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Replica Cluster Variational Method

Profile image of Roberto MuletRoberto Mulet

2010, Journal of Statistical Physics

https://doi.org/10.1007/S10955-010-9938-3
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32 pages

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Abstract

We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi's Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We obtain a set of integral equations for the message functionals. The main difference with the Bethe case is that the functionals appear in the equations in implicit form and are not positive definite, thus standard iterative population dynamic algorithms cannot be used to determine them. In the simplest cases the solution could be obtained iteratively using Fourier transforms.

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  54. The original name was "Cluster Variation Method", but we believe "Cluster Variational Method" to be a better wording.
  55. We adopt the original notation of Ref. [6], which was changed in the more recent Ref. [7]. The ensembles M (r) \ M (s) and M (r, s) corresponds respectively to N (r, s) and M (r, s) defined in [7]. Note however that for us these are ensembles of couples of regions labels instead of ensembles of the corresponding messages as in [6].
  56. Actually one obtain a different set of equations that can be proved to be equivalent to eqs. (6) along the same lines of the proof that the two sets of constraints are equivalent, see theorem 5 in [7]

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Related topics

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