Quantum ergodicity on graphs: From spectral to spatial delocalization

Abstract

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrödinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrödinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrödinger operator. We show that an absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply “quantum ergodicity,” a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies, in particular, to graphs converging to the Anderson model on a regular tree, in the regime of extended states studied by Klein and Aizenman-Warzel.

  • [ATV] M. Abért, A. Thom, and B. Virág, Benjamini-Schramm convergence and pointwise convergence of the spectral measure.
    @MISC{ATV,
      author = {Abért, M. and Thom, A. and Virág, B.},
      title = {Benjamini-{S}chramm convergence and pointwise convergence of the spectral measure},
      note = {author homepage},
      zblnumber = {},
      }
  • [Alt1] Go to document A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, "Anderson localization on the Bethe lattice: Nonergodicity of extended states," Phys. Rev. Lett., vol. 113, p. 046806, 2014.
    @ARTICLE{Alt1,
      author = {De~Luca, A. and Altshuler, B. L. and Kravtsov, V. E. and Scardicchio, A.},
      title = {Anderson localization on the {B}ethe lattice: {N}onergodicity of extended states},
      journal = {Phys. Rev. Lett.},
      volume = {113},
      year = {2014},
      pages = {046806},
      zblnumber = {},
      doi = {10.1103/PhysRevLett.113.046806},
      }
  • [Alt2] A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, Support set of random wave-functions on the Bethe lattice, 2013.
    @MISC{Alt2,
      author = {De~Luca, A. and Altshuler, B. L. and Kravtsov, V. E. and Scardicchio, A.},
      title = {Support set of random wave-functions on the {B}ethe lattice},
      year = {2013},
      note = {preprint},
      arxiv = {1401.0019},
      }
  • [AW2] Go to document M. Aizenman and S. Warzel, "Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs," J. Math. Phys., vol. 53, iss. 9, p. 095205, 2012.
    @ARTICLE{AW2,
      author = {Aizenman, Michael and Warzel, Simone},
      title = {Absolutely continuous spectrum implies ballistic transport for quantum particles in a random potential on tree graphs},
      journal = {J. Math. Phys.},
      fjournal = {Journal of Mathematical Physics},
      volume = {53},
      year = {2012},
      number = {9},
      pages = {095205, 15},
      issn = {0022-2488},
      mrclass = {81Q35},
      mrnumber = {2905787},
      doi = {10.1063/1.4714617},
      url = {https://doi.org/10.1063/1.4714617},
      zblnumber = {1278.81090},
      }
  • [ASW] Go to document M. Aizenman, M. Shamis, and S. Warzel, "Resonances and partial delocalization on the complete graph," Ann. Henri Poincaré, vol. 16, iss. 9, pp. 1969-2003, 2015.
    @ARTICLE{ASW,
      author = {Aizenman, Michael and Shamis, Mira and Warzel, Simone},
      title = {Resonances and partial delocalization on the complete graph},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {16},
      year = {2015},
      number = {9},
      pages = {1969--2003},
      issn = {1424-0637},
      mrclass = {60H25 (30C45 35R60 47A15 47B80 82B26 82B44)},
      mrnumber = {3383321},
      mrreviewer = {Oleksiy Khorunzhiy},
      doi = {10.1007/s00023-014-0366-9},
      url = {https://doi.org/10.1007/s00023-014-0366-9},
      zblnumber = {1329.81192},
      }
  • [AL] Go to document D. Aldous and R. Lyons, "Processes on unimodular random networks," Electron. J. Probab., vol. 12, p. no. 54, 1454-1508, 2007.
    @ARTICLE{AL,
      author = {Aldous, David and Lyons, Russell},
      title = {Processes on unimodular random networks},
      journal = {Electron. J. Probab.},
      fjournal = {Electronic Journal of Probability},
      volume = {12},
      year = {2007},
      pages = {no. 54, 1454--1508},
      issn = {1083-6489},
      mrclass = {60C05 (05C80 60G50)},
      mrnumber = {2354165},
      mrreviewer = {Jean-François Delmas},
      doi = {10.1214/EJP.v12-463},
      url = {https://doi.org/10.1214/EJP.v12-463},
      zblnumber = {1131.60003},
      }
  • [A] Go to document N. Anantharaman, "Quantum ergodicity on regular graphs," Comm. Math. Phys., vol. 353, iss. 2, pp. 633-690, 2017.
