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

To get the complexity, 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.

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 $P(M)=Z_{N}^{-1}e^{-{\frac {N}{4\sigma ^{2}}}{\text{Tr}}M^{2}}.$ 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:

   ${\overline {|{\text{det}}\left(M-p\epsilon \mathbb {I} \right)|}}={\overline {{\text{exp}}\left[(N-1)\left(\int d\lambda \,\rho _{N}(\lambda )\,\log |\lambda -p\epsilon |\right)\right]}},\quad \quad \rho _{N}(\lambda )={\frac {1}{N-1}}\sum _{\alpha =1}^{N-1}\delta (\lambda -\lambda _{\alpha })$

   where $\rho _{N}(\lambda )$ is the empirical eigenvalue density. It can be shown that if $M$ is a GOE matrix, the distribution of the empirical density has a large deviation form (recall TD1) 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

   ${\overline {{\text{exp}}\left[(N-1)\left(\int d\lambda \,\rho _{N}(\lambda )\,\log |\lambda -p\epsilon |\right)\right]}}={\text{exp}}\left[N\left(\int d\lambda \,\rho ^{\text{typ}}(\lambda +p\epsilon )\,\log |\lambda |\right)+o(N)\right]$

   where $\rho ^{\text{typ}}(\lambda )$ is the typical value of the eigenvalue density, which satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g[\rho^{\text{typ}}]=0 } .

3. The semicircle and the complexity. The eigenvalue density of GOE matrices is self-averaging, and it equals to

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N \to \infty}\rho_N (\lambda)=\lim_{N \to \infty} \overline{\rho_N}(\lambda)= \rho^{\text{typ}}(\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{\text{a}}(\epsilon)= \frac{1}{2}\log [4 e (p-1)]- \frac{\epsilon^2}{2}+ I_p(\epsilon), \quad \quad I_p(\epsilon)= \frac{2}{\pi}\int d x \sqrt{1-\left(x- \frac{\epsilon}{ \epsilon_{\text{th}}}\right)^2}\, \log |x| , \quad \quad \epsilon_{\text{th}}= -2\sqrt{\frac{p-1}{p}}. }

     The integral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_p(\epsilon)} can be computed explicitly, and one finds:

     $I_{p}(\epsilon )={\begin{cases}&{\frac {\epsilon ^{2}}{\epsilon _{\text{th}}^{2}}}-{\frac {1}{2}}-{\frac {\epsilon }{\epsilon _{\text{th}}}}{\sqrt {{\frac {\epsilon ^{2}}{\epsilon _{\text{th}}^{2}}}-1}}+\log \left({\frac {\epsilon }{\epsilon _{\text{th}}}}+{\sqrt {{\frac {\epsilon ^{2}}{\epsilon _{\text{th}}^{2}}}-1}}\right)-\log 2\quad {\text{if}}\quad \epsilon \leq \epsilon _{\text{th}}\\&{\frac {\epsilon ^{2}}{\epsilon _{\text{th}}^{2}}}-{\frac {1}{2}}-\log 2\quad {\text{if}}\quad \epsilon >\epsilon _{\text{th}}\end{cases}}$

     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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{\text{typ}}(\lambda+p \epsilon) } for different values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } ; 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 $\epsilon _{\text{th}}=-2{\sqrt {(p-1)/p}}.$ When are the critical points stable local minima? When are they saddles? Why the stationary points at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon= \epsilon_{\text{th}}} are called marginally stable ?

Check out: key concepts

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