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.

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

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

   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 (recall TD1) with speed 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 N^2 } , meaning that 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 P_N[\rho] = e^{-N^2 \, g[\rho]} } where now $g[\cdot ]$ is a functional. Using a saddle point argument, show that this implies

   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 \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_\infty(\lambda+p \epsilon) \, \log |\lambda|\right)+ o(N) \right] }

   where 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) } is the typical value of the eigenvalue density, which satisfies $g[\rho ^{\text{typ}}]=0$.

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 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 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 $\Sigma _{\text{a}}(\epsilon )={\frac {1}{2}}\log[4e(p-1)]-{\frac {\epsilon ^{2}}{2}}+I_{p}(\epsilon ),\quad \quad I_{p}(\epsilon )={\frac {2}{\pi }}\int dx{\sqrt {1-\left(x-{\frac {\epsilon }{\epsilon _{\text{th}}}}\right)^{2}}}\,\log |x|,\quad \quad \epsilon _{\text{th}}=-2{\sqrt {\frac {p-1}{p}}}.$

     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:

     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)= \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 $\rho ^{\text{typ}}(\lambda +p\epsilon )$ 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 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_{\text{th}}= -2\sqrt{(p-1)/p}. } When are the critical points stable local minima? When are they saddles? Why the stationary points at $\epsilon =\epsilon _{\text{th}}$ are called marginally stable ?

Check out: key concepts

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