Communications in Mathematical Physics

, Volume 153, Issue 3, pp 605–646 | Cite as

Limits of infinite interaction radius, dimensionality and the number of components for random operators with off-diagonal randomness

  • A. M. Khorunzhy
  • L. A. Pastur
Article

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.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Green Function 

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References

  1. 1.
    Abou-Chacra, R., Anderson, P., Thouless, D.: A selfconsistent theory of localization. J. Phys.C6, 1734–1752 (1973)Google Scholar
  2. 2.
    Akhiezer, N.: The classical moment problem. London: Oliver and Boyd 1964Google Scholar
  3. 3.
    Berlin, T., Kac, M.: The spherical model of a ferromagnet. Phys. Rev.86, 821–825 (1952)Google Scholar
  4. 4.
    Bogachev, L., Molchanov, S., Pastur, L.: On the density of states of random band matrices (in Russian). Mat. Zametki,50, 31–42 (1991)Google Scholar
  5. 5.
    Brezin, E., Itzykson, C., Parisi, G., Zuber, J.: Planar diagrams. Commun. Math. Phys.59, 35–51 (1978)Google Scholar
  6. 6.
    Casati, G., Molinari, L., Izrailev, F.: Scaling properties of band random matrices. Phys. Rev. Lett.64, 1851–1854 (1990)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Elliot, P., Krumhansl, J., Leath, P.: Theory and properties of randomly disordered crystals and related physical systems. Rev. Mod. Phys.46, 463–510 (1974)Google Scholar
  9. 9.
    Fernandez, R., Fröhlich, J., Sokal, A.: Random walks, random surfaces, critical phenomena and triviality in quantum field theory. Berlin, Heidelberg, New York: Springer, 1992Google Scholar
  10. 10.
    Girko, V.: Spectral theory of random matrices (in Russian). Moscow: Nauka 1988Google Scholar
  11. 11.
    Haake, F.: Quantum signatures of chaos. Berlin, Heidelberg, New York: Springer 1991Google Scholar
  12. 12.
    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 1968Google Scholar
  13. 13.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  14. 14.
    Kubo, R.: Statistical mechanics. Amsterdam, North-Holland 1965Google Scholar
  15. 15.
    Kac, M., Thompson, C.: Spherical model and the infinite spin dimensionality limit. Physica Norwegica5, 163–168 (1971)Google Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    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 1992Google Scholar
  19. 19.
    Lee, P., Ramakrishman, T.: Disordered electronic systems. Rev. Mod. Phys57, 287–337 (1985)Google Scholar
  20. 20.
    Lebowitz, J., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of the liquidvapour transition. J. Math. Phys7, 98–110 (1966)Google Scholar
  21. 21.
    Lifshitz, I., Gredeskul, S., Pastur, L.: Introduction in the theory of disordered systems. New York: Wiley 1988Google Scholar
  22. 22.
    Mehta, M.: Random matrices. New York: Academic Press 1967Google Scholar
  23. 23.
    Pastur, L.: Spectra of random self-adjoint operators. Russ. Math. Surv.28, 1–67 (1973)Google Scholar
  24. 24.
    Pastur, L.: On the spectrum of random matrices (in Russian). Teor. Mat. Fiz.10, 102–112 (1973)Google Scholar
  25. 25.
    Pastur, L., Figotin, A.: Spectra of random and almost periodic operators. Berlin, Heidelberg, New York: Springer 1992Google Scholar
  26. 26.
    Pastur, L., Shcherbina, M.: Infinite correlation radius limit for correlation functions of lattice systems (in Russian). Teor. Mat. Fiz.61, 3–16 (1984)Google Scholar
  27. 27.
    Shcherbina, M.: Spherical limit ofn-vector correlations (in Russian). Teor. Mat. Fiz.77, 460–471 (1988)Google Scholar
  28. 28.
    Stanley, H.: Spherical model as a limit spin dimensionality. Phys. Rev.176, 718–721 (1968)Google Scholar
  29. 29.
    Velicky, B.: Theory of electronic transport in disordered binary alloys: coherent potential approximation. Phys. Rev.184, 614–627 (1969)Google Scholar
  30. 30.
    Vlaming, R., Vollhardt, D.: Controlled mean field theory for disordered electronic systems: single particle properties. Rutgers preprint RWTH/ITP-C 6/91Google Scholar
  31. 31.
    Wegner, F.: Disordered system withn orbitals per site:n=∞ limit. Phys. Rev.B19, 783–792 (1979)Google Scholar
  32. 32.
    Wigner, E.: Random matrices in physics. SIAM Review J.9, 1–23 (1967)Google Scholar
  33. 33.
    Wegner, F., Opperman, R.: Disordered systems withn orbitals per site: 1/n expansion. Z. Phys. B34, 327–348 (1979)Google Scholar
  34. 34.
    Yonezawa, F., Morigaki, K.: Coherent potential approximation. Suppl. Progr. Theor. Phys.53, 1–76 (1973)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • A. M. Khorunzhy
    • 1
  • L. A. Pastur
    • 1
  1. 1.Institute for Low Temperature PhysicsAcademy of Sciences of UkraineKharkivUkraine

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