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

TL;DR

This paper analyzes the low temperature Langevin dynamics in spherical p-spin models, confirming the predicted asymptotic energy and establishing bounds using optimization hardness and local maximum properties.

Contribution

It provides rigorous proofs for the asymptotic energy levels of Langevin dynamics in spherical p-spin glasses, confirming long-standing predictions.

Findings

Asymptotic energy matches the predicted value $E_{} (p)$.

Upper bounds derived from Lipschitz optimization hardness.

Lower bounds show dynamics stay above the lowest local maximum energy.

Abstract

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.