For PEPS states with loop-decay, BP with cluster corrections approximates local observables exponentially accurately, and loop-decay necessarily implies exponential decay of connected correlations, ruling out BP at critical points.
Kikuchi, A Theory of Cooperative Phenomena, Phys- ical Review81, 988 (1951)
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Thermodynamic efficiency of self-organization in nonequilibrium steady states maximizes at phase transitions and diverges from inferential efficiency in proportion to distance from equilibrium.
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Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits
For PEPS states with loop-decay, BP with cluster corrections approximates local observables exponentially accurately, and loop-decay necessarily implies exponential decay of connected correlations, ruling out BP at critical points.
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Thermodynamic efficiency of self-organisation in nonequilibrium steady states
Thermodynamic efficiency of self-organization in nonequilibrium steady states maximizes at phase transitions and diverges from inferential efficiency in proportion to distance from equilibrium.