    @ARTICLE{A,
      author = {Anantharaman, Nalini},
      title = {Quantum ergodicity on regular graphs},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {353},
      year = {2017},
      number = {2},
      pages = {633--690},
      issn = {0010-3616},
      mrclass = {58J51 (05C50 60G50)},
      mrnumber = {3649482},
      mrreviewer = {Dubi Kelmer},
      doi = {10.1007/s00220-017-2879-9},
      url = {https://doi.org/10.1007/s00220-017-2879-9},
      zblnumber = {1368.58015},
      }
  • [A17] N. Anantharaman, Some relations between the spectra of simple and non-backtracking random walks, 2017.
    @MISC{A17,
      author = {Anantharaman, Nalini},
      title = {Some relations between the spectra of simple and non-backtracking random walks},
      year = {2017},
      arxiv = {1703.03852},
      zblnumber = {},
      }
  • [ALM] Go to document N. Anantharaman and E. Le Masson, "Quantum ergodicity on large regular graphs," Duke Math. J., vol. 164, iss. 4, pp. 723-765, 2015.
    @ARTICLE{ALM,
      author = {Anantharaman, Nalini and Le Masson, Etienne},
      title = {Quantum ergodicity on large regular graphs},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {4},
      pages = {723--765},
      issn = {0012-7094},
      mrclass = {11B75 (05C50 58J51 81Q50)},
      mrnumber = {3322309},
      mrreviewer = {César R. de Oliveira},
      doi = {10.1215/00127094-2881592},
      url = {https://doi.org/10.1215/00127094-2881592},
      zblnumber = {1386.58015},
      }
  • [AS] Go to document N. Anantharaman and M. Sabri, "Poisson kernel expansions for Schrödinger operators on trees," J. Spectr. Theory, vol. 9, iss. 1, pp. 243-268, 2019.
    @ARTICLE{AS,
      author = {Anantharaman, Nalini and Sabri, Mostafa},
      title = {Poisson kernel expansions for {S}chrödinger operators on trees},
      journal = {J. Spectr. Theory},
      fjournal = {Journal of Spectral Theory},
      volume = {9},
      year = {2019},
      number = {1},
      pages = {243--268},
      issn = {1664-039X},
      mrclass = {35R02 (05C05 31C20 35J10 39A12 81Q10)},
      mrnumber = {3900786},
      doi = {10.4171/JST/247},
      url = {https://doi.org/10.4171/JST/247},
      zblnumber = {},
      }
  • [AS2] Go to document N. Anantharaman and M. Sabri, "Quantum ergodicity for the Anderson model on regular graphs," J. Math. Phys., vol. 58, iss. 9, p. 091901, 2017.
    @ARTICLE{AS2,
      author = {Anantharaman, Nalini and Sabri, Mostafa},
      title = {Quantum ergodicity for the {A}nderson model on regular graphs},
      journal = {J. Math. Phys.},
      fjournal = {Journal of Mathematical Physics},
      volume = {58},
      year = {2017},
      number = {9},
      pages = {091901, 10},
      issn = {0022-2488},
      mrclass = {81Q10 (05C63)},
      mrnumber = {3707062},
      mrreviewer = {César R. de Oliveira},
      doi = {10.1063/1.5000962},
      url = {https://doi.org/10.1063/1.5000962},
      zblnumber = {1376.82091},
      }
  • [BaSz16] A. Backhausz and B. Szegedy, On the almost eigenvectors of random regular graphs, 2016.
    @MISC{BaSz16,
      author = {Backhausz, A. and Szegedy, B.},
      title = {On the almost eigenvectors of random regular graphs},
      year = {2016},
      arxiv = {1607.04785},
      note={to appear in \emph{Ann. Prob.}},
      }
  • [BKY] Go to document R. Bauerschmidt, A. Knowles, and H. Yau, "Local semicircle law for random regular graphs," Comm. Pure Appl. Math., vol. 70, iss. 10, pp. 1898-1960, 2017.
    @ARTICLE{BKY,
      author = {Bauerschmidt, Roland and Knowles, Antti and Yau, Horng-Tzer},
      title = {Local semicircle law for random regular graphs},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {70},
      year = {2017},
      number = {10},
      pages = {1898--1960},
      issn = {0010-3640},
      mrclass = {05C80 (05C50 60B20)},
      mrnumber = {3688032},
      mrreviewer = {Lyuben R. Mutafchiev},
      doi = {10.1002/cpa.21709},
      url = {https://doi.org/10.1002/cpa.21709},
      zblnumber = {1372.05194},
      }
  • [BHKY] Go to document R. Bauerschmidt, J. Huang, A. Knowles, and H. Yau, "Bulk eigenvalue statistics for random regular graphs," Ann. Probab., vol. 45, iss. 6A, pp. 3626-3663, 2017.
