Message passing and Monte Carlo algorithms: Connecting fixed points with metastable states

Mean field-like approximations (including naive mean-field, Bethe and Kikuchi and more general cluster variational methods) are known to stabilize ordered phases at temperatures higher than the thermodynamical transition. For example, in the Edwards-Anderson model in 2 dimensions these approximations predict a spin glass transition at finite T. Here we show that the spin glass solutions of the Cluster Variational Method (CVM) at plaquette level do describe well the actual metastable states of the system. Moreover, we prove that these states can be used to predict non-trivial statistical quantities, like the distribution of the overlap between two replicas. Our results support the idea that message passing algorithms can be helpful to accelerate Monte Carlo simulations in finite-dimensional systems.

[1]  Riccardo Zecchina,et al.  Coloring random graphs , 2002, Physical review letters.

[2]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[3]  Chuang Wang,et al.  Partition function expansion on region graphs and message-passing equations , 2011, ArXiv.

[4]  Federico Ricci-Tersenghi,et al.  Characterizing and improving generalized belief propagation algorithms on the 2D Edwards–Anderson model , 2011, ArXiv.

[5]  Haijun Zhou,et al.  Partition function loop series for a general graphical model: free-energy corrections and message-passing equations , 2011, ArXiv.

[6]  Signature of the ground-state topology in the low-temperature dynamics of spin glasses. , 2005, Physical review letters.

[7]  Michael Chertkov,et al.  Belief propagation and loop series on planar graphs , 2008, ArXiv.

[8]  Gerard T. Barkema,et al.  Monte Carlo Methods in Statistical Physics , 1999 .

[9]  William T. Freeman,et al.  Constructing free-energy approximations and generalized belief propagation algorithms , 2005, IEEE Transactions on Information Theory.

[10]  S. F. Schifano,et al.  Sample-to-sample fluctuations of the overlap distributions in the three-dimensional Edwards-Anderson spin glass , 2011, 1107.5772.

[11]  Glassy dynamics near zero temperature , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[12]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[13]  S. Stenholm.  Information, Physics and Computation, by Marc Mézard and Andrea Montanari , 2010 .

[14]  R. Zecchina,et al.  Polynomial iterative algorithms for coloring and analyzing random graphs. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M. Mézard,et al.  The Cavity Method at Zero Temperature , 2002, cond-mat/0207121.

[16]  Haijun Zhou,et al.  Region Graph Partition Function Expansion and Approximate Free Energy Landscapes: Theory and Some Numerical Results , 2012, 1204.1818.

[17]  F. Ricci-Tersenghi,et al.  Inference algorithm for finite-dimensional spin glasses: belief propagation on the dual lattice. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  H. Bethe.  Statistical Theory of Superlattices , 1935 .

[19]  Hilbert J. Kappen,et al.  Approximate Inference and Constrained Optimization , 2002, UAI.

[20]  Federico Ricci-Tersenghi,et al.  Replica cluster variational method: the replica symmetric solution for the 2D random bond Ising model , 2012, 1204.0439.

[21]  S. Edwards,et al.  Theory of spin glasses , 1975 .

[22]  Alessandro Pelizzola,et al.  Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models , 2005, ArXiv.

[23]  M. Mézard,et al.  Random K-satisfiability problem: from an analytic solution to an efficient algorithm. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Federico Ricci-Tersenghi,et al.  Replica Cluster Variational Method , 2009, 0906.2695.

[25]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .