T-6
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
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.
- 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 What is the value of ?
- Eigenvalue density and concentration. Let be the eigenvalues of the matrix 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 M }
. Show that the following identity holds:
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{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 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_{N}(\lambda)} is the empirical eigenvalue density. It can be shown that if 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 M } is a GOE matrix, the distribution of the empirical density 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 where now 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[\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^{\text{typ}}(\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 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 } .
- 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
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:
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?
- The threshold and the stability. Sketch 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 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.