Advertisement

Journal of Statistical Physics

, Volume 124, Issue 6, pp 1317–1350 | Cite as

Reconstruction on Trees and Spin Glass Transition

  • Marc Mézard
  • Andrea Montanari
Article

Abstract

Consider an information source generating a symbol at the root of a tree network whose links correspond to noisy communication channels, and broadcasting it through the network. We study the problem of reconstructing the transmitted symbol from the information received at the leaves. In the large system limit, reconstruction is possible when the channel noise is smaller than a threshold.

We show that this threshold coincides with the dynamical (replica symmetry breaking) glass transition for an associated statistical physics problem. Motivated by this correspondence, we derive a variational principle which implies new rigorous bounds on the reconstruction threshold. Finally, we apply a standard numerical procedure used in statistical physics, to predict the reconstruction thresholds in various channels. In particular, we prove a bound on the reconstruction problem for the antiferromagnetic “Potts” channels, which implies, in the noiseless limit, new results on random proper colorings of infinite regular trees.

This relation to the reconstruction problem also offers interesting perspective for putting on a clean mathematical basis the theory of glasses on random graphs.

Key Words

reconstruction spin glasses reconstruction threshold phase transition 

References

  1. C. MacDonald, J. Gibbs and A. Pipkin, Biopolymers 6:1 (1968); C. MacDonald and J. Gibbs, Biopolymers 7:707 (1969).Google Scholar
  2. B. Derrida, E. Domany and D. Mukamel, J. Stat. Phys. 69:667 (1992).MATHCrossRefMathSciNetGoogle Scholar
  3. B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, J. Phys. A: Math. Gen. 26:1493 (1993).MATHCrossRefADSMathSciNetGoogle Scholar
  4. G. M. Schütz and E. Domany, J. Stat. Phys. 72:277 (1993).CrossRefGoogle Scholar
  5. B. Derrida, Phys. Rep. 301:65 (1998).CrossRefMathSciNetGoogle Scholar
  6. G. M. Schütz, in Phase Transition and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic Press, San Diego, 2000).Google Scholar
  7. J. Solomovici, T. Lesnik and C. Reiss, J. Theor. Biol. 185:511 (1997).CrossRefGoogle Scholar
  8. C. M. Stenström, H. Jin, L. L. Major, W. P. Tate and L. A. Isaksson, Gene 263:273 (2001).CrossRefGoogle Scholar
  9. T. Chou and G. Lakatos, Phys. Rev. Lett. 93:198101 (2004).CrossRefADSGoogle Scholar
  10. M. Robinson, R. Lilley, S. Little, J. S. Emtage, G. Yarranton, P. Stephens, A. Millican, M. Eaton and G. Humphreys, Nucl. Acids Res. 12:6663 (1984).Google Scholar
  11. M. A. Sorensen, C. G. Kurland and S. Pedersen, J. Mol. Biol. 207:365 (1989).CrossRefGoogle Scholar
  12. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, in Molecular Biology of the Cell, 4th ed. (Garland Science, New York, NY, 2002)Google Scholar
  13. F. Neidhardt and H. Umbarger, in Escherichia coli and Salmonella, 2nd ed., edited by F. C. Neidhardt (ASM Press, Washington, DC, 1996).Google Scholar
  14. R. Heinrich and T. Rapoport, J. Theor. Biol. 86:279 (1980).CrossRefGoogle Scholar
  15. C. Kang and C. Cantor, J. Mol. Struct. 181:241 (1985).Google Scholar
  16. L. B. Shaw, R. K. P. Zia and K. H. Lee, Phys. Rev. E 68:021910 (2003).CrossRefADSGoogle Scholar
  17. J. J. Dong, B. Schmittmann and R. K. P. Zia, to be published.Google Scholar
  18. A. Kolomeisky, J. Phys. A: Math. Gen. 31:1153 (1998).MATHCrossRefADSGoogle Scholar
  19. S. Janowsky and J. Lebowitz, Phys. Rev. A 45:618 (1992).CrossRefADSGoogle Scholar
  20. S. Janowsky and J. Lebowitz, J. Stat. Phys. 77:35 (1994).MATHCrossRefMathSciNetGoogle Scholar
  21. R. J. Harris and R. B. Stinchcombe, Phys. Rev. E 70:016108 (2004).CrossRefADSMathSciNetGoogle Scholar
  22. M. Ha, J. Timonen and M. den Nijs, Phys. Rev. E 68:056122 (2003). For more details, see also M. Ha, PhD thesis, University of Washington, 2003.Google Scholar
  23. L. B. Shaw, A. B. Kolomeisky, and K. H. Lee, J. Phys. A: Math. Gen. 37:2105 (2004).MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Marc Mézard
    • 1
  • Andrea Montanari
    • 2
  1. 1.Laboratoire de Physique Théorique et Modèles StatistiquesUniversité de Paris-SudOrsay CedexFrance
  2. 2.Laboratoire de Physique Théorique de l’Ecole Normale SupérieureParis Cedex 05France

Personalised recommendations