### Abstract

We consider stochastic processes, S ^{t} ≡ (S _{x} ^{t}: x ∈ ℤ ^{d}) ∈ script capital L sign _{0} ^{ℤd} with script capital L sign _{0} finite, in which spin flips (i.e., changes of S _{x} ^{t}) do not raise the energy. We extend earlier results of Nanda-Newman-Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.

Original language | English (US) |
---|---|

Pages (from-to) | 431-442 |

Number of pages | 12 |

Journal | Journal of Statistical Physics |

Volume | 110 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 2003 |

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### Keywords

- Absorbing state
- Disordered system
- Energy lowering
- Lyapunov function
- Percolation
- Stochastic Ising model
- Stochastic spin system

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*110*(1-2), 431-442. https://doi.org/10.1023/A:1021039200087

**Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems.** / De Santis, Emilio; Newman, Charles.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 110, no. 1-2, pp. 431-442. https://doi.org/10.1023/A:1021039200087

}

TY - JOUR

T1 - Convergence in Energy-Lowering (Disordered) Stochastic Spin Systems

AU - De Santis, Emilio

AU - Newman, Charles

PY - 2003/1

Y1 - 2003/1

N2 - We consider stochastic processes, S t ≡ (S x t: x ∈ ℤ d) ∈ script capital L sign 0 ℤd with script capital L sign 0 finite, in which spin flips (i.e., changes of S x t) do not raise the energy. We extend earlier results of Nanda-Newman-Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.

AB - We consider stochastic processes, S t ≡ (S x t: x ∈ ℤ d) ∈ script capital L sign 0 ℤd with script capital L sign 0 finite, in which spin flips (i.e., changes of S x t) do not raise the energy. We extend earlier results of Nanda-Newman-Stein that each site x has almost surely only finitely many flips that strictly lower the energy and thus that in models without zero-energy flips there is convergence to an absorbing state. In particular, the assumption of finite mean energy density can be eliminated by constructing a percolation-theoretic Lyapunov function density as a substitute for the mean energy density. Our results apply to random energy functions with a translation-invariant distribution and to quite general (not necessarily Markovian) dynamics.

KW - Absorbing state

KW - Disordered system

KW - Energy lowering

KW - Lyapunov function

KW - Percolation

KW - Stochastic Ising model

KW - Stochastic spin system

UR - http://www.scopus.com/inward/record.url?scp=0037210304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037210304&partnerID=8YFLogxK

U2 - 10.1023/A:1021039200087

DO - 10.1023/A:1021039200087

M3 - Article

VL - 110

SP - 431

EP - 442

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -