### Abstract

The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

Original language | English |
---|---|

Pages (from-to) | 1658-1683 |

Number of pages | 26 |

Journal | Journal of Statistical Physics |

Volume | 160 |

Issue number | 6 |

DOIs | |

Publication status | Published - 14 Sep 2015 |

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### Keywords

- Lattice system
- Low-temperature Gibbs measure
- XY models

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*160*(6), 1658-1683. https://doi.org/10.1007/s10955-015-1288-8

}

*Journal of Statistical Physics*, vol. 160, no. 6, pp. 1658-1683. https://doi.org/10.1007/s10955-015-1288-8

**Sensitive Dependence of Gibbs Measures at Low Temperatures.** / Coronel, Daniel; Rivera-Letelier, Juan.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Sensitive Dependence of Gibbs Measures at Low Temperatures

AU - Coronel, Daniel

AU - Rivera-Letelier, Juan

PY - 2015/9/14

Y1 - 2015/9/14

N2 - The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

AB - The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

KW - Lattice system

KW - Low-temperature Gibbs measure

KW - XY models

UR - http://www.scopus.com/inward/record.url?scp=84939262225&partnerID=8YFLogxK

U2 - 10.1007/s10955-015-1288-8

DO - 10.1007/s10955-015-1288-8

M3 - Article

AN - SCOPUS:84939262225

VL - 160

SP - 1658

EP - 1683

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -