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We show that the only solutions of the TAP equations for theSherrington-Kirkpatrick model of Ising spin glasses which can be found byiteration are those whose free energy lies on the border between replicasymmetric and broken replica symmetric states, when the number of spins $N$ islarge. Convergence to this same borderline also happens in quenches from a hightemperature initial state to a locally stable state where each spin is parallelto its local field; both are examples of self-organized criticality. At thisborderline the band of eigenvalues of the Hessian associated with a solutionextends to zero, so the states reached have marginal stability. We have alsoinvestigated the factors which determine the free energy difference between astationary solution corresponding to a saddle point and its associated minimum,which is the barrier which has to be surmounted to escape from the vicinity ofa TAP minimum or pure state.

The realizable solutions of the TAP equations