    @ARTICLE{BHKY,
      author = {Bauerschmidt, Roland and Huang, Jiaoyang and Knowles, Antti and Yau, Horng-Tzer},
      title = {Bulk eigenvalue statistics for random regular graphs},
      journal = {Ann. Probab.},
      fjournal = {The Annals of Probability},
      volume = {45},
      year = {2017},
      number = {6A},
      pages = {3626--3663},
      issn = {0091-1798},
      mrclass = {05C80 (05C50 15B52 60B20)},
      mrnumber = {3729611},
      mrreviewer = {Steven Joel Miller},
      doi = {10.1214/16-AOP1145},
      url = {https://doi.org/10.1214/16-AOP1145},
      zblnumber = {1379.05098},
      }
  • [BHY] Go to document R. Bauerschmidt, J. Huang, and H. T. Yau, Local Kesten–McKay law for random regular graphs.
    @MISC{BHY,
      author = {Bauerschmidt, Roland and Huang, Jiaoyang and Yau, H.T.},
      title = {Local {K}esten--{M}c{K}ay law for random regular graphs},
      sortyear = {2019},
      note={\emph{Comm. Math. Phys.},
      published online 28 {F}ebruary 2019},
      doi = {10.1007/s00220-019-03345-3},
     }
  • [BS] Go to document I. Benjamini and O. Schramm, "Recurrence of distributional limits of finite planar graphs," Electron. J. Probab., vol. 6, p. 23, 2001.
    @ARTICLE{BS,
      author = {Benjamini, Itai and Schramm, Oded},
      title = {Recurrence of distributional limits of finite planar graphs},
      journal = {Electron. J. Probab.},
      fjournal = {Electronic Journal of Probability},
      volume = {6},
      year = {2001},
      pages = {no. 23, 13},
      issn = {1083-6489},
      mrclass = {82B41 (05C80 52C26 60G50)},
      mrnumber = {1873300},
      mrreviewer = {Olle Häggström},
      doi = {10.1214/EJP.v6-96},
      url = {https://doi.org/10.1214/EJP.v6-96},
      zblnumber = {1010.82021},
      }
  • [B] Go to document I. Benjamini, Coarse Geometry and Randomness, Springer, Cham, 2013, vol. 2100.
    @BOOK{B,
      author = {Benjamini, Itai},
      title = {Coarse Geometry and Randomness},
      series = {Lecture Notes in Math.},
      volume = {2100},
      note = {lecture notes from the 41st Probability Summer School held in Saint-Flour, 2011, Chapter 5 is due to Nicolas Curien, Chapter 12 was written by Ariel Yadin, and Chapter 13 is joint work with Gady Kozma, \'{E}cole d'\'{E}té de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]},
      publisher = {Springer, Cham},
      year = {2013},
      pages = {viii+129},
      isbn = {978-3-319-02575-9; 978-3-319-02576-6},
      mrclass = {05-06 (05C10 05C25 05C50 05C62 05C80)},
      mrnumber = {3156647},
      doi = {10.1007/978-3-319-02576-6},
      url = {https://doi.org/10.1007/978-3-319-02576-6},
      zblnumber = {1282.05001},
      }
  • [BT] Go to document M. Berry and M. Tabor, "Level clustering in the regular spectrum," Proc. Royal Soc. A, vol. 356, pp. 375-394, 1977.
    @ARTICLE{BT,
      author = {Berry, M. and Tabor, M.},
      title = {Level clustering in the regular spectrum},
      journal = {Proc. Royal Soc. A},
      volume = {356},
      year = {1977},
      pages = {375--394},
      zblnumber = {1119.81395},
      doi = {10.1098/rspa.1977.0140},
      }
  • [BGS1] Go to document O. Bohigas, M. -J. Giannoni, and C. Schmit, "Characterization of chaotic quantum spectra and universality of level fluctuation laws," Phys. Rev. Lett., vol. 52, iss. 1, pp. 1-4, 1984.
    @ARTICLE{BGS1,
      author = {Bohigas, O. and Giannoni, M.-J. and Schmit, C.},
      title = {Characterization of chaotic quantum spectra and universality of level fluctuation laws},
      journal = {Phys. Rev. Lett.},
      fjournal = {Physical Review Letters},
      volume = {52},
      year = {1984},
      number = {1},
      pages = {1--4},
      issn = {0031-9007},
      mrclass = {58F05 (58F13 81B05)},
      mrnumber = {0730191},
      mrreviewer = {Dieter H. Mayer},
      doi = {10.1103/PhysRevLett.52.1},
      url = {https://doi.org/10.1103/PhysRevLett.52.1},
      zblnumber = {1119.81326},
      }
  • [BGS2] Go to document O. Bohigas, M. Giannoni, and C. Schmit, "Spectral fluctuations, random matrix theories and chaotic motion," in Stochastic Processes in Classical and Quantum Systems, Springer, Berlin, 1986, vol. 262, pp. 118-138.
