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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Random systems
- directed polymers
- extensive quantities
- static disorder