  We study the Langevin dynamics for spherical $p$-spin models, focusing on the
short time regime described by the Cugliandolo-Kurchan equations. Confirming a
prediction of [Cugliandolo-Kurchan, Phys. Rev. Lett. 1993], we show the
asymptotic energy achieved is exactly $E_{\infty}(p)=2\sqrt{\frac{p-1}{p}}$ in
the low temperature limit. The upper bound uses hardness results for Lipschitz
optimization algorithms and applies for all temperatures. For the lower bound,
we prove the dynamics reaches and stays above the lowest energy of any
approximate local maximum. In fact the latter behavior holds for any
Hamiltonian obeying natural smoothness estimates, even with disorder-dependent
initialization and on exponential time-scales.
