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To test the stability of the Parisi solution near T=0, we study the spectrumof the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvaluesare the masses of the bare propagators in the expansion around the mean-fieldsolution. In the limit $T\ll 1$ two regions can be identified. In the firstregion, for $x$ close to 0, where $x$ is the Parisi replica symmetry breakingscheme parameter, the spectrum of the Hessian is not trivial and maintains thestructure of the full replica symmetry breaking state found at highertemperatures. In the second region $T\ll x \leq 1$ as $T\to 0$, the componentsof the Hessian become insensitive to changes of the overlaps and the bandstypical of the full replica symmetry breaking state collapse. In this regiononly two eigenvalues are found: a null one and a positive one, ensuringstability for $T\ll 1$. In the limit $T\to 0$ the width of the first regionshrinks to zero and only the positive and null eigenvalues survive. Asbyproduct we enlighten the close analogy between the static Parisi replicasymmetry breaking scheme and the multiple time-scales approach of dynamics, andcompute the static susceptibility showing that it equals the static limit ofthe dynamic susceptibility computed via the modified fluctuation dissipationtheorem.

Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T=0