Communications in Mathematical Physics

, Volume 55, Issue 1, pp 37–45 | Cite as

In one and two dimensions, every stationary measure for a stochastic Ising Model is a Gibbs state

  • R. A. Holley
  • D. W. Stroock


It is shown that one and two dimensional (generalized) stochastic Ising models with finite range potentials have only Gibbs states as their stationary measures. This is true even if the stationary measure or the potential is not translation invariant. This extends previously known results which are restricted to translation invariant stationary measures and potentials. In particular if the potential has only one Gibbs state the stochastic Ising Model must be ergodic.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Stationary Measure 
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  1. 1.
    Higuchi, Y., Shiga, T.: Some results on Markov processes in infinite lattice spin systems. J. Math. Kyoto Univ.15, 211–229 (1975)Google Scholar
  2. 2.
    Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. math. Phys.23, 87–99 (1971)Google Scholar
  3. 3.
    Holley, R.: Recent results on the stochastic Ising model. Rocky Mountain J. Math.4, 479–496 (1974)Google Scholar
  4. 4.
    Holley, R., Stroock, D.: A martingale approach to infinite systems of interacting process. Ann. Prob.4, 195–228 (1976)Google Scholar
  5. 5.
    Holley, R., Stroock, D.: Applications of the stochastic Ising model to the Gibbs states. Commun. math. Phys.48, 249–265 (1976)Google Scholar
  6. 6.
    Holley, R., Stroock, D.:L 2 theory for the stochastic Ising model. Z. Wahrscheinlichkeitstheorie verw. Geb.35, 87–101 (1976)Google Scholar
  7. 7.
    Moulin Ollagnier, J., Pinchon, D.: Free energy in spin-flip processes is non-increasing. Commun. math. Phys.55, 29–35 (1977)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • R. A. Holley
    • 1
  • D. W. Stroock
    • 1
  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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