Equivalence of statistical-mechanical and dynamic descriptions of the infinite-range Ising spin-glass

We prove the equivalence between the dynamic mean-field theory of the Ising spin-glass and the statistical-mechanical theory of Thouless, Anderson, and Palmer (TAP). Individual low-free-energy TAP solutions describe short-time properties, whereas thermodynamic equilibrium corresponds to averaging over all such solutions. The square of the staggered magnetization associated with the largest eigenvalue of the interaction matrix scales as $N^{\frac{5}{6}}$ ($N$ is the number of spins). Results are confirmed by Monte Carlo simulation and numerical solution of the TAP equations.