Electronic Journal of Probability

GOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z} ^3$ with deformed Laplacian

Christian Sadel

Full-text: Open access

Abstract

Sequences of certain finite graphs - special types of antitrees - are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\mathrm{Sine} }_1$ process. The Anderson model on the graph is a random matrix being the sum of the adjacency matrix and a random diagonal matrix with independent identically distributed entries along the diagonal. The strength of the randomness stays fixed, there is no re-scaling with matrix size. These considered random matrices giving GOE statistics can also be viewed as random Schrödinger operators $\mathcal{P} \Delta +\mathcal{V} $ on thin finite boxes in $\mathbb{Z} ^3$ where the Laplacian $\Delta $ is deformed by a projection $\mathcal{P} $ commuting with $\Delta $.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 76, 24 pp.

Dates
Received: 24 November 2017
Accepted: 11 June 2018
First available in Project Euclid: 8 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1533715241

Digital Object Identifier
doi:10.1214/18-EJP187

Mathematical Reviews number (MathSciNet)
MR3858904

Zentralblatt MATH identifier
06924688

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60H25: Random operators and equations [See also 47B80] 15B52: Random matrices

Keywords
Anderson model universality local statistics

Rights
Creative Commons Attribution 4.0 International License.

Citation

Sadel, Christian. GOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z} ^3$ with deformed Laplacian. Electron. J. Probab. 23 (2018), paper no. 76, 24 pp. doi:10.1214/18-EJP187. https://projecteuclid.org/euclid.ejp/1533715241


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References

  • [ASW] M. Aizenman, R. Sims and S. Warzel, Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs, Prob. Theor. Rel. Fields (2006) 136:363–394
  • [AM] M. Aizenman and S. Molchanov, Localization at large disorder and extreme energies: an elementary derivation, Commun. Math. Phys. (1993) 157:245–278
  • [AW] M. Aizenman and S. Warzel, Resonant delocalization for random Schrödinger operators on tree graphs, J. Eur. Math. Soc. (2013) 15(4):1167–1222
  • [AEK] O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Th. Rel. Fields (2016) published online DOI: 10.1007/s00440-016-0740-2
  • [And] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. (1958) 109:1492–1505
  • [AGZ] G. W. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, (Cambridge Studies in Advanced Mathematics) Cambridge University Press, Cambridge, 2009
  • [DLS] F. Delyon, Y. Levy and B. Souillard, Anderson localization for multidimensional systems at large disorder or low energy, Commun. Math. Phys. (1985) 100:463-470 (1985)
  • [DG] P. Deift, D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, Courant Lect. Notes Math., vol. 18, American Mathematical Society, Providence, RI, 2009.
  • [EKS] L. Erdős, T. Krüger and D Schröder, Random Matrices with Slow Correlation Decay, preprint (2017) arXiv:1705.10661
  • [ESY] L. Erdös, B. Schlein and H.-T. Yau, Universality of random matrices and local relaxation flow., Inventiones Math. (2011) 185:75-119 (2011)
  • [ESYY] L. Erdös, B. Schlein, H.-T. Yau and J. Yin, The local relaxation flow approach to universality of the local statistics for random matrices, An. Inst. Henri Poincare Prob. Stat. (2012) 48:1–46
  • [For] P. J. Forrester, Log-gases and Random Matrices (London Mathematical Society Monographs), Princeton University Press
  • [FHH] R. Froese, F. Halasan and D. Hasler, Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph, J. Funct. Analysis (2012) 262:1011–1042
  • [FHS] R. Froese, D. Hasler and W. Spitzer, Absolutely continuous spectrum for the Anderson Model on a tree: A geometric proof of Klein’s Theorem, Commun. Math. Phys. (2007) 269:239–257
  • [FS] J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. (1983) 88:151–184
  • [GK] F. Germinet and F. Klopp, Spectral statistics for random Schrödinger operators in the localized regime, J. Eur. Math. Soc. (2014) 16:1967–2031
  • [GMP] Ya. Gol’dsheid, S. Molchanov and L. Pastur, Pure point spectrum of stochastic one dimensional Schrödinger operators, Funct. Anal. Appl. (1977) 11:1–10
  • [KLW] M. Keller, D. Lenz and S. Warzel, Absolutely continuous spectrum for random operators on trees of finite cone type, J. D’ Analyse Math. (2012) 118:363–396
  • [KLWo] M. Keller, D. Lenz and R. K. Wojciechowski, Volume Growth, Spectrum and Stochastic Completeness of Infinite Graphs Math. Z. (2013) 274:905–932
  • [Kle] A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. (1994) 1:399–407
  • [KLS] A. Klein, J. Lacroix and A. Speis, Localization for the Anderson model on a strip with singular potentials, J. Funct. Anal. (1990) 94:135–155
  • [KS] A. Klein and C. Sadel, Absolutely Continuous Spectrum for Random Schrödinger Operators on the Bethe Strip, Math. Nachr. (2012) 285:5–26
  • [Klo] F. Klopp, Weak disorder localization and Lifshitz tails, Commun. Math. Phys. (2002) 232:125–155
  • [KuS] H. Kunz and B. Souillard, Sur le spectre des operateurs aux differences finies aleatoires, Commun. Math. Phys. (1980) 78: 201–246
  • [Meh] M. Mehta, Random matrices, Elsevier/Academic Press, Amsterdam, 2004
  • [Min] N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Commun. Math. Phys. (1996) 177:709–725
  • [Sa1] C. Sadel, Absolutely continuous spectrum for random Schrödinger operators on tree-strips of finite cone type, Annales Henri Poincaré (2013) 14:737–773
  • [Sa2] C. Sadel, Absolutely continuous spectrum for random Schrödinger operators on the Fibbonacci and similar tree-strips, Math. Phys. Anal. Geom. (2014) 17:409–440
  • [Sa3] C. Sadel, Anderson transition at two-dimensional growth rate on antitrees and spectral theory for Operators with one propagating channel, Annales Henri Poincre (2016) 17:1631–1675
  • [Sa4] C. Sadel, Spectral Theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models, J. Funct. Anal. (2018) 274(8):2205-2244
  • [SV] C. Sadel and B. Virág, A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes, Commun. Math. Phys. (2016) 343:881–919
  • [TV] T. Tao and V. Vu, Random matrices: Universality of local eigenvalue statistics, Annals of Prob. (2012) 40:1285–1315
  • [VV] B. Valko and B. Virág, Random Schrödinger Operators on long boxes, noise explosion and the GOE, Trans. Amer. Math. Soc. (2014) 366:3709–3728
  • [Wan] W.-M Wang, Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder, Invent. Math. (2001) 146:365–398
  • [Wig] E. P. Wigner, Gatlinberg Conference on Neutron Physics, Oak Ridge National Laboratory Report, ORNL 2309:59