    @INCOLLECTION{BGS2,
      author = {Bohigas, Oriol and Giannoni, Marie-Joya and Schmit, Charles},
      title = {Spectral fluctuations, random matrix theories and chaotic motion},
      booktitle = {Stochastic Processes in Classical and Quantum Systems},
      venue = {{A}scona, 1985},
      series = {Lecture Notes in Phys.},
      volume = {262},
      pages = {118--138},
      publisher = {Springer, Berlin},
      year = {1986},
      mrclass = {81C10 (58F11 81C20)},
      mrnumber = {0870168},
      doi = {10.1007/3540171665_59},
      url = {https://doi.org/10.1007/3540171665_59},
      zblnumber = {},
      }
  • [Bor15] Go to document C. Bordenave, "On quantum percolation in finite regular graphs," Ann. Henri Poincaré, vol. 16, iss. 11, pp. 2465-2497, 2015.
    @ARTICLE{Bor15,
      author = {Bordenave, Charles},
      title = {On quantum percolation in finite regular graphs},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {16},
      year = {2015},
      number = {11},
      pages = {2465--2497},
      issn = {1424-0637},
      mrclass = {60K35 (05C80 60J80 81P45 82B43)},
      mrnumber = {3411739},
      mrreviewer = {Jakob E. Björnberg},
      doi = {10.1007/s00023-014-0382-9},
      url = {https://doi.org/10.1007/s00023-014-0382-9},
      zblnumber = {1332.82054},
      }
  • [BourgadeYau13] Go to document P. Bourgade and H. -T. Yau, "The eigenvector moment flow and local quantum unique ergodicity," Comm. Math. Phys., vol. 350, iss. 1, pp. 231-278, 2017.
    @ARTICLE{BourgadeYau13,
      author = {Bourgade, P. and Yau, H.-T.},
      title = {The eigenvector moment flow and local quantum unique ergodicity},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {350},
      year = {2017},
      number = {1},
      pages = {231--278},
      issn = {0010-3616},
      mrclass = {58J51 (60B20 60H30)},
      mrnumber = {3606475},
      mrreviewer = {Jiang Hu},
      doi = {10.1007/s00220-016-2627-6},
      url = {https://doi.org/10.1007/s00220-016-2627-6},
      zblnumber = {1379.58014},
      }
  • [BL] Go to document S. Brooks and E. Lindenstrauss, "Non-localization of eigenfunctions on large regular graphs," Israel J. Math., vol. 193, iss. 1, pp. 1-14, 2013.
    @ARTICLE{BL,
      author = {Brooks, Shimon and Lindenstrauss, Elon},
      title = {Non-localization of eigenfunctions on large regular graphs},
      journal = {Israel J. Math.},
      fjournal = {Israel Journal of Mathematics},
      volume = {193},
      year = {2013},
      number = {1},
      pages = {1--14},
      issn = {0021-2172},
      mrclass = {11B75 (05C50)},
      mrnumber = {3038543},
      mrreviewer = {Song Guo},
      doi = {10.1007/s11856-012-0096-y},
      url = {https://doi.org/10.1007/s11856-012-0096-y},
      zblnumber = {1317.05110},
      }
  • [BLML] Go to document S. Brooks, E. Le Masson, and E. Lindenstrauss, "Quantum ergodicity and averaging operators on the sphere," Int. Math. Res. Not. IMRN, iss. 19, pp. 6034-6064, 2016.
    @ARTICLE{BLML,
      author = {Brooks, Shimon and Le Masson, Etienne and Lindenstrauss, Elon},
      title = {Quantum ergodicity and averaging operators on the sphere},
      journal = {Int. Math. Res. Not. IMRN},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2016},
      number = {19},
      pages = {6034--6064},
      issn = {1073-7928},
      mrclass = {35P20 (28D05 37A45 58J51)},
      mrnumber = {3567266},
      mrreviewer = {Anton Deitmar},
      doi = {10.1093/imrn/rnv337},
      url = {https://doi.org/10.1093/imrn/rnv337},
      zblnumber = {1404.35307},
      }
  • [CdV85] Go to document Y. Colin de Verdière, "Ergodicité et fonctions propres du laplacien," Comm. Math. Phys., vol. 102, iss. 3, pp. 497-502, 1985.
