Probing the scale-free hierarchy of the $p=3$ spherical spin glass via persistent Langevin dynamics

Abstract

How does a persistent random walker perceive a complex energy landscape? We address this question by studying the persistent Langevin dynamics of the p=3 spherical spin glass, a paradigmatic mean-field model with a scale-free hierarchical landscape. By tuning the persistence time tau_p, which controls the walker's inertia and effectively sets its energy resolution delta E proportional to 1/tau_p, we measure the energy correlation time tau_corr. At temperature T=1.0, we find tau_corr scales as tau_p to the power alpha with alpha = 0.337 plus or minus 0.035 for tau_p in the range [2,32] (for N=64), in excellent agreement with the Kardar-Parisi-Zhang (KPZ) universality class prediction alpha = 1/3. Finite-size scaling using N = 16,32,48,64,128 yields the thermodynamic limit alpha(infinity) = 0.3333 plus or minus 0.0134, fully consistent with 1/3. Thus, tau_p acts as a tunable probe that reveals the predicted scale-free hierarchy of the landscape. Moreover, the temperature dependence alpha(T) for T = 0.5,1.0,1.5,2.0 exhibits a clear U-shaped curve, identifying three dynamical regimes: ballistic/inertial, KPZ, and noise-dominated. Our results establish persistent Langevin dynamics as a powerful tool for uncovering hidden landscape topology and demonstrate that the p=3 spherical spin glass belongs to the KPZ universality class.