T-6

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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 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 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 ?



  1. 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 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 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 .



  1. 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

      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?


  1. 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 are called marginally stable ?


Check out: key concepts

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