How to Compute Loop Corrections to Bethe Approximation

Abstract

We introduce a method for computing corrections to Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well-known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.

16 pages, 2 eps figures

References in corpus (1)

• The high temperature region of the Viana-Bray diluted spin glass model

Cited by in corpus (51)

• Critical phenomena in complex networks
• Loop series for discrete statistical models on graphs
• The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods
• Loop Calculus in Statistical Physics and Information Science
• On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
• Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
• Belief propagation for networks with loops
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Networking - A Statistical Physics Perspective
• Finite size correction to the spectrum of regular random graphs: an analytical solution
• Loop expansion around the Bethe-Peierls approximation for lattice models
• On the number of circuits in random graphs
• Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results
• Partition function loop series for a general graphical model: free energy corrections and message-passing equations
• Tensor Networks contraction and the Belief Propagation algorithm
• Finite size corrections to disordered Ising models on Random Regular Graphs
• Spectral density of random graphs with topological constraints
• Loop expansion around the Bethe approximation through the $M$-layer construction
• Distribution of shortest cycle lengths in random networks
• Finite size corrections to disordered systems on Erdös-Rényi random graphs
• Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart
• On $k$-Core Percolation in Four Dimensions
• Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model
• A mean field method with correlations determined by linear response
• Stochastic optimization by message passing
• Block belief propagation algorithm for two-dimensional tensor networks
• Communication and correlation among communities
• On Cavity Approximations for Graphical Models
• Large Deviations of Correlation Functions in Random Magnets
• Replica methods for loopy sparse random graphs
• Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices
• Resummed mean-field inference for strongly coupled data
• Simulating quantum dynamics in two-dimensional lattices with tensor network influence functional belief propagation
• Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models
• Boltzmann distribution of free energies in a finite-connectivity spin-glass system and the cavity approach
• Random-link matching problems on random regular graphs
• Sums over geometries and improvements on the mean field approximation
• Long-range frustration in T=0 first-step replica-symmetry-broken solutions of finite-connectivity spin glasses
• Loop corrections in spin models through density consistency
• Computing loop corrections by message passing
• Susceptibility Propagation for Constraint Satisfaction Problems
• Ising spin glass models versus Ising models: an effective mapping at high temperature III. Rigorous formulation and detailed proof for general graphs
• Non-convex image reconstruction via Expectation Propagation
• Cycle-based Cluster Variational Method for Direct and Inverse Inference
• Anomalous finite size corrections in random field models
• Statistical physics of loopy interactions: Independent-loop approximation and beyond
• Loop-corrected belief propagation for lattice spin models
• Correcting beliefs in the mean-field and Bethe approximations using linear response
• Decimation flows in constraint satisfaction problems
• Gauge-free cluster variational method by maximal messages and moment matching
• On one-step replica symmetry breaking in the Edwards-Anderson spin glass model