×

Diluted banded random matrices: scaling behavior of eigenfunction and spectral properties. (English) Zbl 1381.15030

Summary: We demonstrate that the normalized localization length \(\beta\) of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law \(\beta=x^*/(1+x^*)\) . The scaling parameter of the model is defined as \(x^*\propto(b_{\mathrm{eff}}^2/N)^\delta\), where \(b_{\mathrm{eff}}\) is the average number of non-zero elements per matrix row, \(N\) is the matrix size, and \(\delta\sim 1\) . Additionally, we show that \(x^*\) also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Metha M L 2004 Random Matrices (Amsterdam: Elsevier)
[2] Akemann G, Baik J and Di Francesco P (ed) 2011 The Oxford Handbook of Random Matrix Theory (New York: Oxford University Press)
[3] Kravtsov V E, Khaymovich I M, Cuevas E and Amini M 2015 A random matrix model with localization and ergodic transitions New J. Phys.17 122002 · doi:10.1088/1367-2630/17/12/122002
[4] Wigner E P 1955 Characteristic vectors of bordered matrices with infinite dimensions Ann. Math.62 548 · Zbl 0067.08403 · doi:10.2307/1970079
[5] Wigner E P 1957 Ann. Math.65 203 · doi:10.2307/1969956
[6] Wigner E P 1967 SIAM Rev.9 1 · doi:10.1137/1009001
[7] Wilkinson M, Feingold M and Leitner D M 1991 Localization and spectral statistics in a banded random matrix ensemble J. Phys. A: Math. Gen.24 175 · doi:10.1088/0305-4470/24/1/025
[8] Wilkinson M, Feingold M and Leitner D M 1991 Spectral statistics in semiclasical random-matrix ensembles Phys. Rev. Lett.66 986 · Zbl 0968.82515 · doi:10.1103/PhysRevLett.66.986
[9] Feingold M, Gioletta A, Izrailev F M and Molinari L 1993 Two parameter scaling in the Wigner ensemble Phys. Rev. Lett.70 2936 · doi:10.1103/PhysRevLett.70.2936
[10] Casati G, Chirikov B V, Guarneri I and Izrailev F M 1993 Band-random-matrix model for quantum localization in conservative systems Phys. Rev. E 48 R1613 · doi:10.1103/PhysRevE.48.R1613
[11] Casati G, Chirikov B V, Guarneri I and Izrailev F M 1996 Quantum ergodicity and localization in conservative systems: the Wigner band random matrix model Phys. Lett. A 223 430 · Zbl 1037.82525 · doi:10.1016/S0375-9601(96)00784-0
[12] Feingold M 1997 Localization in strongly chaotic systems J. Phys. A: Math. Gen.30 3603 · Zbl 0926.37011 · doi:10.1088/0305-4470/30/10/032
[13] Wang W 2000 Perturbative and nonperturbative parts of eigenstates and local spectral density of states: the Wigner-band random-matrix model Phys. Rev. E 61 952 · doi:10.1103/PhysRevE.61.952
[14] Wang W 2001 Approach to energy eigenvalues and eigenfunctions from nonperturbative regions of eigenfunctions Phys. Rev. E 63 036215 · doi:10.1103/PhysRevE.63.036215
[15] Mirlin A D, Fyodorov Y V, Dittes F-M, Quezada J and Seligman T H 1996 Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices Phys. Rev. E 54 3221 · doi:10.1103/PhysRevE.54.3221
[16] Evers F and Mirlin A D 2008 Anderson transitions Rev. Mod. Phys.80 1355 · doi:10.1103/RevModPhys.80.1355
[17] Casati G, Molinari L and Izrailev F M 1990 Scaling properties of band random matrices Phys. Rev. Lett.64 1851 · Zbl 1050.82500 · doi:10.1103/PhysRevLett.64.1851
[18] Evangelou S N and Economou E N 1990 Eigenvector statistics and multifractal scaling of band random matrices Phys. Lett. A 151 345 · doi:10.1016/0375-9601(90)90295-Y
[19] Fyodorov Y F and Mirlin A D 1991 Scaling properties of localization in random band matrices: a σ-model approach Phys. Rev. Lett.67 2405 · Zbl 0990.82529 · doi:10.1103/PhysRevLett.67.2405
[20] Bogachev L V, Molchanov S A and Pastur L A 1991 On the level density of random band matrices Math. Notes50 1232 · Zbl 0778.60008 · doi:10.1007/BF01158263
[21] Molchanov S A, Pastur L A and Khorunzhii A M 1992 Limiting eigenvalue distribution for band random matrices Theor. Math. Phys.90 108 · doi:10.1007/BF01028434
[22] Izrailev F M 1995 Scaling properties of spectra and eigenfunctions for quantum dynamical and disordered systems Chaos Solitons Fractals5 1219 · Zbl 0900.81043 · doi:10.1016/0960-0779(94)E0063-U
[23] Fyodorov Y F and Mirlin A D 1992 Analytical derivation of the scaling law for the inverse participation ratio in quasi-one-dimensional disordered systems Phys. Rev. Lett.69 1093 · doi:10.1103/PhysRevLett.69.1093
[24] Mirlin A D and Fyodorov Y F 1993 The statistics of eigenvector components of random band matrices: analytical results J. Phys. A: Math. Gen.26 L551 · doi:10.1088/0305-4470/26/12/012
[25] Fyodorov Y F and Mirlin A D 1993 Level-to-level fluctuations of the inverse participation ratio in finite quasi 1D disordered systems Phys. Rev. Lett.71 412 · doi:10.1103/PhysRevLett.71.412
[26] Fyodorov Y F and Mirlin A D 1994 Statistical properties of eigenfunctions of random quasi 1D one-particle Hamiltonians Int. J. Mod. Phys. B 8 3795 · doi:10.1142/S0217979294001640
[27] Casati G, Izrailev F M and Molinari L 1991 Scaling properties of the eigenvalue spacing distribution for band random matrices J. Phys. A: Math. Gen.24 4755 · doi:10.1088/0305-4470/24/20/011
[28] Kottos T, Politi A, Izrailev F M and Ruffo S 1996 Scaling properties of Lyapunov spectra for the band random matrix model Phys. Rev. E 53 R5553 · doi:10.1103/PhysRevE.53.R5553
[29] Casati G, Guarneri I and Maspero G 1997 Landauer and Thouless conductance: a band random matrix approach J. Phys. I 7 729
[30] Kottos T, Politi A and Izrailev F M 1998 Finite-size corrections to Lyapunov spectra for band random matrices J. Phys.: Condens. Matter10 5965 · doi:10.1088/0953-8984/10/26/021
[31] Kottos T, Izrailev F M and Politi A 1999 Finite-length Lyapunov exponents and conductance for quasi-1D disordered solids Physica D 131 155 · Zbl 1076.82517 · doi:10.1016/S0167-2789(98)00226-7
[32] Shukla P 2001 Eigenvalue correlations for banded matrices Physica E 9 548 · doi:10.1016/S1386-9477(00)00261-7
[33] Wang W 2002 Localization in band random matrix models with and without increasing diagonal elements Phys. Rev. E 65 066207 · doi:10.1103/PhysRevE.65.066207
[34] Mon K K and French J B 1975 Statistical properties of many-particle spectra Ann. Phys.95 90 · doi:10.1016/0003-4916(75)90045-7
[35] Benet L and Weidenmüller H A 2003 Review of the k-body embedded ensembles of Gaussian random matrices J. Phys. A: Math. Gen.36 3569 · Zbl 1041.82005 · doi:10.1088/0305-4470/36/12/340
[36] Kota V K B 2014 Embedded Random Matrix Ensembles in Quantum Physics(Lecture Notes in Physics vol 884) (London: Springer) · doi:10.1007/978-3-319-04567-2
