Journal of Statistical Physics

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

Reconstruction on Trees and Spin Glass Transition

  • Marc Mézard
  • Andrea Montanari


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 


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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

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