### Abstract

In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.

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

Article number | 123301 |

Journal | Journal of Mathematical Physics |

Volume | 48 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2007 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Mathematical Physics*,

*48*(12), [123301]. https://doi.org/10.1063/1.2812417

**Large deviations and Chernoff bound for certain correlated states on a spin chain.** / Hiai, Fumio; Mosonyi, M.; Ogawa, Tomohiro.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 48, no. 12, 123301. https://doi.org/10.1063/1.2812417

}

TY - JOUR

T1 - Large deviations and Chernoff bound for certain correlated states on a spin chain

AU - Hiai, Fumio

AU - Mosonyi, M.

AU - Ogawa, Tomohiro

PY - 2007

Y1 - 2007

N2 - In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.

AB - In this paper we extend the results of Lenci and Rey-Bellet [J. Stat. Phys. 119, 715 (2005)] on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods, the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state, the spectral theory of positive maps is applied to prove the full large deviation principle.

UR - http://www.scopus.com/inward/record.url?scp=37649013048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37649013048&partnerID=8YFLogxK

U2 - 10.1063/1.2812417

DO - 10.1063/1.2812417

M3 - Article

AN - SCOPUS:37649013048

VL - 48

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

M1 - 123301

ER -