[37] Cohen D and Kottos T 2001 Parametric dependent Hamiltonians, wave functions, random matrix theory, and quantal-classical correspondence Phys. Rev. E 63 036203 · doi:10.1103/PhysRevE.63.036203
[38] Cohen D and Heller E J 2000 Unification of perturbation theory, random matrix theory, and semiclassical considerations in the study of parametrically dependent eigenstates Phys. Rev. Lett.84 2841 · doi:10.1103/PhysRevLett.84.2841
[39] Feingold M and Peres A 1986 Distribution of matrix elements of chaotic systems Phys. Rev. A 34 591 · doi:10.1103/PhysRevA.34.591
[40] Shepelyansky D L 1994 Coherent propagation of two interacting particles in a random potential Phys. Rev. Lett.73 2607 · doi:10.1103/PhysRevLett.73.2607
[41] Fyodorov Y V and Mirlin A D 1995 Statistical properties of random banded matrices with strongly fluctuating diagonal elements Phys. Rev. B 52 R11580 · doi:10.1103/PhysRevB.52.R11580
[42] Fyodorov Y V and Mirlin A D 1995 Analytical results for random band matrices with preferential basis Europhys. Lett.32 385 · doi:10.1209/0295-5075/32/5/001
[43] Silvestrov P G 1997 Summing graphs for random band matrices Phys. Rev. E 55 6419 · doi:10.1103/PhysRevE.55.6419
[44] Disertori M, Pinson H and Spencer T 2002 Density of states for random band matrices Commun. Math. Phys.232 83 · Zbl 1019.15014 · doi:10.1007/s00220-002-0733-0
[45] Khorunzhy A and Kirsch W 2002 On asymptotic expansions and scales of spectral universality in band random matrix ensembles Commun. Math. Phys.231 223 · Zbl 1034.82022 · doi:10.1007/s00220-002-0711-6
[46] Schenker J 2009 Eigenvector localization for random band matrices with power law band width Commun. Math. Phys.290 1065 · Zbl 1179.82079 · doi:10.1007/s00220-009-0798-0
[47] Sodin S 2010 The spectral edge of some random band matrices Ann. Math.172 2223 · Zbl 1210.15039 · doi:10.4007/annals.2010.172.2223
[48] Prosen T and Robnik M 1993 Energy level statistics and localization in sparsed banded random matrix ensemble J. Phys. A: Math. Gen.26 1105 · Zbl 0772.60100 · doi:10.1088/0305-4470/26/5/029
[49] Fyodorov Y V, Chubykalo O A, Izrailev F M and Casati G 1996 Wigner random banded matrices with sparse structure: local spectral density of states Phys. Rev. Lett.76 1603 · doi:10.1103/PhysRevLett.76.1603
[50] Cao X, Rosso A, Bouchaud J-P and LeDoussal P 2017 Genuine localisation transition in a long-range hopping model Phys. Rev. E 95 062118 · doi:10.1103/physreve.95.062118
[51] Mendez-Bermudez J A, Rodrigues F A and Vega-Oliveros D A Multifractality in random networks with long-range spatial correlations (unpublished)
[52] Mendez-Bermudez J A, Ferraz-de-Arruda G, Rodrigues F A and Moreno Y 2017 Scaling properties of multilayer random networks Phys. Rev. E 96 012307 · doi:10.1103/PhysRevE.96.012307
[53] Rodgers G J and Bray A J 1988 Density of states of a sparse random matrix Phys. Rev. B 37 3557 · doi:10.1103/PhysRevB.37.3557
[54] Fyodorov Y V and Mirlin A D 1991 On the density of states of sparse random matrices J. Phys. A: Math. Gen.24 2219 · doi:10.1088/0305-4470/24/9/027
[55] Fyodorov Y F and Mirlin A D 1991 Localization in ensemble of sparse random matrices Phys. Rev. Lett.67 2049 · doi:10.1103/PhysRevLett.67.2049
