Bounding flows for spherical spin glass dynamics

Abstract

We introduce a new approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, we obtain an approximate phase diagram for the evolution of the energy $H$ and its gradient under Langevin dynamics for spherical $p$-spin models. We then derive several consequences of this phase diagram. For example, at any temperature, uniformly over all starting points, the process must reach and remain in an absorbing region of large negative values of $H$ and large (in norm) gradients in order 1 time. Furthermore, if the process starts in a neighborhood of a critical point of $H$ with negative energy, then both the gradient and energy must increase macroscopically under this evolution, even if this critical point is a saddle with index of order $N$. As a key technical tool, we estimate Sobolev norms of spin glass Hamiltonians, which are of independent interest.

arXiv comment32 pages, 6 figures

References in corpus (16)

• Theoretical perspective on the glass transition and amorphous materials
• Analytical Solution of the Off-Equilibrium Dynamics of a Long Range Spin-Glass Model
• Phase space geometry and slow dynamics
• Complexity of random smooth functions on the high-dimensional sphere
• Metastable states in glassy systems
• Aging in the random energy model
• The geometry of the Gibbs measure of pure spherical spin glasses
• Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models
• Dynamical TAP approach to mean field glassy systems
• Low Temperature Asymptotics in Spherical Mean Field Spin Glasses
• Activated Aging Dynamics and Effective Trap Model Description in the Random Energy Model
• Spectral characterization of aging: the rem-like trap model
• Convergence to equilibrium for finite Markov processes, with application to the Random Energy Model
• On the Spectral Gap of Spherical Spin Glass Dynamics
• Convergence of clock processes in random environments and ageing in the p-spin SK model
• Limiting dynamics for spherical models of spin glasses at high temperature

Cited by in corpus (5)

• Algorithmic thresholds for tensor PCA
• Functional renormalization group for multilinear disordered Langevin dynamics I: Formalism and first numerical investigations at equilibrium
• Functional renormalization group for multilinear disordered Langevin dynamics II: Revisiting the $p=2$ spin dynamics for Wigner and Wishart ensembles
• Dynamics for spherical spin glasses: disorder dependent initial conditions
• High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models