Finite-connectivity spin-glass phase diagrams and low-density parity check codes

Gabriele Migliorini*, David Saad

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review


    We obtain phase diagrams of regular and irregular finite-connectivity spin glasses. Contact is first established between properties of the phase diagram and the performance of low-density parity check (LDPC) codes within the replica symmetric (RS) ansatz. We then study the location of the dynamical and critical transition points of these systems within the one step replica symmetry breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass problem. We observe that the location of the dynamical transition line does change within the RSB theory, in comparison with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel we find, at zero temperature and rate R=14, an RS critical transition point at pc 0.67 while the critical RSB transition point is located at pc 0.7450±0.0050, to be compared with the corresponding Shannon bound 1-R. For the binary symmetric channel we show that the low temperature reentrant behavior of the dynamical transition line, observed within the RS ansatz, changes its location when the RSB ansatz is employed; the dynamical transition point occurs at higher values of the channel noise. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed. © 2006 The American Physical Society.

    Original languageEnglish
    Article number026122
    Number of pages16
    JournalPhysical Review E
    Issue number2
    Early online date21 Feb 2006
    Publication statusPublished - 2 Mar 2006

    Bibliographical note

    Copyright of the American Physical Society


    • finite connectivity spin-glasses
    • low density parity check codes
    • Replica Symmetric
    • replica symmetry breaking theory


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