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The spin glass behavior near zero temperature is a complicated matter. To getan easier access to the spin glass order parameter $Q(x)$ and, at the sametime, keep track of $Q_{ab}$, its matrix aspect, and hence of the Hessiancontrolling stability, we investigate an expansion of the replicated freeenergy functional around its ``spherical'' approximation. This expansion isobtained by introducing a constraint-field and a (double) Legendre Transformexpressed in terms of spin correlators and constraint-field correlators. Thespherical approximation has the spin fluctuations treated with a globalconstraint and the expansion of the Legendre Transformed functional brings themcloser and closer to the Ising local constraint. In this paper we examine thefirst contribution of the systematic corrections to the spherical startingpoint.

Low temperature spin glass fluctuations: expanding around a spherical approximation