0 40 80 02 v 1 2 A ug 2 00 4 spin - glass stochastic stability : a rigorous proof

We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spin-glass quenched state. We show that stochastic stability holds for both the SherringtonKirkpatrick model in terms of the square of the overlap function and for the EdwardsAnderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is at least V −1/2. The conjecture of differentiability w.r.t. the temperature of the quenched state is still open and we show that its proof would be equivalent to a rate of V −1.