T-6

Goal: Complete the characterisation of the energy landscape of the spherical $p$-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 $p$-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 discussion in the Tutorial and Exercise 4 .

1. Gaussian Random matrices. Show that the matrix $M$ is a GOE matrix, i.e. a matrix taken from the Gaussian Orthogonal Ensemble, meaning that it is a symmetric matrix with distribution $P_N(M)=Z_N^{-1}\text{exp}(-\frac{N}{4{\sigma }^2}\text{Tr}{M}^2)$ where $Z_N$ is a normalization. What is the value of ${\sigma }^2$?

2. Eigenvalue density and concentration. Let ${\lambda }_{\alpha }$ be the eigenvalues of the matrix $M$. Show that the following identity holds: $$\mathbb{𝔼}[|\text{det}(M-pϵ\mathbb{𝕀})|]=\mathbb{𝔼}[\text{exp}[(N-1)(\int d\lambda {\rho }_{N-1}(\lambda )\log |\lambda -pϵ|)]],{\rho }_{N-1}(\lambda )=\frac{1}{N-1}\sum _{\alpha =1}^{N-1}\delta (\lambda -{\lambda }_{\alpha })$$ where ${\rho }_{N-1}(\lambda )$ is the empirical eigenvalue distribution. It can be shown that if $M$ is a GOE matrix, the distribution of the empirical distribution has a large deviation form with speed $N^2$, meaning that $P_N[\rho ]=e^{-N^{2}g[\rho ]}$ where now $g[\cdot ]$ is a functional. Using a saddle point argument, show that this implies $$\mathbb{𝔼}[\text{exp}[(N-1)(\int d\lambda {\rho }_{N-1}(\lambda )\log |\lambda -pϵ|)]]=\text{exp}[N(\int d\lambda {\rho }_{\mathrm{\infty }}(\lambda +pϵ)\log |\lambda |)+o(N)]$$ where ${\rho }_{\mathrm{\infty }}(\lambda )$ is the typical value of the eigenvalue density, which satisfies $g[{\rho }_{\mathrm{\infty }}]=0$.

3. The semicircle and the complexity. The eigenvalue density of GOE matrices is self-averaging, and it equals to $$\underset{N\to \mathrm{\infty }}{\lim }{\rho }_N(\lambda )=\underset{N\to \mathrm{\infty }}{\lim }\overline{{\rho }_N}(\lambda )={\rho }_{\mathrm{\infty }}(\lambda )=\frac{1}{2\pi {\sigma }^2}\sqrt{4{\sigma }^2-{\lambda }^2}$$
   • Check this numerically: generate matrices for various values of $N$, 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 $${\mathrm{\Sigma }}_{\text{a}}(ϵ)=\frac{1}{2}\log [4e(p-1)]-\frac{{ϵ}^2}{2}+I_p(ϵ),I_p(ϵ)=\frac{2}{\pi }\int dx\sqrt{1-{(x-\frac{ϵ}{{ϵ}_{\text{th}}})}^2}\log |x|,{ϵ}_{\text{th}}=-2\sqrt{\frac{p-1}{p}}.$$ The integral $I_p(ϵ)$ can be computed explicitly, and one finds: $$I_p(ϵ)=\{\begin{array}{ll}& \frac{{ϵ}^2}{{ϵ}_{\text{th}}^2}-\frac{1}{2}-\frac{ϵ}{{ϵ}_{\text{th}}}\sqrt{\frac{{ϵ}^2}{{ϵ}_{\text{th}}^2}-1}+\log (\frac{ϵ}{{ϵ}_{\text{th}}}+\sqrt{\frac{{ϵ}^2}{{ϵ}_{\text{th}}^2}-1})-\log 2\text{if}ϵ\le {ϵ}_{\text{th}}\\ & \frac{{ϵ}^2}{{ϵ}_{\text{th}}^2}-\frac{1}{2}-\log 2\text{if}ϵ>{ϵ}_{\text{th}}\end{array}$$ Plot the annealed complexity, and determine numerically where it vanishes: why is this a lower bound or the ground state energy density?

4. The threshold and the stability. Sketch ${\rho }_{\mathrm{\infty }}(\lambda +pϵ)$ for different values of $ϵ$; 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 ${ϵ}_{\text{th}}=-2\sqrt{(p-1)/p}.$ When are the critical points stable local minima? When are they saddles? Why the stationary points at $ϵ={ϵ}_{\text{th}}$ are called marginally stable ?

Check out: key concepts

Metastable states, Hessian matrices, random matrix theory, landscape’s complexity.