    @ARTICLE{CdV85,
      author = {Colin de Verdière, Y.},
      title = {Ergodicité et fonctions propres du laplacien},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {102},
      year = {1985},
      number = {3},
      pages = {497--502},
      issn = {0010-3616},
      mrclass = {58G25 (35P20)},
      mrnumber = {0818831},
      mrreviewer = {Jürgen Eichhorn},
      doi = {10.1007/BF01209296},
      zblnumber = {0592.58050},
      }
  • [Diac91] Go to document P. Diaconis and D. Stroock, "Geometric bounds for eigenvalues of Markov chains," Ann. Appl. Probab., vol. 1, iss. 1, pp. 36-61, 1991.
    @ARTICLE{Diac91,
      author = {Diaconis, Persi and Stroock, Daniel},
      title = {Geometric bounds for eigenvalues of {M}arkov chains},
      journal = {Ann. Appl. Probab.},
      fjournal = {The Annals of Applied Probability},
      volume = {1},
      year = {1991},
      number = {1},
      pages = {36--61},
      issn = {1050-5164},
      mrclass = {60J10},
      mrnumber = {1097463},
      mrreviewer = {Gregory F. Lawler},
      doi = {10.1214/aoap/1177005980},
      zblnumber = {0731.60061},
      }
  • [Dumitriu] Go to document I. Dumitriu and S. Pal, "Sparse regular random graphs: spectral density and eigenvectors," Ann. Probab., vol. 40, iss. 5, pp. 2197-2235, 2012.
    @ARTICLE{Dumitriu,
      author = {Dumitriu, Ioana and Pal, Soumik},
      title = {Sparse regular random graphs: spectral density and eigenvectors},
      journal = {Ann. Probab.},
      fjournal = {The Annals of Probability},
      volume = {40},
      year = {2012},
      number = {5},
      pages = {2197--2235},
      issn = {0091-1798},
      mrclass = {60B20 (05C50 05C80 60C05)},
      mrnumber = {3025715},
      mrreviewer = {Steven Joel Miller},
      doi = {10.1214/11-AOP673},
      url = {https://doi.org/10.1214/11-AOP673},
      zblnumber = {1255.05173},
      }
  • [EKYY] Go to document L. ErdHos, A. Knowles, H. Yau, and J. Yin, "Spectral statistics of Erdős-Rényi graphs I: Local semicircle law," Ann. Probab., vol. 41, iss. 3B, pp. 2279-2375, 2013.
    @ARTICLE{EKYY,
      author = {Erd{{ő}}s, L\'{a}szló and Knowles, Antti and Yau, Horng-Tzer and Yin, Jun},
      title = {Spectral statistics of {E}rd{ő}s-{R}ényi graphs {I}: {L}ocal semicircle law},
      journal = {Ann. Probab.},
      fjournal = {The Annals of Probability},
      volume = {41},
      year = {2013},
      number = {3B},
      pages = {2279--2375},
      issn = {0091-1798},
      mrclass = {60B20 (05C80)},
      mrnumber = {3098073},
      mrreviewer = {Longmin Wang},
      doi = {10.1214/11-AOP734},
      url = {https://doi.org/10.1214/11-AOP734},
      zblnumber = {1272.05111},
      }
  • [ESY09] Go to document L. ErdHos, B. Schlein, and H. Yau, "Local semicircle law and complete delocalization for Wigner random matrices," Comm. Math. Phys., vol. 287, iss. 2, pp. 641-655, 2009.
    @ARTICLE{ESY09,
      author = {Erd{ő}s, L\'{a}szló and Schlein, Benjamin and Yau, Horng-Tzer},
      title = {Local semicircle law and complete delocalization for {W}igner random matrices},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {287},
      year = {2009},
      number = {2},
      pages = {641--655},
      issn = {0010-3616},
      mrclass = {60B20 (60E05 82B44)},
      mrnumber = {2481753},
      mrreviewer = {Olivier Marchal},
      doi = {10.1007/s00220-008-0636-9},
      url = {https://doi.org/10.1007/s00220-008-0636-9},
      zblnumber = {1186.60005},
      }
  • [ESY09-2] Go to document L. ErdHos, B. Schlein, and H. Yau, "Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices," Ann. Probab., vol. 37, iss. 3, pp. 815-852, 2009.
