Bounding flows for spherical spin glass dynamics

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.