Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists

Abstract

We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community.

10 pages, 2 figures

References in corpus (1)

• Critical phenomena in complex networks

Cited by in corpus (52)

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• Evolutionary potential games on lattices
• On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field
• A universal tensor network algorithm for any infinite lattice
• Quantum correlations at infinite temperature: the dynamical Nagaoka effect
• Some Relativistic and Gravitational Properties of the Wolfram Model
• Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree
• Anderson Localization on the Bethe Lattice using Cages and the Wegner Flow
• Quantum Walks on Trees with Disorder: Decay, Diffusion, and Localization
• On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree
• QSWalk: a Mathematica package for quantum stochastic walks on arbitrary graphs
• Quantum Markov States on Cayley trees
• Featureless quantum spin liquid, 1/3-magnetization plateau state and exotic thermodynamic properties of spin-1/2 frustrated Heisenberg antiferromagnet on an infinite Husimi lattice
• Renormalization method in $p$-adic $\lambda$-model on the Cayley tree
• First encounters on Bethe Lattices and Cayley Trees
• Heisenberg antiferromagnet on the Husimi lattice
• Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
• Phase diagram of the isotropic spin-3/2 model on the z=3 Bethe lattice
• Percolation Thresholds in Hyperbolic Lattices
• A counterintuitive role of geometry in transport by quantum walks
• Mapping the $q$-voter model: From a single chain to complex networks
• Analysis of fluctuations in the first return times of random walks on regular branched networks
• On Ground States and Phase Transition for $\lambda$-Model with the Competing Potts Interactions on Cayley Trees
• Gibbs Measures with memory of length 2 on an arbitrary order Cayley tree
• Refinement of quantum Markov states on trees
• Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices
• Cooperative sequential adsorption models on a Cayley tree: analytical results and applications
• Thermal Tensor Network Simulations of the Heisenberg Model on the Bethe Lattice
• The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid
• Polymer models with competing collapse interactions on Husimi and Bethe lattices
• Phase diagrams of lattice models on Cayley tree and chandelier network: a review
• Local cluster-size statistics in the critical phase of bond percolation on the Cayley tree
• Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process
• Surface growth on treelike lattices and the upper critical dimension of the KPZ class
• Quantum annealing of Cayley-tree Ising spins at small scales
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs
• Universality in minor-closed graph classes
• Thermal Transport in Phononic Cayley Tree Networks
• Extended Falicov-Kimball model at weak onsite and intersite Coulomb interactions
• Entanglement Entropy on the Cayley Tree
• Ising Spins on Randomly Multi-Branched Husimi Square Lattice: Thermodynamics and Phase Transition in Cross-dimensional Range
• Finite temperature quantum condensations in the space of states: General Proof
• Magnetization of Antiferromagnetic Cactus Graph Model with Exact Dimer Ground State
• Exact results for the Ising model on a small-world network
• Inferring Tree Structure with Hidden Traps from First Passage Times
• Linked cluster expansion on trees
• Modelling brain-wide neuronal morphology via rooted Cayley trees
• The general Albertson irregularity index of graphs
• Harmonic oscillator kicked by spin measurements: a Floquet-like system without classical analogous
• Semianalytical solutions of Ising-like and Potts-like magnetic polymers on the Bethe lattice
• An introduction to stochastic self-assembly: theory, simulation and experimental applications