Skip to main content
SpringerLink
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. Holley, R.: Free energy in a Markovian model of a lattice spin system. Commun. math. Phys.23, 87–99 (1971)

    Google Scholar 

  3. Holley, R.: Recent results on the stochastic Ising model. Rocky Mountain J. Math.4, 479–496 (1974)

    Google Scholar 

  4. Holley, R., Stroock, D.: A martingale approach to infinite systems of interacting process. Ann. Prob.4, 195–228 (1976)

    Google Scholar 

  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. Holley, R., Stroock, D.:L 2 theory for the stochastic Ising model. Z. Wahrscheinlichkeitstheorie verw. Geb.35, 87–101 (1976)

    Google Scholar 

  7. Moulin Ollagnier, J., Pinchon, D.: Free energy in spin-flip processes is non-increasing. Commun. math. Phys.55, 29–35 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Colorado, 80309, Boulder, Colorado, USA

    R. A. Holley & D. W. Stroock

Authors
  1. R. A. Holley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. W. Stroock
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J. L. Lebowitz

Research supported in part by N.S.F. Grant MPS 74-18926

Alfred P. Sloan Fellow

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holley, R.A., Stroock, D.W. In one and two dimensions, every stationary measure for a stochastic Ising Model is a Gibbs state. Commun.Math. Phys. 55, 37–45 (1977). https://doi.org/10.1007/BF01613147

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01613147

Keywords

Search

Navigation