Abstract
An extensive quantity is a family of functionsΨ v of random parameters, indexed by the finite regionsV (subsets of ℤd) over whichΨ 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 limΨ 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 ofV. Of particular interest is the free energy of a system with random couplings; for suchΨ v bounds are derived also for the generating functionE(e tΨ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.
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Wehr, J., Aizenman, M. Fluctuations of extensive functions of quenched random couplings. J Stat Phys 60, 287–306 (1990). https://doi.org/10.1007/BF01314921
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DOI: https://doi.org/10.1007/BF01314921