Propagating beliefs in spin-glass models

We investigate dynamics of an inference algorithm termed the belief propagation (BP) when employed in spin glass (SG) models and show that its macroscopic behaviors can be traced by recursive updates of certain auxiliary field distributions the stationary state of which reproduces the replica symmetric solution offered by the equilibrium analysis. We further provide a compact expression for instability condition of the BP's fixed point which turns out to be identical to that of instability for breaking the replica symmetry in equilibrium when the number of couplings per spin is infinite. This correspondence is extended to the case of finite connectivity to determine the phase diagram, which is numerically validated.

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