### 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) |
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Pages (from-to) | 287-306 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 60 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1 1990 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

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

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## Cite this

*Journal of Statistical Physics*,

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