T-5

Goal: So far we have discussed the equilibrium properties of disordered systems, that are encoded in their partition function and free energy. In this set of problems, we characterize the energy landscape of the spherical $p$-spin, by determining the number of its stationary points.

Key concepts: gradient descent, out-of-equilibrium dynamics, metastable states, Hessian matrices, random matrix theory, Langevin dynamics,?

Dynamics, optimization, trapping local minima

• Energy landscapes. Consider the spherical 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} -spin model discussed in the Problems 2 and 3; The function $E({\vec {\sigma }})$ is an energy landscape: it is a random function defined on configuration space, which is the space all configurations ${\vec {\sigma }}$ belong to. This landscape has its global minimum(a) at the ground state configuration(s): the energy density of the ground state(s) can be obtained studying the partition function $Z$ in the limit $\beta \to \infty$. Besides the ground state(s), the energy landscape can have other local minima; the fully-connected models of glasses are characterized by the fact that there are plenty of these local minima, see SKETCH.


• Gradient descent and stationary points. Suppose that we are interested in finding the configurations of minimal energy of some model with energy landscape $E({\vec {\sigma }})$, starting from an arbitrary initial configuration ${\vec {\sigma }}_{0}$: we can think about a dynamics in which we progressively update the configuration of the system moving towards lower and lower values of the energy, hoping to eventually converge to the ground state(s). The simplest dynamics of this sort is gradient descent, ${\frac {d{\vec {\sigma }}(t)}{dt}}=-\nabla _{\perp }E({\vec {\sigma }})$

  where the configuration changes in time moving in the direction of the gradient of the energy landscape restricted to the sphere, $\nabla _{\perp }E({\vec {\sigma }})$. The dynamics stops when it reaches a stationary point , i.e. a configuration where $\nabla _{\perp }E({\vec {\sigma }})=0$. If the landscape has a simple, convex structure, this will be the ground state one is seeking for; if the energy landscape is very non-convex like in glasses, the end point of this algorithm will be a local minimum at energies much higher than the ground state. SKETCH



• The landscape’s complexity. To understand the structure of the energy landscape and to guess where gradient descent dynamics (or its variation) are expected to converge, it is useful to characterize the distribution of the stationary points, i.e. the number ${\mathcal {N}}(\epsilon )$ of such configuration having a given energy density $\epsilon$. In fully-connected models of glasses, this quantity has an exponential scaling, ${\mathcal {N}}(\epsilon )\sim {\text{exp}}\left(N\Sigma (\epsilon )\right)$, where $\Sigma (\epsilon )$ is the complexity of the landscape. ^[1]

Problem 5.1: the Kac-Rice method and the complexity

In this Problem, we set up the computation of the annealed complexity of the spherical 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} -spin model, which is defined by

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)= \lim_{N \to \infty}\frac{1}{N}\log \overline{\mathcal{N}(\epsilon)} , \quad \quad \mathcal{N}(\epsilon)= \left\{ \text{number stat. points of energy density } \epsilon\right\} }

1. The Kac-Rice formula. Consider first a random function of one variable $f(x)$ defined on an interval 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 [a,b]} , and let 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 \mathcal{N}} be the number of points $x$ such 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 f(x)=0} . Justify why

   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{\mathcal{N}}= \int_a^b dx \,p_0(x) , \quad \quad p_0(x)=\overline{\delta(f(x)) |f'(x)|} }

   where $p_{0}(x)$ is the probability density 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 x } is a zero of the function. In particular, why is the derivative of the function appearing in this formula? Consider now the number of stationary points 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 \mathcal{N}(\epsilon)} of the $p$-spin energy landscape, which satisfy 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 \nabla_\perp E(\vec{\sigma})=0} . Justify why the generalization of the formula above gives

   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{\mathcal{N}(\epsilon)}= \int_{S_N} d \vec{\sigma} \,p_{\epsilon}(\vec{\sigma}) , \quad \quad p_{\epsilon}(\vec{\sigma})=\overline{|\text{det} \nabla_\perp^2 E (\vec{\sigma})|\,\, \delta(\nabla_\perp E(\vec{\sigma})=0) \, \,\delta(E(\vec{\sigma})- N \epsilon)} }

   where $p_{\epsilon }({\vec {\sigma }})$ is the probability density 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 \vec \sigma } is a stationary point of 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 } , and $\nabla _{\perp }^{2}E({\vec {\sigma }})$ is the Hessian matrix of the function 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 E (\vec{\sigma}) } restricted to the sphere.^[2]

2. Statistical rotational invariance. Recall the expression of the correlations of the energy landscape of the 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} -spin computed in Problem 2.1: in which sense the correlation function is rotationally invariant? Justify why rotational invariance implies that

   ${\overline {{\mathcal {N}}(\epsilon )}}=(2\pi e)^{\frac {N}{2}}\,p_{\epsilon }({\vec {1}})$

   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 \vec{1}=(1,1,1, \cdots, 1) } is one fixed vector belonging to the surface of the sphere. Where does the prefactor arise from?

