T-5

Goal: So far we have discussed the equilibrium properties of disordered systems, that are encoded in their partition function and free energy. In this set of problems, we characterize the energy landscape of a prototypical model, the spherical $p$-spin.

Key concepts: Langevin dynamics, gradient descent, oout-of-equilibrium dynamics, metastable states, Hessian matrices, random matrix theory.

Dynamics, optimization, trapping local minima

• Energy landscapes. Consider the spherical $p$-spin model; The function $E({\vec {\sigma }})$ defines the energy landscape of the model: this is a random function defined on configuration space, which is the space all configurations ${\vec {\sigma }}$ belong to. This landscape has its global minima in the ground state configurations: the energy density of the ground states can be obtained studying the partition function $Z$ in the limit $\beta \to \infty$. Besides the ground state(s), the energy landscape can have other local minima; the models of glasses are characterize by the fact that there are plenty of these local minima, see the sketch.


• Gradient descent and stationary points. Suppose that we are interested in finding the configurations of minimal energy of some model with energy landscape $E({\vec {\sigma }})$, starting from an arbitrary initial configuration ${\vec {\sigma }}_{0}$: we can think about a dynamics in which we progressively update the configuration of the system moving towards lower and lower values of the energy, hoping to eventually converge to the ground state(s). The simplest dynamics of this sort is gradient descent, where the configurations change in time moving in the direction of the gradient of the energy landscape: ${\frac {d{\vec {\sigma }}(t)}{dt}}=-\nabla E({\vec {\sigma }})$

  Under this dynamics, the



• Noise, Langevin dynamics and activation.

- moves towards configs where energy is smaller. dynamics is stuck when reach a conf stationary point (local minima, max, saddles). If the landscape is convex, then go to GS. If not, can end up in other stationary point, in particular local minima.

- can add a bit of noise: Langevin dynamics.