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Renormalization Group Analysis of the Hierarchical Anderson Model

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Abstract

We apply Feshbach–Krein–Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension \( d > 2 \), which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.

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References

  1. Aizenman, M., Shamis, M., Warzel, S.: Resonances and partial delocalization on the complete graph. Ann. Henri Poincaré 16(9), 1969–2003 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Aizenman, M., Warzel, S.: On the ubiquity of the Cauchy distribution in spectral problems. Probab. Theory Relat Fields 163(1–2), 61–87 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizenman, M., Warzel, S.: Random Operators: Disorder Effects on Quantum Spectra and Dynamics. Graduate Studies in Mathematics, vol. 168. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  4. Bauerschmidt, R., Brydges, D.C., Slade, G.: Introduction to a renormalisation group method for critical phenomena. Springer, New York (in preparation)

  5. Bleher, P.M., Sinai, J.G.: Investigation of the critical point in models of the type of Dyson’s hierarchical models. Commun. Math. Phys. 33(1), 23–42 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bleher, P.M., Sinai, Y.G.: Critical indices for Dyson’s asymptotically-hierarchical models. Commun. Math. Phys. 45(3), 247–278 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bovier, A.: The density of states in the Anderson model at weak disorder: a renormalization group analysis of the hierarchical model. J. Stat. Phys. 59(3–4), 745–779 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Combes, F., Germinet, J.-M., Klein, A.: Generalized eigenvalue-counting for the Anderson model. J. Stat. Phys. 135, 201–216 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer, New York (1988)

    MATH  Google Scholar 

  10. Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12(2), 91–107 (1969)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Dyson, F.J.: An Ising ferromagnet with discontinuous long-range order. Commun. Math. Phys. 21, 269–283 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  12. Evers, F., Mirlin, A.D.: Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008)

    Article  ADS  Google Scholar 

  13. Fyodorov, Y., Kupiainen, A., Webb, C.: Towards rigorous analysis of the Levitov-Mirlin-Evers recursion. Preprint arXiv:1509.01366 (2015)

  14. Fyodorov, Y.V., Ossipov, A., Rodriguez, A.: The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices. J. Stat. Mech. Theory Exp. 2009(12), L12001 (2009)

    Article  Google Scholar 

  15. Hislop, P.D., Müller, P.: A lower bound for the density of states of the lattice Anderson model. Proc. Am. Math. Soc. 136(8), 2887–2893 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York), 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  17. Kritchevski, E.: Hierarchical Anderson model. Probability and mathematical physics. In: CRM Proceedings and Lecture Notes, vol. 42, pp. 309–322. American Mathematical Society, Providence (2007)

  18. Kritchevski, E.: Spectral localization in the hierarchical Anderson model. Proc. Am. Math. Soc. 135(5), 1431–1440 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kritchevski, E.: Poisson statistics of eigenvalues in the hierarchical Anderson model. Ann. Henri Poincaré 9(4), 685–709 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Kuttruf, S., Müller, P.: Lifshits tails in the hierarchical Anderson model. Ann. Henri Poincaré 13(3), 525–541 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Metz, F.L., Leuzzi, L., Parisi, G.: Renormalization flow of the hierarchical Anderson model at weak disorder. Phys. Rev. B 89, 064201 (2014)

    Article  ADS  Google Scholar 

  22. Metz, F.L., Leuzzi, L., Parisi, G., Sacksteder, V.: Transition between localized and extended states in the hierarchical Anderson model. Phys. Rev. B 88, 045103 (2013)

    Article  ADS  Google Scholar 

  23. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Molchanov, S.: Hierarchical random matrices and operators. Application to Anderson model. Multidimensional statistical analysis and theory of random matrices (Bowling Green. OH, 1996), pp. 179–194. VSP, Utrecht (1996)

  25. Monthus, C., Garel, T.: A critical Dyson hierarchical model for the Anderson localization transition. J. Stat. Mech. P05005, 1–27 (2011)

    Google Scholar 

  26. Rushkin, I., Ossipov, A., Fyodorov, Y.V.: Universal and non-universal features of the multifractality exponents of critical wavefunctions. J. Stat. Mech. Theory Exp. 2011(03), L03001 (2011)

    Article  Google Scholar 

  27. Sadel, C.: Anderson transition at two-dimensional growth rate on antitrees and spectral theory for operators with one propagating channel. Ann. Henri Poincaré 17(7), 1631–1675 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Simon, B.: Trace ideals and their applications. Mathematical Surveys and Monographs, 2nd edn, vol. 120. American Mathematical Society, Providence (2005)

  29. Simon, B., Wolff, T.: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians. Commun. Pure Appl. Math. 39(1), 75–90 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747, Garching, Germany

    Per von Soosten & Simone Warzel

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  1. Per von Soosten
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  2. Simone Warzel
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Correspondence to Simone Warzel.

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Communicated by Anton Bovier.

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von Soosten, P., Warzel, S. Renormalization Group Analysis of the Hierarchical Anderson Model. Ann. Henri Poincaré 18, 1919–1947 (2017). https://doi.org/10.1007/s00023-016-0549-7

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