    @ARTICLE{ESY09-2,
      author = {Erd{ő}s, L\'{a}szló and Schlein, Benjamin and Yau, Horng-Tzer},
      title = {Semicircle law on short scales and delocalization of eigenvectors for {W}igner random matrices},
      journal = {Ann. Probab.},
      fjournal = {The Annals of Probability},
      volume = {37},
      year = {2009},
      number = {3},
      pages = {815--852},
      issn = {0091-1798},
      mrclass = {15B52 (15A18 15A42 47B80 82B44)},
      mrnumber = {2537522},
      mrreviewer = {Razvan Teodorescu},
      doi = {10.1214/08-AOP421},
      url = {https://doi.org/10.1214/08-AOP421},
      zblnumber = {1175.15028},
      }
  • [Geisinger] Go to document L. Geisinger, "Convergence of the density of states and delocalization of eigenvectors on random regular graphs," J. Spectr. Theory, vol. 5, iss. 4, pp. 783-827, 2015.
    @ARTICLE{Geisinger,
      author = {Geisinger, Leander},
      title = {Convergence of the density of states and delocalization of eigenvectors on random regular graphs},
      journal = {J. Spectr. Theory},
      fjournal = {Journal of Spectral Theory},
      volume = {5},
      year = {2015},
      number = {4},
      pages = {783--827},
      issn = {1664-039X},
      mrclass = {60B20 (05C50 05C80 35A08 35P20 35R60)},
      mrnumber = {3433288},
      doi = {10.4171/JST/114},
      url = {https://doi.org/10.4171/JST/114},
      zblnumber = {1384.60024},
      }
  • [Keating] Go to document J. P. Keating, "Quantum graphs and quantum chaos," in Analysis on Graphs and its Applications, Amer. Math. Soc., Providence, RI, 2008, vol. 77, pp. 279-290.
    @INCOLLECTION{Keating,
      author = {Keating, J. P.},
      title = {Quantum graphs and quantum chaos},
      booktitle = {Analysis on Graphs and its Applications},
      series = {Proc. Sympos. Pure Math.},
      volume = {77},
      pages = {279--290},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2008},
      mrclass = {81Q50 (81Q10 81U15)},
      mrnumber = {2459875},
      mrreviewer = {Steven Joel Miller},
      doi = {10.1090/pspum/077/2459875},
      url = {https://doi.org/10.1090/pspum/077/2459875},
      zblnumber = {1153.81501},
      }
  • [Klein] Go to document A. Klein, "Extended states in the Anderson model on the Bethe lattice," Adv. Math., vol. 133, iss. 1, pp. 163-184, 1998.
    @ARTICLE{Klein,
      author = {Klein, Abel},
      title = {Extended states in the {A}nderson model on the {B}ethe lattice},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {133},
      year = {1998},
      number = {1},
      pages = {163--184},
      issn = {0001-8708},
      mrclass = {82B44 (47B80 47N55 81Q10 82D30)},
      mrnumber = {1492789},
      mrreviewer = {Aernout C. D. van Enter},
      doi = {10.1006/aima.1997.1688},
      url = {https://doi.org/10.1006/aima.1997.1688},
      zblnumber = {0899.60088},
      }
  • [Klenke] Go to document A. Klenke, Probability Theory. A Comprehensive Course, Springer, London, 2014.
    @BOOK{Klenke,
      author = {Klenke, Achim},
      title = {Probability Theory. A Comprehensive Course},
      series = {Universitext},
      publisher = {Springer, London},
      year = {2014},
      pages = {xii+638},
      isbn = {978-1-4471-5360-3; 978-1-4471-5361-0},
      mrclass = {60-01 (60Fxx 60Gxx 60H10 60Jxx)},
      mrnumber = {3112259},
      doi = {10.1007/978-1-4471-5361-0},
      url = {https://doi.org/10.1007/978-1-4471-5361-0},
      zblnumber = {1295.60001},
      }
  • [Mark] Go to document J. Marklof, "Pair correlation densities of inhomogeneous quadratic forms," Ann. of Math. (2), vol. 158, iss. 2, pp. 419-471, 2003.
    @ARTICLE{Mark,
      author = {Marklof, Jens},
      title = {Pair correlation densities of inhomogeneous quadratic forms},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {158},
      year = {2003},
      number = {2},
      pages = {419--471},
      issn = {0003-486X},
      mrclass = {11F72 (11E45)},
      mrnumber = {2018926},
      mrreviewer = {Zeév Rudnick},
      doi = {10.4007/annals.2003.158.419},
      url = {https://doi.org/10.4007/annals.2003.158.419},
      zblnumber = {1106.11018},
      }
  • [Mey01] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
    @BOOK{Mey01,
      author = {Meyer, Carl D.},
      title = {Matrix Analysis and Applied Linear Algebra},
      publisher = {Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA},
      year = {2000},
      pages = {xii+718},
      isbn = {0-89871-454-0},
      mrclass = {15-01},
      mrnumber = {1777382},
      zblnumber = {0962.15001},
      }
  • [PP] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Math. Soc. France, 1990, vol. 187-188.
