The Sherrington–Kirkpatrick model near Tc and near T=0

Some recent results concerning the Sherrington-Kirkpatrick model are reported. For T near the critical temperature T-c, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in powers of delta Q(ab) = (Q(ab) - Q(ab)(RS)) about the replica symmetric solution Q(ab)(RS). The expansion is kept up to fourth order in delta Q where a Parisi solution Q(ab) = Q(x) emerges, but only if one remains close enough to T-c. For T near zero we show how to separate contributions from x << T << 1 where the Hessian maintains the standard structure of Parisi replica symmetry breaking with bands of eigenvalues bounded below by zero modes. For T << x <= 1 the bands collapse and only two eigenvalues, a null one and a positive one, are found. In this region the solution stands in what can be called a droplet-like regime.