Goal:
Complete the characterisation of the energy landscape of the spherical
-spin.
Techniques: saddle point, random matrix theory.
Problems
Problem 6: the Hessian at the stationary points, and random matrix theory
This is a continuation of problem 5. To get the complexity of the spherical
-spin, it remains to compute the expectation value of the determinant of the Hessian matrix: this is the goal of this problem. We will do this exploiting results from random matrix theory.
- Gaussian Random matrices. Show that the matrix
is a GOE matrix, i.e. a matrix taken from the Gaussian Orthogonal Ensemble, meaning that it is a symmetric matrix with distribution
where
is a normalization. What is the value of
?
- Eigenvalue density and concentration. Let
be the eigenvalues of the matrix
. Show that the following identity holds:
where
is the empirical eigenvalue distribution. It can be shown that if
is a GOE matrix, the distribution of the empirical distribution has a large deviation form (recall TD1) with speed
, meaning that
where now
is a functional. Using a saddle point argument, show that this implies
where
is the typical value of the eigenvalue density, which satisfies
.
- The semicircle and the complexity. The eigenvalue density of GOE matrices is self-averaging, and it equals to
- Check this numerically: generate matrices for various values of
, plot their empirical eigenvalue density and compare with the asymptotic curve. Is the convergence faster in the bulk, or in the edges of the eigenvalue density, where it vanishes?
- Combining all the results, show that the annealed complexity is
The integral
can be computed explicitly, and one finds:
Plot the annealed complexity, and determine numerically where it vanishes: why is this a lower bound or the ground state energy density?
- The threshold and the stability.
Sketch
for different values of
; recalling that the Hessian encodes for the stability of the stationary points, show that there is a transition in the stability of the stationary points at the critical value of the energy density
When are the critical points stable local minima? When are they saddles? Why the stationary points at
are called marginally stable ?
Check out: key concepts
Metastable states, Hessian matrices, random matrix theory, landscape’s complexity.