### Abstract

An extensive quantity is a family of functions Ψ_{v} of random parameters, indexed by the finite regions V (subsets of ℤ^{d}) over which Ψ_{v} are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim Ψ_{v}/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume of V. Of particular interest is the free energy of a system with random couplings; for such Ψ_{v} bounds are derived also for the generating function E(e^{tΨ}). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.

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

Pages (from-to) | 287-306 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 60 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1990 |

Externally published | Yes |

### Fingerprint

### Keywords

- directed polymers
- extensive quantities
- fluctuations
- Random systems
- static disorder

### ASJC Scopus subject areas

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

### Cite this

*Journal of Statistical Physics*,

*60*(3-4), 287-306. https://doi.org/10.1007/BF01314921

**Fluctuations of extensive functions of quenched random couplings.** / Wehr, Jan; Aizenman, Michael.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 60, no. 3-4, pp. 287-306. https://doi.org/10.1007/BF01314921

}

TY - JOUR

T1 - Fluctuations of extensive functions of quenched random couplings

AU - Wehr, Jan

AU - Aizenman, Michael

PY - 1990/8

Y1 - 1990/8

N2 - An extensive quantity is a family of functions Ψv of random parameters, indexed by the finite regions V (subsets of ℤd) over which Ψv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim Ψv/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume of V. Of particular interest is the free energy of a system with random couplings; for such Ψv bounds are derived also for the generating function E(etΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.

AB - An extensive quantity is a family of functions Ψv of random parameters, indexed by the finite regions V (subsets of ℤd) over which Ψv are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim Ψv/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume of V. Of particular interest is the free energy of a system with random couplings; for such Ψv bounds are derived also for the generating function E(etΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.

KW - directed polymers

KW - extensive quantities

KW - fluctuations

KW - Random systems

KW - static disorder

UR - http://www.scopus.com/inward/record.url?scp=0000733381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000733381&partnerID=8YFLogxK

U2 - 10.1007/BF01314921

DO - 10.1007/BF01314921

M3 - Article

AN - SCOPUS:0000733381

VL - 60

SP - 287

EP - 306

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -