T-6

Goal:

Key concepts: gradient descent, rugged landscapes, metastable states, Hessian matrices, random matrix theory, landscape’s complexity.

Problem

Consider the REM discussed in Problems 1. Assume that the $2^{N}$ configurations are organised in an hypercube of connectivity $N$: each configuration has $N$ neighbours, that are obtained flipping one spin of the first configuration.

- Consider now the smallest energy values $E_{\alpha }$ among the $2^{N}$ ones, those with energy density $\epsilon \sim -{\sqrt {\log 2}}$: what is their distribution? (Hint: recall extreme value statistics discussed in Lecture 1)

- Assume to be in a configuration of given (very small) energy $E_{\alpha }$: what is the minimal energy density among the neighbouring configurations? Does it depend on $E_{\alpha }$? In which sense the energy landscape of the REM has a golf course structure?

The trap model is an effective model for the dynamics, which mimics the exploration of energy landscapes in which there are plenty of minima separated by high energy barriers.