3. Gaussianity and correlations. Determine the distribution of the quantity 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 E (\vec{1})} . Show that the components of the vector 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 \nabla E (\vec{1})} are Gaussian random variables with zero mean and covariances

   ${\overline {(\nabla E)_{i}\,(\nabla E)_{j}}}={\frac {p}{2}}\,\delta _{ij}+O\left({\frac {1}{N}}\right)$

   The quantity 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 \nabla_\perp E (\vec{1})} can be shown to be uncorrelated 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 E (\vec{1}), \nabla^2_\perp E (\vec{1})} . Moreover, in the notation of ^[2] , $\nabla _{\perp }^{2}E({\vec {1}})={\hat {\Pi }}({\vec {1}})\,\nabla ^{2}E({\vec {1}})\,{\hat {\Pi }}({\vec {1}})-pE({\vec {1}})\mathbb {I}$. The entries of the 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-1)\times (N-1) } 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 \nabla_\perp^2 E (\vec{\sigma}) } are also Gaussian variables. Computing their correlation, one finds that the matrix conditioned to the fact that $E({\vec {1}})=N\epsilon$ can be written as

   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 [\nabla_\perp^2 E(\vec{1})]_{\alpha \beta}= \left[\hat \Pi \nabla^2 E (\vec{1}) \hat \Pi - N^{-1}p \, E (\vec{\sigma}) \mathbb{I} \right]_{\alpha \beta}= G_{\alpha \beta}- p \epsilon\, \delta_{\alpha \beta}, }

   where 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 G } has random entries with zero average and correlations

   ${\overline {{G}_{\alpha \beta }\,{G}_{\gamma \delta }}}={\frac {p(p-1)}{2N}}\left(\delta _{\alpha \gamma }\delta _{\beta \delta }+\delta _{\alpha \delta }\delta _{\beta \gamma }\right)$

   Combining everything, 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{\mathcal{N}(\epsilon)}= (2 \pi e)^{\frac{N}{2}} \,\frac{1}{(\pi\, p)^{\frac{N-1}{2}}}\; \sqrt{\frac{N}{\pi}}} e^{-N \epsilon^2}\;\overline{|\text{det} \left( G- p \epsilon \mathbb{1} \right)|}. }

   It remains to compute the expectation value of the determinant of the Hessian matrix: this is the goal of the next problem.

Problem 5.2: the Hessian and random matrix theory

In this problem, we determine the average of the determinant of the Hessian matrix and conclude the calculation of the annealed complexity.

1. Random matrices. Show that 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 G } is a GOE matrix, i.e. a matrix taken from the Gaussian Orthogonal Ensemble, meaning that its distribution 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 P(G)= \frac{1}{Z_N} e^{-\frac{N}{4 \sigma^2} \text{Tr} G^2}, \quad \quad \sigma^2=\frac{p (p-1)}{2} }

   What is the value 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 \sigma^2 } ? How can these matrices be generated numerically?

2. Eigenvalue density and concentration. Let 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 \lambda_\alpha } 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 G } . 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(\hat \Pi \nabla^2 E (\vec{1}) \hat \Pi - 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 $G$ 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 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[\rho] = e^{-N^2 \, g[\rho]} } 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 (a function of a function). 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{ty}}(\lambda) \, \log |\lambda - p \epsilon|\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{ty}}(\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{ty}}]=0 } .

3. The semicircle, the threshold and the ground state. 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{ty}}(\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 extreme of the support of the density?
   • Combining all the results of the previous problems, 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)]- \epsilon^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}}= \sqrt{\frac{2(p-1)}{p}}. }
   • Show 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 \epsilon_{\text{th}}} is a critical value where a transition occurs for the complexity. What happens to the eigenvalue density in 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)} when this value of energy is attained? Recalling that this is the eigenvalue density is that of the Hessian matrix of the landscape, explain why the stationary points at energy 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}}} are marginally stable .
   • 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 the corresponding energy density equal to the ground state energy density?

Notes

[1] - This quantity looks similar to the entropy 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 S(\epsilon) } we computed for the REM in Problem 1.1. However, while the entropy counts all configurations at a given energy density, the complexity 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(\epsilon) } accounts only for the stationary points.

[2] - We define with 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 \hat \Pi(\vec{\sigma}) } the projector on the tangent plane to the sphere 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 \vec{\sigma}} : this is the plane orthogonal to the vector 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 \vec{\sigma}} . The gradient 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 \nabla_\perp E(\vec{\sigma}) } is a 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-1)} -dimensional vector that is obtained projecting the gradient 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 [\nabla E(\vec{\sigma})]_i=\partial E/\partial \sigma_i } on the tangent plane, 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 \nabla_\perp E(\vec{\sigma})=\hat \Pi \nabla E(\vec{\sigma})} . The Hessian 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 \nabla^2_\perp E(\vec{\sigma}) } is a 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-1) \times (N-1)} -dimensional matrix that is obtained from the Hessian 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 [\nabla^2 E(\vec{\sigma})]_{ij}=\partial^2 E/\partial \sigma_i \partial \sigma_j } as 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 \nabla^2_\perp E(\vec{\sigma})= \hat \Pi(\vec{\sigma}) \, \nabla^2 E(\vec{\sigma}) \, \hat\Pi(\vec{\sigma}) - N^{-1}\nabla E(\vec{\sigma}) \cdot \vec{\sigma} \mathbb{I}} 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 \mathbb{I}} is the identity matrix.