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We study the Langevin dynamics for spherical $p$-spin models, focusing on theshort time regime described by the Cugliandolo-Kurchan equations. Confirming a1999 conjecture of Biroli, we show the asymptotic energy achieved is exactly$E_{\infty}(p)=2\sqrt{\frac{p-1}{p}}$ in the low temperature limit. The upperbound uses hardness results for Lipschitz optimization algorithms and appliesfor all temperatures. For the lower bound, we prove the dynamics reaches andstays above the lowest energy of any approximate local maximum. In fact thelatter behavior holds for any Hamiltonian obeying natural smoothness estimates,even with disorder-dependent initialization and on exponential time-scales.

The Threshold Energy of Low Temperature Langevin Dynamics for Pure Spherical Spin Glasses