- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Spectral properties of disordered fully connected graphs

S. N. Taraskin*

  • St. Catharine’s College and Department of Chemistry, University of Cambridge, Cambridge, United Kingdom

  • *Electronic address: snt1000@cam.ac.uk
PDF

Phys. Rev. E 72, 056126 – Published 21 November, 2005

DOI: https://doi.org/10.1103/PhysRevE.72.056126

Abstract

The spectral properties of disordered fully connected graphs with a special type of node-node interactions are investigated. The approximate analytical expression for the ensemble-averaged spectral density for the Hamiltonian defined on the fully connected graph is derived and analyzed for both the electronic and vibrational problems which can be related to the contact process and to the problem of stochastic diffusion, respectively. It is demonstrated how to evaluate the extreme eigenvalues and use them for finding the lower-bound estimates of the critical parameter for the contact process on the disordered fully connected graphs.

View figure in article

Article Text

References (51)

  1. S. Dorogovtsev and J. Mendes, Adv. Phys. 51, 1075 (2002).
  2. R. Albert and A.-L. Barabáshi, Rev. Mod. Phys. 74, 47 (2002).
  3. S. Dorogovtsev and J. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003).
  4. J.-Y. Fortin, J. Phys. A 38, L57 (2005).
  5. K. Iguchi and H. Yamada, Phys. Rev. E 71, 037102 (2005).
  6. K. Iguchi and H. Yamada, Phys. Rev. E 71, 036144 (2005).
  7. S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, and A. N. Samukhin, Phys. Rev. E 68, 046109 (2003).
  8. M. B. Hastings, Phys. Rev. Lett. 90, 148702 (2003).
  9. I. J. Farkas, I. Derényi, A.-L. Barabási, and T. Vicsek, Phys. Rev. E 64, 026704 (2001).
  10. K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. E 64, 051903 (2001).
  11. R. Monasson, Eur. Phys. J. B 12, 555 (1999).
  12. G. Biroli and R. Monasson, J. Phys. A 32, L255 (1999).
  13. J. Stäring, B. Mehlig, Y. V. Fyodorov, and J. M. Luck, Phys. Rev. E 67, 047101 (2003).
  14. Y. V. Fyodorov, J. Phys. A 32, 7429 (1999).
  15. A. A. Maradudin, E. W. Montroll, G. Weiss, and I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation (Academic Press, New York, 1971).
  16. H. Böttger and V. Bryksin, Hopping Conduction in Solids (VCH, Berlin, 1985).
  17. J. Rudnick and G. Gaspari, Elements of the Random Walk (Cambridge University Press, Cambridge, England, 2004).
  18. T. M. Liggett, Interacting Particle Systems (Springer-Verlag, New York, 1985).
  19. J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models (Cambridge University Press, Cambridge, England, 1999).
  20. S. N. Taraskin and S. R. Elliott, J. Phys.: Condens. Matter 14, 3143 (2002).
  21. H. Ehrenreich and L. Schwarts, Solid State Phys. 31, 149 (1976).
  22. E. Bogomolny, U. Gerland, and C. Schmit, Phys. Rev. E 63, 036206 (2001).
  23. K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
  24. B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993).
  25. S. Wiseman and E. Domany, Phys. Rev. Lett. 81, 22 (1988).
  26. M. Kawasaki, T. Odagaki, and K. W. Kehr, Phys. Rev. B 67, 134203 (2003).
  27. M. C. W. van Rossum, T. M. Nieuwenhuizen, E. Hofstetter, and M. Schreiber, Phys. Rev. B 49, 13377 (1994).
  28. Computational Physics, edited by K. H. Hoffmann and M. Schreiber (Springer-Verlag, Berlin, 1996).
  29. S. R. Elliott, Physics of Amorphous Materials, 2nd ed. (Longman, New York, 1990).
  30. A. D. Mirlin, Phys. Rep. 326, 259 (2000).
  31. T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Phys. Rep. 299, 189 (1998).
  32. V. Dotsenko, Introduction to the Replica Theory of Disordered Statistical Systems (Cambridge University Press, Cambridge, England, 2000).
  33. K. B. Efetov, Adv. Phys. 32, 53 (1983).
  34. T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981).
  35. M. L. Mehta, Random Matrices, 2nd ed. (Academic, New York, 1991).
  36. L. A. Pastur, Theor. Math. Phys. 10, 67 (1972).
  37. P. Lloyd, J. Phys. C 2, 1717 (1969).
  38. G. Lehmann, J. Phys. C 6, 1881 (1973).
  39. M. Mezard, G. Parisi, and A. Zee, Nucl. Phys. B 559, 689 (1999).
  40. S. N. Taraskin, J. J. Ludlam, C. J. Neugebauer, and C. A. Gilligan, Phys. Rev. E 72, 016111 (2005).
  41. J. Hooyberghs, F. Iglói, and C. Vanderzande, Phys. Rev. Lett. 90, 100601 (2003).
  42. J. Hooyberghs, F. Iglói, and C. Vanderzande, Phys. Rev. E 69, 066140 (2004).
  43. T. Vojta, Phys. Rev. E 70, 026108 (2004).
  44. I. Webman, Phys. Rev. Lett. 47, 1496 (1981).
  45. T. Odagaki and M. Lax, Phys. Rev. B 24, 5284 (1981).
  46. M. S. Friedrichs and R. A. Friesner, Phys. Rev. B 37, 308 (1988).
  47. E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer, Heidelberg, 1983).
  48. P. Neut and R. Speicher, J. Phys. A 28, L79 (1995).
  49. A. Bohr and B. R. Mottelson, Nuclear Structure (Benjamin, New York, 1989), Vol. 1.
  50. M. I. Klinger and S. N. Taraskin, Phys. Rev. B 47, 10235 (1993).
  51. R. Guimera, A. Arenas, A. Diaz-Guilera, and F. Giralt, Phys. Rev. E 66, 026704 (2002).

Outline

Information

Phys. Rev. E 72, 056126– Published 21 November, 2005
  • Received 1 August 2005

© 2005 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation