2000 character limit reached
The planted XY model: thermodynamics and inference (2208.06488v2)
Published 12 Aug 2022 in cond-mat.dis-nn, cond-mat.stat-mech, cs.IR, math.PR, and stat.CO
Abstract: In this paper we study a fully connected planted spin glass named the planted XY model. Motivation for studying this system comes both from the spin glass field and the one of statistical inference where it models the angular synchronization problem. We derive the replica symmetric (RS) phase diagram in the temperature, ferromagnetic bias plane using the approximate message passing (AMP) algorithm and its state evolution (SE). While the RS predictions are exact on the Nishimori line (i.e. when the temperature is matched to the ferromagnetic bias), they become inaccurate when the parameters are mismatched, giving rise to a spin glass phase where AMP is not able to converge. To overcome the defects of the RS approximation we carry out a one-step replica symmetry breaking (1RSB) analysis based on the approximate survey propagation (ASP) algorithm. Exploiting the state evolution of ASP, we count the number of metastable states in the measure, derive the 1RSB free entropy and find the behavior of the Parisi parameter throughout the spin glass phase.
- J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, Journal of Physics C: Solid State Physics 6, 1181 (1973).
- D. Sherrington, The infinite-ranged m-vector spin glass, in Heidelberg Colloquium on Spin Glasses (Springer, 1983) pp. 125–136.
- A. Singer, Angular synchronization by eigenvectors and semidefinite programming, Applied and computational harmonic analysis 30, 20 (2011).
- A. Giridhar and P. R. Kumar, Distributed clock synchronization over wireless networks: Algorithms and analysis, in Proceedings of the 45th IEEE Conference on Decision and Control (IEEE, 2006) pp. 4915–4920.
- A. Agrawal, R. Raskar, and R. Chellappa, What is the range of surface reconstructions from a gradient field?, in European conference on computer vision (Springer, 2006) pp. 578–591.
- J. Rubinstein and G. Wolansky, Reconstruction of optical surfaces from ray data, Optical review 8, 281 (2001).
- A. S. Bandeira, N. Boumal, and A. Singer, Tightness of the maximum likelihood semidefinite relaxation for angular synchronization, Mathematical Programming 163, 145 (2017).
- A. Javanmard, A. Montanari, and F. Ricci-Tersenghi, Phase transitions in semidefinite relaxations, Proceedings of the National Academy of Sciences 113, E2218 (2016).
- D. L. Donoho, A. Maleki, and A. Montanari, Message-passing algorithms for compressed sensing, Proceedings of the National Academy of Sciences 106, 18914 (2009).
- L. Zdeborová and F. Krzakala, Statistical physics of inference: Thresholds and algorithms, Advances in Physics 65, 453 (2016).
- C. Ebner and D. Stroud, Diamagnetic susceptibility of superconducting clusters: Spin-glass behavior, Phys. Rev. B 31, 165 (1985).
- Y. Ozeki and H. Nishimori, Phase diagram of gauge glasses, Journal of Physics A: Mathematical and General 26, 3399 (1993).
- H. Nishimori, Statistical physics of spin glasses and information processing: an introduction, 111 (Clarendon Press, 2001).
- Y. Iba, The nishimori line and bayesian statistics, Journal of Physics A: Mathematical and General 32, 3875 (1999).
- C. Lupo and F. Ricci-Tersenghi, Approximating the xy model on a random graph with a q-state clock model, Physical Review B 95, 054433 (2017).
- S. Morita, Y. Ozeki, and H. Nishimori, Gauge theory for quantum spin glasses, Journal of the Physical Society of Japan 75, 014001 (2006).
- F.-F. Song and G.-M. Zhang, Tensor network approach to the two-dimensional fully frustrated xy model and a bosonic metallic phase with chirality order (2021), arXiv:2112.03550.
- D. J. Thouless, P. W. Anderson, and R. G. Palmer, Solution of’solvable model of a spin glass’, Philosophical Magazine 35, 593 (1977).
- L. Miolane, Fundamental limits of low-rank matrix estimation: the non-symmetric case, arXiv preprint arXiv:1702.00473 (2017).
- E. Mossel, J. Neeman, and A. Sly, Reconstruction and estimation in the planted partition model, Probability Theory and Related Fields 162, 431 (2015).
- E. Bolthausen, An iterative construction of solutions of the tap equations for the sherrington–kirkpatrick model, Communications in Mathematical Physics 325, 333 (2014).
- A. Coja-Oghlan and W. Perkins, Bethe states of random factor graphs, Communications in Mathematical Physics 366, 173 (2019).
- A. J. Bray and M. A. Moore, Metastable states in spin glasses, Journal of Physics C: Solid State Physics 13, L469 (1980).
- F. Zamponi, Mean field theory of spin glasses, arXiv preprint arXiv:1008.4844 (2010).
- M. Mézard, G. Parisi, and R. Zecchina, Analytic and algorithmic solution of random satisfiability problems, Science 297, 812 (2002).
- C. Lucibello, L. Saglietti, and Y. Lu, Generalized approximate survey propagation for high-dimensional estimation, in International Conference on Machine Learning (PMLR, 2019) pp. 4173–4182.
- E. Gardner, Spin glasses with p-spin interactions, Nuclear Physics B 257, 747 (1985).
- T. Castellani and A. Cavagna, Spin-glass theory for pedestrians, Journal of Statistical Mechanics: Theory and Experiment 2005, P05012 (2005).
- A. Montanari, G. Parisi, and F. Ricci-Tersenghi, Instability of one-step replica-symmetry-broken phase in satisfiability problems, Journal of Physics A: Mathematical and General 37, 2073 (2004).
- S. R. Jammalamadaka and A. Sengupta, Topics in circular statistics, Vol. 5 (world scientific, 2001).
- M. Bayati and A. Montanari, The dynamics of message passing on dense graphs, with applications to compressed sensing, IEEE Transactions on Information Theory 57, 764 (2011).