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 \(4\leq d_ c^{loc}\leq 6\).


60H25 Random operators and equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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[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.).
[2] H. Kunz and B. Souillard, Sur le spectre des opérateurs aux differences finies aléatoires,Commun. Math. Phys. 78:201 (1980). · Zbl 0449.60048
[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
[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
[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).
[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
[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).
[11] E. Brézin and G. Parisi, Exponential tail of the electronic density of levels in a random potential,J. Phys. C. 13:L307 (1980).
[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).
[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).
[14] F. Wegner, Bounds on the density of states in disordered systems,Z. Phys. B, Condensed Matter 44:9 (1981).
[15] C. Cammarota, Decay of correlations for infinite range interactions in unbounded spin systems,Commun. Math. Phys. 85:517 (1982).
[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).
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