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?