Loop expansion around the Bethe approximation through the $M$-layer construction

Abstract

For every physical model defined on a generic graph or factor graph, the Bethe $M$-layer construction allows building a different model for which the Bethe approximation is exact in the large $M$ limit and it coincides with the original model for $M=1$. The $1/M$ perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat-diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice but are either qualitatively different or absent in the corresponding fully connected case. In this case the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters close to the Bethe critical point where strong deviations from mean-field behavior will be observed. In this region, the $1/M$ expansion for the corrections diverges and it can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated associating Gaussian propagators to its lines as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model.

References in corpus (17)

• Loop series for discrete statistical models on graphs
• Energy exponents and corrections to scaling in Ising spin glasses
• Loop Calculus in Statistical Physics and Information Science
• How to Compute Loop Corrections to Bethe Approximation
• Anderson model on Bethe lattices: density of states, localization properties and isolated eigenvalue
• Thermodynamic glass transition in a spin glass without time-reversal symmetry
• The three dimensional Ising spin glass in an external magnetic field: the role of the silent majority
• Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function
• Dynamical transition in the D = 3 Edwards-Anderson spin glass in an external magnetic field
• Non-backtracking random walk
• Finite size corrections to disordered Ising models on Random Regular Graphs
• Diluted Mean-Field Spin-Glass Models at Criticality
• On the finite size corrections to some random matching problems
• On Cavity Approximations for Graphical Models
• One-dimensional disordered Ising models by replica and cavity methods
• Sums over geometries and improvements on the mean field approximation
• A Multispin Algorithm for the Kob-Andersen Stochastic Dynamics on Regular Lattices

Cited by in corpus (18)

• Universality in Anderson localization on random graphs with varying connectivity
• Solvable Models of Supercooled Liquids in Three Dimensions
• Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature
• The fate of the bootstrap percolation hybrid critical point in finite dimension
• Solving the fully-connected spherical $p$-spin model with the cavity method: equivalence with the replica results
• New loop expansion for the Random Magnetic Field Ising Ferromagnets at zero temperature
• Spin-glass dynamics: experiment, theory and simulation
• One-loop topological expansion for spin glasses in the large connectivity limit
• Dynamics of sparse Boolean networks with multi-node and self-interactions
• Bethe $M$-layer construction for the percolation problem
• The Ising spin glass on random graphs at zero temperature: not all spins are glassy in the glassy phase
• Analytical solution to Heisenberg spin glass models on sparse random graphs and their de Almeida-Thouless line
• Thermodynamics of the Fredrickson-Andersen Model on the Bethe Lattice
• Bethe $M$-layer construction on the Ising model
• Loop corrections in spin models through density consistency
• Critical exponents of the spin glass transition in a field at zero temperature
• Theory of Kinetically-Constrained-Models Dynamics
• Long-range spin glass in a field at zero temperature