    @BOOK{PP,
      author = {Parry, William and Pollicott, Mark},
      title = {Zeta functions and the periodic orbit structure of hyperbolic dynamics},
      series = {Astérisque},
      volume = {187-188},
      year = {1990},
      pages = {268},
      issn = {0303-1179},
      mrclass = {58F20 (58F11 58F15)},
      publisher={Math. Soc. France},
      mrnumber = {1085356},
      mrreviewer = {Nicolaĭ T. A. Haydn},
      zblnumber = {0726.58003},
      }
  • [Salez] J. Salez, Some implications of local weak convergence for sparse random graphs.
    @MISC{Salez,
      author = {Salez, J.},
      title = {Some implications of local weak convergence for sparse random graphs},
      note = {Hal Id: tel-00637130},
      zblnumber = {},
      }
  • [SS] Go to document C. Schumacher and F. Schwarzenberger, "Approximation of the integrated density of states on sofic groups," Ann. Henri Poincaré, vol. 16, iss. 4, pp. 1067-1101, 2015.
    @ARTICLE{SS,
      author = {Schumacher, Christoph and Schwarzenberger, Fabian},
      title = {Approximation of the integrated density of states on sofic groups},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {16},
      year = {2015},
      number = {4},
      pages = {1067--1101},
      issn = {1424-0637},
      mrclass = {47A10 (20F65)},
      mrnumber = {3317792},
      mrreviewer = {Gilles Cassier},
      doi = {10.1007/s00023-014-0342-4},
      url = {https://doi.org/10.1007/s00023-014-0342-4},
      zblnumber = {1316.82015},
      }
  • [SarnakSchur] P. Sarnak, "Arithmetic quantum chaos," in The Schur Lectures (1992), Bar-Ilan Univ., Ramat Gan, 1995, vol. 8, pp. 183-236.
    @INCOLLECTION{SarnakSchur,
      author = {Sarnak, Peter},
      title = {Arithmetic quantum chaos},
      booktitle = {The {S}chur Lectures (1992)},
      venue = {{T}el {A}viv)},
      series = {Israel Math. Conf. Proc.},
      volume = {8},
      pages = {183--236},
      publisher = {Bar-Ilan Univ., Ramat Gan},
      year = {1995},
      mrclass = {11F72 (11M41 81Q50)},
      mrnumber = {1321639},
      mrreviewer = {Jens Bolte},
      zblnumber = {0831.58045},
      }
  • [SarnakPoisson] P. Sarnak, "Values at integers of binary quadratic forms," in Harmonic Analysis and Number Theory, Amer. Math. Soc., Providence, RI, 1997, vol. 21, pp. 181-203.
    @INCOLLECTION{SarnakPoisson,
      author = {Sarnak, Peter},
      title = {Values at integers of binary quadratic forms},
      booktitle = {Harmonic Analysis and Number Theory},
      venue = {{M}ontreal, {PQ},
      1996},
      series = {CMS Conf. Proc.},
      volume = {21},
      pages = {181--203},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1997},
      mrclass = {11E16},
      mrnumber = {1472786},
      mrreviewer = {Jens Marklof},
      zblnumber = {0911.11032},
      }
  • [Smi07] Go to document U. Smilansky, "Quantum chaos on discrete graphs," J. Phys. A, vol. 40, iss. 27, p. f621-f630, 2007.
    @ARTICLE{Smi07,
      author = {Smilansky, Uzy},
      title = {Quantum chaos on discrete graphs},
      journal = {J. Phys. A},
      fjournal = {Journal of Physics. A. Mathematical and Theoretical},
      volume = {40},
      year = {2007},
      number = {27},
      pages = {F621--F630},
      issn = {1751-8113},
      mrclass = {81Q50 (05C50 11M41 11Z05)},
      mrnumber = {2369953},
      mrreviewer = {Alexander Strohmaier},
      doi = {10.1088/1751-8113/40/27/F07},
      url = {https://doi.org/10.1088/1751-8113/40/27/F07},
      zblnumber = {1124.81024},
      }
  • [Smi10] Go to document U. Smilansky, "Discrete graphs—a paradigm model for quantum chaos," in Chaos, Birkhäuser/Springer, Basel, 2013, vol. 66, pp. 97-124.
