×

zbMATH — the first resource for mathematics

Analyticity of the density of states and replica method for random Schrödinger operators on a lattice. (English) Zbl 0591.60060
Summary: We analyze the density of states and some aspects of the replica method for Anderson’s tight binding model on a lattice of arbitrary dimension, with diagonal disorder. We give heuristic arguments for the conjectures that the classical value of the exponent v of the localization length is 1/2 and that the upper critical dimension, \(d_ c^{loc}\), is bounded by \(4\leq d_ c^{loc}\leq 6\).

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] P. W. Anderson, Absence of diffusion in certain random lattices,Phys. Rev. 109:1492 (1958); For reviews see D. J. Thouless, inIll Condensed Matter, R. Balian, R. Maynard, and G. Toulouse, eds. (North-Holland, Amsterdam, 1979), pp. 1-62; D. J. Thouless, E. Abrahams, and F. Wegner,Contributions Phys. Rep. 67:No. 1 (1980) (E. Brézin, J.-L. Gervais, and G. Toulouse, eds.). · doi:10.1103/PhysRev.109.1492
[2] H. Kunz and B. Souillard, Sur le spectre des opérateurs aux differences finies aléatoires,Commun. Math. Phys. 78:201 (1980). · Zbl 0449.60048 · doi:10.1007/BF01942371
[3] I. Ya. Goldsheid, S. A. Molchanov, and L. A. Pastur, A pure point spectrum of the stochastic one-dimensional Schrödinger operator,Funct. Anal. App. 11:1 (1977). S. A. Molchanov, The structure of the eigenfunctions of one-dimensional unordered structures,Math. USSR Izvestija 12:69 (1978). L. A. Pastur, Spectral properties of disordered systems in the one-body approximation,Commun. Math. Phys. 75:179 (1980). · Zbl 0368.34015 · doi:10.1007/BF01135526
[4] J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy,Commun. Math. Phys. 88:151-184 (1983). J. Fröhlich and T. Spencer, Existence of localized states for random Schrödinger operators on ?d, in preparation. · Zbl 0519.60066 · doi:10.1007/BF01209475
[5] M. Fukushima, On asymptotics of spectra of Schrödinger operators, in Aspects statistiques et aspects physiques des processus Gaussiens, Colloques internationaux du Centre National de la Recherche Scientifique, Saint-Flour, 22-29 June 1980.
[6] J. L. van Hemmen, On thermodynamic observables and the Anderson model,J. Phys. A: Math. Gen. 15:3891 (1982). · doi:10.1088/0305-4470/15/12/038
[7] G. Gallavotti, A. Martin-Löf, and S. Miracle-Sole,Statistical Mechanics, Proceedings of the Battelle Rencontre, 1970 (Lecture Notes in Physics, No. 6, Springer, Berlin, 1971).
[8] V. Malyshev, Uniform cluster estimates for lattice models,Commun. Math. Phys. 64:131 (1979). · Zbl 0409.46068 · doi:10.1007/BF01197510
[9] E. Seiler,Gauge theories as a problem of constructive quantum field theory and statistical mechanics (Lecture Notes in Physics, No. 159, Springer, Berlin, 1982).
[10] J. T. Edwards and D. J. Thouless, Regularity of the density of states in Anderson’s localized electron model,J. Phys. C: Solid State Phys. 4:453 (1971); D. J. Thouless, Electrons in disordered systems and the theory of localisation,Phys. Rep. 13:93 (1974). · doi:10.1088/0022-3719/4/4/007
[11] E. Brézin and G. Parisi, Exponential tail of the electronic density of levels in a random potential,J. Phys. C. 13:L307 (1980). · doi:10.1088/0022-3719/13/12/005
[12] D. Brydges, J. Fröhlich, and T. Spencer, The random walk representation of classical spin systems and correlation inequalities,Commun. Math. Phys. 83:125 (1982). · doi:10.1007/BF01947075
[13] J. Fröhlich, A. Mardin and V. Rivasseau, Borel summability of the 1/N expansion for the N-vector [O(N) non-linear ?] models,Commun. Math. Phys. 86:87 (1982). · doi:10.1007/BF01205663
[14] F. Wegner, Bounds on the density of states in disordered systems,Z. Phys. B, Condensed Matter 44:9 (1981). · doi:10.1007/BF01292646
[15] C. Cammarota, Decay of correlations for infinite range interactions in unbounded spin systems,Commun. Math. Phys. 85:517 (1982). · doi:10.1007/BF01403502
[16] W. Froese and I. Herbst, preprint, University of Virginia, 1982.
[17] A. Dvoretsky, P. Erdos, and S. Kakutani, Brownian motion inn-space,Acta Sci. Math. (Szeged)12B:75 (1950).
[18] L. Carleson,Selected Problems on Exceptional Sets, (Van Nostrand, Princeton, 1967), Section VII.3, pp. 95-98; J. Serrin, Removable singularities of solutions of elliptic equations.Archive Rat. Mech. Analys. 17:67 (1964), Theorem 1, p. 68. The mathematical details have been explained to us by H. Brézis and B. Simon whom we wish to thank for interesting discussions.
[19] H. Kunz and B. Souillard, On the upper critical dimension and the critical exponents of the localization transition, preprint, Spring 1983.
[20] P. Lloyd, Exactly solvable model of electronic states in a three-dimensional disordered Hamiltonian: non-existence of localized states,J. Phys. C2:1717 (1969).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.