[56] Mirlin A D and Fyodorov Y V 1991 Universality of level correlation function of sparse random matrices J. Phys. A: Math. Gen.24 2273 · Zbl 0760.15017 · doi:10.1088/0305-4470/24/10/016
[57] Evangelou S N and Economou E N 1992 Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble Phys. Rev. Lett.68 361 · doi:10.1103/PhysRevLett.68.361
[58] Evangelou S N 1992 A numerical study of sparse random matrices J. Stat. Phys.69 361 · Zbl 0888.65046 · doi:10.1007/BF01053797
[59] Jackson A D, Mejia-Monasterio C, Rupp T, Saltzer M and Wilke T 2001 Spectral ergodicity and normal modes in ensembles of sparse matrices Nucl. Phys. A 687 405 · Zbl 0982.15029 · doi:10.1016/S0375-9474(00)00576-5
[60] Khorunzhy O, Shcherbina M and Vengerovsky V 2004 Eigenvalue distribution of large weighted random graphs J. Math. Phys.45 1648 · Zbl 1068.05062 · doi:10.1063/1.1667610
[61] Kühn R 2008 Spectra of sparse random matrices J. Phys. A: Math. Theor.41 295002 · Zbl 1188.15037 · doi:10.1088/1751-8113/41/29/295002
[62] Sodin S 2009 The Tracy-Widom law for some sparse random matrices J. Stat. Phys.136 834 · Zbl 1177.82066 · doi:10.1007/s10955-009-9813-2
[63] Semerjian G and Cugliandolo L F 2002 Sparse random matrices: the eigenvalue spectrum revisited J. Phys. A: Math. Gen.35 4837 · Zbl 1066.82019 · doi:10.1088/0305-4470/35/23/303
[64] Erdös L, Knowles A, Yau H-T and Yin J 2013 Spectral statistics of Erdös-Rényi graphs I: local semicircle law Ann. Probab.41 2279 · Zbl 1272.05111 · doi:10.1214/11-AOP734
[65] Erdös L, Knowles A, Yau H-T and Yin J 2012 Spectral statistics of Erdös-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues Commun. Math. Phys.314 587 · Zbl 1251.05162 · doi:10.1007/s00220-012-1527-7
[66] Mendez-Bermudez J A, Alcazar-Lopez A, Martinez-Mendoza A J, Rodrigues F A and Peron T K 2015 Universality in the spectral and eigenfunction properties of random networks Phys. Rev. E 91 032122 · doi:10.1103/PhysRevE.91.032122
[67] Casati G, Guarneri I, Izrailev F M and Scharf R 1990 Scaling behavior of localization in quantum chaos Phys. Rev. Lett.64 5 · doi:10.1103/PhysRevLett.64.5
[68] Izrailev F M 1990 Simple models of quantum chaos: spectrum and eigenfunctions Phys. Rep.196 299 · doi:10.1016/0370-1573(90)90067-C
[69] Casati G, Guarneri I, Izrailev F M, Fishman S and Molinari L 1992 Scaling of the information length in 1D tight-binding models J. Phys.: Condens. Matter4 149 · doi:10.1088/0953-8984/4/1/024
[70] Izrailev F M 1993 Quantum chaos Proc. Int. School of Physics ‘Enrico Fermi’(Course CXIX, Varenna, 1991) ed G Casati et al (Amsterdam: North-Holland) p 265
[71] Izrailev F M 1989 Intermediate statistics of the quasi-energy spectrum and quantum localisation of classical chaos J. Phys. A: Math. Gen.22 865 · doi:10.1088/0305-4470/22/7/017
[72] Sorathia S, Izrailev F M, Zelevinsky V G and Celardo G L 2012 From closed to open one-dimensional Anderson model: transport versus spectral statistics Phys. Rev. E 86 011142 · doi:10.1103/PhysRevE.86.011142
[73] Flores J, Gutierrez L, Mendez-Sanchez R A, Monsivais G, Mora P and Morales A 2013 Anderson localization in finite disordered vibrating rods Europhys. Lett.101 67002 · doi:10.1209/0295-5075/101/67002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.