseille Université & CNRS, Institut de Mathématiques de Marseille, 3 Pl. Victor Hugo, 13331 MARSEILLE Cedex 03, Marseille, France

    Adam Arras & Charles Bordenave

Authors
  1. Adam Arras
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles Bordenave
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Charles Bordenave.

Additional information

Communicated by J. Ding.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arras, A., Bordenave, C. Existence of Absolutely Continuous Spectrum for Galton–Watson Random Trees. Commun. Math. Phys. 403, 495–527 (2023). https://doi.org/10.1007/s00220-023-04798-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-023-04798-3

Search

Navigation