Communications in Mathematical Physics

, Volume 125, Issue 2, pp 337–353 | Cite as

The Ising model and percolation on trees and tree-like graphs

  • Russell Lyons


We calculate the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field. In the same setting, we calculate the critical temperature for spin percolation. The same problems are solved for the diluted models and for more general random interaction strengths. In the case of no interaction, we generalize to percolation on certain tree-like graphs. This last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory.


Neural Network Phase Transition Statistical Physic Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Russell Lyons
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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