Upper bounds on susceptibilities and correlations of quenched impurity systems

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Abstract
It is shown that for any dynamical quantity phi associated with an impurity system the variance, var ( phi )p calculated after averaging the partition function, is greater than or equal to the corresponding variance var ( phi )q calculated according to the usual quenched prescription in which the free energy is averaged. In particular, if chi p, chi q denote the respective isothermal susceptibilities subject to the same conditions, then chi p>or= chi q. This is a general result valid for classical and quantum systems in which the susceptibility is proportional to a variance, and it implies that if a quenched spin system displays a phase transition at a temperature of Tcq at which a certain variance, say the susceptibility chi q diverges, then chi p must also be infinite at the same temperature. It is also shown that for a large class of spin S ferromagnetic Ising systems which satisfy the conditions of Griffiths' correlation inequalities and other specified conditions, the covariance between any two spins si and sj is similarly ordered, i.e. cov (si,sj)p>or=cov (si,sj)q.

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