    @incollection {Smi10,
      author = {Smilansky, Uzy},
      TITLE = {Discrete graphs---a paradigm model for quantum chaos},
      BOOKTITLE = {Chaos},
      venue={Poincaré seminar 2010. Proceedings, Paris, France, June 5, 2010},
      SERIES = {Prog. Math. Phys.},
      VOLUME = {66},
      PAGES = {97--124},
      PUBLISHER = {Birkhäuser/Springer, Basel},
      YEAR = {2013},
      MRCLASS = {37D45 (81Q50)},
      MRNUMBER = {3204183},
      MRREVIEWER = {Brian L. Burrows},
      DOI = {10.1007/978-3-0348-0697-8_3},
      URL = {https://doi.org/10.1007/978-3-0348-0697-8_3},
      ZBLNUMBER = {1319.81047},
      }
  • [Sni] Go to document A. I. vSnirelcprimeman, "Ergodic properties of eigenfunctions," Uspehi Mat. Nauk, vol. 29, iss. 6(180), pp. 181-182, 1974.
    @ARTICLE{Sni,
      author = {Šnirel{\cprime}man, A. I.},
      title = {Ergodic properties of eigenfunctions},
      journal = {Uspehi Mat. Nauk},
      fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      volume = {29},
      year = {1974},
      number = {6(180)},
      pages = {181--182},
      issn = {0042-1316},
      mrclass = {58G99 (35P20)},
      mrnumber = {0402834},
      mrreviewer = {D. Newton},
      zblnumber = {0324.58020},
      url = {http://mi.mathnet.ru/eng/umn/v29/i6/p181},
     }
  • [TMS] Go to document K. S. Tikhonov, A. D. Mirlin, and M. A. Skvortsov, "Anderson localization and ergodicity on random regular graphs," Phys. Rev. B, vol. 94, iss. 22, p. 220203, 2016.
    @ARTICLE{TMS,
      author = {Tikhonov, K. S. and Mirlin, A. D. and Skvortsov, M. A.},
      title = {Anderson localization and ergodicity on random regular graphs},
      journal = {Phys. Rev. B},
      volume = {94},
      number = {22},
      year = {2016},
      pages = {220203},
      zblnumber = {},
      doi = {10.1103/PhysRevB.94.220203},
      url = {https://link.aps.org/doi/10.1103/PhysRevB.94.220203},
      }
  • [TranVu] Go to document L. V. Tran, V. H. Vu, and K. Wang, "Sparse random graphs: eigenvalues and eigenvectors," Random Structures Algorithms, vol. 42, iss. 1, pp. 110-134, 2013.
    @ARTICLE{TranVu,
      author = {Tran, Linh V. and Vu, Van H. and Wang, Ke},
      title = {Sparse random graphs: eigenvalues and eigenvectors},
      journal = {Random Structures Algorithms},
      fjournal = {Random Structures \& Algorithms},
      volume = {42},
      year = {2013},
      number = {1},
      pages = {110--134},
      issn = {1042-9832},
      mrclass = {05C80 (05C50 60B20)},
      mrnumber = {2999215},
      mrreviewer = {Jihyeok Choi},
      doi = {10.1002/rsa.20406},
      url = {https://doi.org/10.1002/rsa.20406},
      zblnumber = {1257.05089},
      }
  • [Zel87] Go to document S. Zelditch, "Uniform distribution of eigenfunctions on compact hyperbolic surfaces," Duke Math. J., vol. 55, iss. 4, pp. 919-941, 1987.
    @ARTICLE{Zel87,
      author = {Zelditch, Steven},
      title = {Uniform distribution of eigenfunctions on compact hyperbolic surfaces},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {55},
      year = {1987},
      number = {4},
      pages = {919--941},
      issn = {0012-7094},
      mrclass = {58G25 (58G30)},
      mrnumber = {0916129},
      mrreviewer = {Alejandro Uribe},
      doi = {10.1215/S0012-7094-87-05546-3},
      url = {https://doi.org/10.1215/S0012-7094-87-05546-3},
      zblnumber = {0643.58029},
      }
  • [ZelC] Go to document S. Zelditch, "Quantum ergodicity of $C^*$ dynamical systems," Comm. Math. Phys., vol. 177, iss. 2, pp. 507-528, 1996.
    @ARTICLE{ZelC,
      author = {Zelditch, Steven},
      title = {Quantum ergodicity of {$C^*$} dynamical systems},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {177},
      year = {1996},
      number = {2},
      pages = {507--528},
      issn = {0010-3616},
      mrclass = {46L55 (46L50 82C10)},
      mrnumber = {1384146},
      doi = {10.1007/BF02101904},
      zblnumber = {0856.58019},
      }

Authors

Nalini Anantharaman

Université de Strasbourg, CNRS, IRMA UMR 7501, Strasbourg, France

Mostafa Sabri

Department of Mathematics, Faculty of Science, Cairo University, Cairo, Egypt and Université de Strasbourg, CNRS, IRMA UMR 7501, Strasbourg, France