Abstract
We consider random one-body operators that are analogs of the statistical mechanics Hamiltonians with a varying interaction radiusR, the dimensionality of spaced and the number of the field components (orbitals)n. We prove that all the moments of the Green functions for nonreal energies of these operators converge asR, d, n→∞ to the products of the average Green functions, just as in the mean field approximation of statistical mechanics. We find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity, which is nonzero on the whole support of the integrated density of states.
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Abou-Chacra, R., Anderson, P., Thouless, D.: A selfconsistent theory of localization. J. Phys.C6, 1734–1752 (1973)
Akhiezer, N.: The classical moment problem. London: Oliver and Boyd 1964
Berlin, T., Kac, M.: The spherical model of a ferromagnet. Phys. Rev.86, 821–825 (1952)
Bogachev, L., Molchanov, S., Pastur, L.: On the density of states of random band matrices (in Russian). Mat. Zametki,50, 31–42 (1991)
Brezin, E., Itzykson, C., Parisi, G., Zuber, J.: Planar diagrams. Commun. Math. Phys.59, 35–51 (1978)
Casati, G., Molinari, L., Izrailev, F.: Scaling properties of band random matrices. Phys. Rev. Lett.64, 1851–1854 (1990)
Constantinescu, F., Felder, C., Gawedzki, K., Kupiainen, A.: Analyticity of density of states in a gauge invariant model of disordered systems. J. Stat. Phys.48, 365–391 (1987)
Elliot, P., Krumhansl, J., Leath, P.: Theory and properties of randomly disordered crystals and related physical systems. Rev. Mod. Phys.46, 463–510 (1974)
Fernandez, R., Fröhlich, J., Sokal, A.: Random walks, random surfaces, critical phenomena and triviality in quantum field theory. Berlin, Heidelberg, New York: Springer, 1992
Girko, V.: Spectral theory of random matrices (in Russian). Moscow: Nauka 1988
Haake, F.: Quantum signatures of chaos. Berlin, Heidelberg, New York: Springer 1991
Kac, M.: Mathematical mechanisms of the phase transitions. In: Chretien, M., Deser, S. (eds.): Statistical physics, phase transitions and superfluidity Vol. I, pp. 241–301. New York: Gordon and Breach 1968
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Kubo, R.: Statistical mechanics. Amsterdam, North-Holland 1965
Kac, M., Thompson, C.: Spherical model and the infinite spin dimensionality limit. Physica Norwegica5, 163–168 (1971)
Khorunzhy, A., Pastur, L.: On the eigenvalue distribution of the deformed Wigner ensemble of random matrices. In: Operator theory and related topics. AMS (in press)
Khorunzhy, A., Molchanov, S., Pastur, L.: On the eigenvalue distribution of band random matrices in the limit of their infinite order (in Russian). Teor. Mat. Fiz.90, 163–178 (1992)
Khorunzhy, A., Khoruzhenko, B., Pastur, L., Shcherbina, M.: The large-n limit in statistical mechanics and the spectral theory of disordered systems. In: Domb, C., Lebowitz, J. (eds.): Phase transitions and critical phenomena Vol. 15, pp. 73–239. New York: Academic Press 1992
Lee, P., Ramakrishman, T.: Disordered electronic systems. Rev. Mod. Phys57, 287–337 (1985)
Lebowitz, J., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of the liquidvapour transition. J. Math. Phys7, 98–110 (1966)
Lifshitz, I., Gredeskul, S., Pastur, L.: Introduction in the theory of disordered systems. New York: Wiley 1988
Mehta, M.: Random matrices. New York: Academic Press 1967
Pastur, L.: Spectra of random self-adjoint operators. Russ. Math. Surv.28, 1–67 (1973)
Pastur, L.: On the spectrum of random matrices (in Russian). Teor. Mat. Fiz.10, 102–112 (1973)
Pastur, L., Figotin, A.: Spectra of random and almost periodic operators. Berlin, Heidelberg, New York: Springer 1992
Pastur, L., Shcherbina, M.: Infinite correlation radius limit for correlation functions of lattice systems (in Russian). Teor. Mat. Fiz.61, 3–16 (1984)
Shcherbina, M.: Spherical limit ofn-vector correlations (in Russian). Teor. Mat. Fiz.77, 460–471 (1988)
Stanley, H.: Spherical model as a limit spin dimensionality. Phys. Rev.176, 718–721 (1968)
Velicky, B.: Theory of electronic transport in disordered binary alloys: coherent potential approximation. Phys. Rev.184, 614–627 (1969)
Vlaming, R., Vollhardt, D.: Controlled mean field theory for disordered electronic systems: single particle properties. Rutgers preprint RWTH/ITP-C 6/91
Wegner, F.: Disordered system withn orbitals per site:n=∞ limit. Phys. Rev.B19, 783–792 (1979)
Wigner, E.: Random matrices in physics. SIAM Review J.9, 1–23 (1967)
Wegner, F., Opperman, R.: Disordered systems withn orbitals per site: 1/n expansion. Z. Phys. B34, 327–348 (1979)
Yonezawa, F., Morigaki, K.: Coherent potential approximation. Suppl. Progr. Theor. Phys.53, 1–76 (1973)
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Communicated by Ya. G. Sinai
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Khorunzhy, A.M., Pastur, L.A. Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness. Commun.Math. Phys. 153, 605–646 (1993). https://doi.org/10.1007/BF02096955
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DOI: https://doi.org/10.1007/BF02096955