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Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists

Authors
Journal
Physica A Statistical Mechanics and its Applications
0378-4371
Publisher
Elsevier
Publication Date
Volume
391
Issue
12
Identifiers
DOI: 10.1016/j.physa.2012.01.038
Keywords
  • Rigorous Results In Statistical Mechanics
  • Solvable Lattice Models
  • Exact Results
  • Message-Passing Algorithms
Disciplines
  • Mathematics
  • Physics

Abstract

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.

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