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In this paper we study a fully connected planted spin glass named the plantedXY model. Motivation for studying this system comes both from the spin glassfield and the one of statistical inference where it models the angularsynchronization problem. We derive the replica symmetric (RS) phase diagram inthe temperature, ferromagnetic bias plane using the approximate message passing(AMP) algorithm and its state evolution (SE). While the RS predictions areexact on the Nishimori line (i.e. when the temperature is matched to theferromagnetic bias), they become inaccurate when the parameters are mismatched,giving rise to a spin glass phase where AMP is not able to converge. Toovercome the defects of the RS approximation we carry out a one-step replicasymmetry breaking (1RSB) analysis based on the approximate survey propagation(ASP) algorithm. Exploiting the state evolution of ASP, we count the number ofmetastable states in the measure, derive the 1RSB free entropy and find thebehavior of the Parisi parameter throughout the spin glass phase.

The planted XY model: thermodynamics and inference