Replica Cluster Variational Method

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 begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature in all the three regular 2D lattices. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonians. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation.

32 pages, 14 figures. In the revised version the content has been reorganized in order to imporve readability

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• A simple analytical description of the non-stationary dynamics in Ising spin systems
• The Quantum Cluster Variational Method and the Phase Diagram of the quantum ferromagnetic $J_1$-$J_2$ model
• Message passing and Monte Carlo algorithms: connecting fixed points with metastable states
• Statistical Analysis of Loopy Belief Propagation in Random Fields
• Partition Function Expansion on Region-Graphs and Message-Passing Equations
• Message-passing algorithm of quantum annealing with nonstoquastic Hamiltonian
• Simplifying Generalized Belief Propagation on Redundant Region Graphs
• Anomalous finite size corrections in random field models
• Quantum Cluster Variational Method and Message Passing Algorithms Revisited
• Loop-corrected belief propagation for lattice spin models
• Empirical Bayes Method for Boltzmann Machines
• Gauge-free cluster variational method by maximal messages and moment matching
• On one-step replica symmetry breaking in the Edwards-Anderson spin glass model
• Low Auto-correlation Binary Sequences explored using Warning Propagation
• Efficient inference in the transverse field Ising model