Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models

The cluster variation method (CVM) is a hierarchy of approximate variational techniques for discrete (Ising--like) models in equilibrium statistical mechanics, improving on the mean--field approximation and the Bethe--Peierls approximation, which can be regarded as the lowest level of the CVM. In recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models. The foundations of the CVM are briefly reviewed, and the relations with similar techniques are discussed. The main properties of the method are considered, with emphasis on its exactness for particular models and on its asymptotic properties. The problem of the minimization of the variational free energy, which arises in the CVM, is also addressed, and recent results about both provably convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure

CVM ANALYSIS OF CROSSOVER IN THE SEMI-INFINITE ISING MODEL

The crossover behavior of the semi--infinite three dimensional Ising model is investigated by means of Pad\'e approximant analysis of cluster variation method results. We give estimates for ordinary critical as well as for multicritical exponents, which are in very good agreement with extensive Monte Carlo simulations.Comment: RevTeX, 7 pages + 2 uuencoded PostScript figures. To be published in J. Magn. Magn. Mat. (substituted raw with encoded PostScript

Variational approximations for stochastic dynamics on graphs

We investigate different mean-field-like approximations for stochastic dynamics on graphs, within the framework of a cluster-variational approach. In analogy with its equilibrium counterpart, this approach allows one to give a unified view of various (previously known) approximation schemes, and suggests quite a systematic way to improve the level of accuracy. We compare the different approximations with Monte Carlo simulations on a reversible (susceptible-infected-susceptible) discrete-time epidemic-spreading model on random graphs.Comment: 29 pages, 5 figures. Minor revisions. IOP-style

Generalized belief propagation for the magnetization of the simple cubic Ising model

A new approximation of the cluster variational method is introduced for the three-dimensional Ising model on the simple cubic lattice. The maximal cluster is, as far as we know, the largest ever used in this method. A message-passing algorithm, generalized belief propagation, is used to minimize the variational free energy. Convergence properties and performance of the algorithm are investigated. The approximation is used to compute the spontaneous magnetization, which is then compared to previous results. Using the present results as the last step in a sequence of three cluster variational approximations, an extrapolation is obtained which captures the leading critical behavior with a good accurac

Pathways of mechanical unfolding of FnIII10: Low force intermediates

We study the mechanical unfolding pathways of the $FnIII_{10}$ domain of fibronectin by means of an Ising--like model, using both constant force and constant velocity protocols. At high forces and high velocities our results are consistent with experiments and previous computational studies. Moreover, the simplicity of the model allows us to probe the biologically relevant low force regime, where we predict the existence of two intermediates with very close elongations. The unfolding pathway is characterized by stochastic transitions between these two intermediates

Kinetics of the Wako-Saito-Munoz-Eaton Model of Protein Folding

We consider a simplified model of protein folding, with binary degrees of freedom, whose equilibrium thermodynamics is exactly solvable. Based on this exact solution, the kinetics is studied in the framework of a local equilibrium approach, for which we prove that (i) the free energy decreases with time, (ii) the exact equilibrium is recovered in the infinite time limit, and (iii) the folding rate is an upper bound of the exact one. The kinetics is compared to the exact one for a small peptide and to Monte Carlo simulations for a longer protein, then rates are studied for a real protein and a model structure.Comment: 4 pages, 4 figure

Six vertex model with domain-wall boundary conditions in the Bethe-Peierls approximation

We use the Bethe-Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.Comment: 19 pages, 14 figure

Dynamical transition in the TASEP with Langmuir kinetics: mean-field theory

We develop a mean-field theory for the totally asymmetric simple exclusion process (TASEP) with open boundaries, in order to investigate the so-called dynamical transition. The latter phenomenon appears as a singularity in the relaxation rate of the system toward its non-equilibrium steady state. In the high-density (low-density) phase, the relaxation rate becomes independent of the injection (extraction) rate, at a certain critical value of the parameter itself, and this transition is not accompanied by any qualitative change in the steady-state behavior. We characterize the relaxation rate by providing rigorous bounds, which become tight in the thermodynamic limit. These results are generalized to the TASEP with Langmuir kinetics, where particles can also bind to empty sites or unbind from occupied ones, in the symmetric case of equal binding/unbinding rates. The theory predicts a dynamical transition to occur in this case as well.Comment: 37 pages (including 16 appendix pages), 6 figures. Submitted to Journal of Physics

NEW TOPOLOGIES IN THE PHASE DIAGRAM OF THE SEMI-INFINITE BLUME-CAPEL MODEL

The phase diagram of the Blume--Capel model on a semi--infinite simple cubic lattice with a (100) free surface is studied in the pair approximation of the cluster variation method. Six main topologies are found, of which two are new, due to the occurrence of a first order surface transition in the phase with ordered bulk, separating two phases with large and small surface order parameters. The latter is a new phase and is studied in some detail, giving the behaviour of the order parameter profiles in two typical cases. A comparison is made with the results of a low temperature expansion, where these are available, showing a great increase in accuracy with respect to the mean field approximation.Comment: RevTeX, 13 pages + 7 uuencoded PostScript figures (substituted raw with encoded PostScript