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Non-ergodic delocalization in the Rosenzweig–Porter model

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Abstract

We consider the Rosenzweig–Porter model \(H = V + \sqrt{T}\, \varPhi \), where V is a \(N \times N\) diagonal matrix, \(\varPhi \) is drawn from the \(N \times N\) Gaussian Orthogonal Ensemble, and \(N^{-1} \ll T \ll 1\). We prove that the eigenfunctions of H are typically supported in a set of approximately NT sites, thereby confirming the existence of a previously conjectured non-ergodic delocalized phase. Our proof is based on martingale estimates along the characteristic curves of the stochastic advection equation satisfied by the local resolvent of the Brownian motion representation of H.

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References

  1. Altshuler, B.L., Cuevas, E., Ioffe, L.B., Kravtsov, V.E.: Nonergodic phases in strongly disordered random regular graphs. Phys. Rev. Lett. 117, 156601 (2016)

    Article  ADS  Google Scholar 

  2. Amini, M.: Spread of wave packets in disordered hierarchical lattices. EPL (Eur. Lett.) 117(3), 30003 (2017)

    Article  ADS  Google Scholar 

  3. Benigni, L.: Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices. Preprint available at arXiv:1711.07103 (2017)

  4. Biane, P.: Processes with free increments. Math. Z. 227(1), 143–174 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biroli, G., Ribeiro-Teixera, A.C., Tarzia, M.: Difference between level statistics, ergodicity and localization transitions on the Bethe lattice. Preprint available at arXiv:1211.7334 (2012)

  6. Bourgade, P., Huang, J., Yau, H.-T.: Eigenvector statistics of sparse random matrices. Electron. J. Probab. 22, 38 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bourgade, P., Yau, H.-T.: The eigenvector moment flow and local quantum unique ergodicity. Commun. Math. Phys. 350(1), 231–278 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. De Luca, A., Altshuler, B.L., Kravtsov, V.E., Scardicchio, A.: Anderson localization on the Bethe lattice: nonergodicity of extended states. Phys. Rev. Lett. 113, 046806 (2014)

    Article  ADS  Google Scholar 

  9. Duits, M., Johansson, K.: On Mesoscopic Equilibrium for Linear Statistics in Dyson’s Brownian Motion, vol. 255, p. 1222. Memoirs of the American Mathematical Society, Providence (2018)

    MATH  Google Scholar 

  10. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Erdős, L., Knowles, A., Yau, H.-T.: Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincaré 14(8), 1837–1926 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Erdős, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287(2), 641–655 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Erdős, L., Yau, H.-T.: A Dynamical Approach to Random Matrix Theory, Volume 28 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New York (2017)

    Book  Google Scholar 

  14. Facoetti, D., Vivo, P., Biroli, G.: From non-ergodic eigenvectors to local resolvent statistics and back: a random matrix perspective. EPL (Eur. Lett.) 115(4), 47003 (2016)

    Article  ADS  Google Scholar 

  15. Huang, J., Landon, B.: Local law and mesoscopic fluctuations of Dyson Brownian motion for general \(\beta \) and potential. arXiv:1612.06306 (2017)

  16. Kravtsov, V.E., Khaymovich, I.M., Cuevas, E., Amini, M.: A random matrix model with localization and ergodic transitions. New J. Phys. 17(12), 122002 (2015)

    Article  ADS  Google Scholar 

  17. Landon, B., Sosoe, P., Yau, H.-T.: Fixed energy universality for Dyson Brownian motion. Preprint available at arXiv:1609.09011 (2016)

  18. Landon, B., Yau, H.-T.: Convergence of local statistics of Dyson Brownian motion. Commun. Math. Phys. 355(3), 949–1000 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Lee, J.O., Schnelli, K.: Local deformed semicircle law and complete delocalization for Wigner matrices with random potential. J. Math. Phys. 54(10), 103504 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Pastur, L.A.: The spectrum of random matrices. Teor. Mat. Fiz. 10(1), 102–112 (1972)

    Article  MathSciNet  Google Scholar 

  21. Rosenzweig, N., Porter, C.E.: “Repulsion of energy levels” in complex atomic spectra. Phys. Rev. 120, 1698–1714 (1960)

    Article  ADS  Google Scholar 

  22. Scheutzow, M.: A stochastic Gronwall lemma. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(2), 1350019 (2013). 4

    Article  MathSciNet  MATH  Google Scholar 

  23. Tikhonov, K.S., Mirlin, A.D., Skvortsov, M.A.: Anderson localization and ergodicity on random regular graphs. Phys. Rev. B 94, 220203 (2016)

    Article  ADS  Google Scholar 

  24. Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory. I. Commun. Math. Phys. 155(1), 71–92 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. von Renesse, M.-K., Scheutzow, M.: Existence and uniqueness of solutions of stochastic functional differential equations. Random Oper. Stoch. Equ. 18(3), 267–284 (2010)

    MathSciNet  MATH  Google Scholar 

  26. von Soosten, P., Warzel, S.: The phase transition in the ultrametric ensemble and local stability of Dyson Brownian motion. Electron. J. Probab. 23, 24 (2018)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Zentrum Mathematik, TU München, Boltzmannstraße 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.

Additional information

It is a pleasure to thank L. Benigni for interesting discussions about the Rosenzweig–Porter model during the PCMI Summer Session (funded by NSF Grant DMS:1441467). We also thank an anonymous referee for removing a superfluous assumption from an earlier version of this paper. This work was supported by the DFG (WA 1699/2-1).

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von Soosten, P., Warzel, S. Non-ergodic delocalization in the Rosenzweig–Porter model. Lett Math Phys 109, 905–922 (2019). https://doi.org/10.1007/s11005-018-1131-7

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  • DOI: https://doi.org/10.1007/s11005-018-1131-7

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