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| -rw-r--r-- | 2-point.bib | 55 | ||||
| -rw-r--r-- | 2-point.tex | 65 |
2 files changed, 101 insertions, 19 deletions
diff --git a/2-point.bib b/2-point.bib index 4ac1b61..ea6e074 100644 --- a/2-point.bib +++ b/2-point.bib @@ -1,3 +1,58 @@ +@article{Kac_1943_On, + author = {Kac, M.}, + title = {On the average number of real roots of a random algebraic equation}, + journal = {Bulletin of the American Mathematical Society}, + publisher = {American Mathematical Society}, + year = {1943}, + month = {4}, + number = {4}, + volume = {49}, + pages = {314--320}, + url = {https://projecteuclid.org:443/euclid.bams/1183505112} +} + +@article{Rice_1944_Mathematical, + author = {Rice, S. O.}, + title = {Mathematical Analysis of Random Noise}, + journal = {Bell System Technical Journal}, + publisher = {Institute of Electrical and Electronics Engineers (IEEE)}, + year = {1944}, + month = {7}, + number = {3}, + volume = {23}, + pages = {282--332}, + url = {https://doi.org/10.1002%2Fj.1538-7305.1944.tb00874.x}, + doi = {10.1002/j.1538-7305.1944.tb00874.x} +} + +@article{Ros_2019_Complex, + author = {Ros, Valentina and Ben Arous, Gérard and Biroli, Giulio and Cammarota, Chiara}, + title = {Complex Energy Landscapes in Spiked-Tensor and Simple Glassy Models: Ruggedness, Arrangements of Local Minima, and Phase Transitions}, + journal = {Physical Review X}, + publisher = {American Physical Society (APS)}, + year = {2019}, + month = {1}, + number = {1}, + volume = {9}, + pages = {011003}, + url = {https://doi.org/10.1103%2Fphysrevx.9.011003}, + doi = {10.1103/physrevx.9.011003} +} + +@article{Ros_2019_Complexity, + author = {Ros, V. and Biroli, G. and Cammarota, C.}, + title = {Complexity of energy barriers in mean-field glassy systems}, + journal = {EPL (Europhysics Letters)}, + publisher = {IOP Publishing}, + year = {2019}, + month = {5}, + number = {2}, + volume = {126}, + pages = {20003}, + url = {https://doi.org/10.1209%2F0295-5075%2F126%2F20003}, + doi = {10.1209/0295-5075/126/20003} +} + @article{Ros_2020_Distribution, author = {Ros, Valentina}, title = {Distribution of rare saddles in the {$p$}-spin energy landscape}, diff --git a/2-point.tex b/2-point.tex index 32d88a8..6e208cd 100644 --- a/2-point.tex +++ b/2-point.tex @@ -25,7 +25,7 @@ \begin{document} \title{ - The character of nearby minima and saddles in the mixed $p$-spin energy landscape + Arrangement of nearby minima and saddles in the mixed $p$-spin energy landscape } \author{Jaron Kent-Dobias} @@ -35,32 +35,61 @@ \begin{abstract} \end{abstract} -\cite{Ros_2020_Distribution} +\cite{Ros_2020_Distribution, Ros_2019_Complex, Ros_2019_Complexity} -We introduce the Kac--Rice measure +The mixed $p$-spin models are defined by the Hamiltonian +\begin{equation} \label{eq:hamiltonian} + H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} +\end{equation} +where the vectors $\mathbf s\in\mathbb R^N$ are confined to the sphere +$\|\mathbf s\|^2=N$. The coupling coefficients $J$ are fully-connected and random, with +zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ scaled so that +the energy is typically extensive. The overbar denotes an average +over the coefficients $J$. The factors $a_p$ in the variances are freely chosen +constants that define the particular model. For instance, the `pure' +$p$-spin model has $a_{p'}=\delta_{p'p}$. This class of models encompasses all +statistically isotropic gaussian random Hamiltonians defined on the +hypersphere. + +The covariance between the energy at two different points is a function of the overlap, or dot product, between those points, or +\begin{equation} \label{eq:covariance} + \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right) +\end{equation} +where the function $f$ is defined from the coefficients $a_p$ by \begin{equation} - d\nu_H(s)=ds\,\delta\big(\nabla H(s)\big)\,\big|\det\operatorname{Hess}H(s)\big| + f(q)=\frac12\sum_pa_pq^p \end{equation} -which counts stationary points of the function $H$. More interesting is the measure conditioned on a given energy density $E$ and stability $\mu$, +In this paper, we will focus on models with a replica symmetric complexity. + +We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure \begin{equation} - d\nu_H(s\mid E,\mu)=d\nu_H(s)\, - \delta\big(NE-H(s)\big)\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big) + d\nu_H(\mathbf s) + =d\mathbf s\,\delta\big(\nabla H(\mathbf s)\big)\, + \big|\det\operatorname{Hess}H(\mathbf s)\big| +\end{equation} +which counts stationary points of the function $H$. More interesting is the +measure conditioned on the energy density $E$ and stability $\mu$, +\begin{equation} + d\nu_H(\mathbf s\mid E,\mu) + =d\nu_H(\mathbf s)\, + \delta\big(H(\mathbf s)-NE\big)\, + \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s)\big) \end{equation} -We want to count the number of stationary points with given energy density -$E_2$ and stability $\mu_2$ of overlap $q$ with a reference stationary point of -energy density $E_1$ and stability $\mu_1$. +We want the typical number of stationary points with energy density +$E_2$ and stability $\mu_2$ that lie a fixed overlap $q$ from a reference +stationary point of energy density $E_1$ and stability $\mu_1$. \begin{align*} \Sigma_{12} - &=\frac1N\overline{\int\frac{d\nu_H(s_0\mid E_0,\mu_0)}{\int d\nu_H(s_0'\mid E_0,\mu_0)}\, - \log\bigg(\int d\nu_H(s_1\mid E_1,\mu_1)\,\delta(Nq-s_0\cdot s_1)\bigg)} + &=\frac1N\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\, + \log\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot\mathbf s_1)\bigg)} \end{align*} \begin{align*} \Sigma_{12} - &=\frac1N\lim_{n\to0}\overline{\int\frac{d\nu_H(s_0\mid E_0,\mu_0)}{\int d\nu_H(s_0'\mid E_0,\mu_0)}\, - \frac\partial{\partial n}\bigg(\int d\nu_H(s_1\mid E_1,\mu_1)\,\delta(Nq-s_0\cdot s_1)\bigg)^n}\\ - &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\overline{\int\frac{d\nu_H(s_0\mid E_0,\mu_0)}{\int d\nu_H(s_0'\mid E_0,\mu_0)}\int\prod_{a=1}^nd\nu_H(s_a\mid E_1,\mu_1)\,\delta(Nq-s_0\cdot s_a)} + &=\frac1N\lim_{n\to0}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\, + \frac\partial{\partial n}\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_1)\bigg)^n}\\ + &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\int\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_a)} \end{align*} \begin{equation} \overline{\big|\det\operatorname{Hess}H(s)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big)} @@ -88,10 +117,8 @@ energy density $E_1$ and stability $\mu_1$. \end{equation} \begin{align*} - &\Sigma_{12}-\Sigma_1(E_0,\mu_0) - =\mathcal D(\mu_0)+\mathcal D(\mu_1)+\hat\beta_0E_0+\hat\beta_1E_1-\frac12\hat\mu_1-\mu_0r_{00} - +\frac12\left[\hat\beta_0^2f(1)+(2\hat\beta_0r_{00}^2-d_{00})f'(1)+r_{00}^2f''(1)\right] - \\& + &\Sigma_{12} + =\frac1N\frac{e^{-\hat\beta_0E_0-r_0\mu_0+\frac12\left[\hat\beta_0^2f(1)-(2\hat\beta_0r_0^2+d_0)f'(1)+r_0^2f''(1)\right]+\mathcal D(\mu_0)}}{e^{N\Sigma(E_0,\mu_0)}}+\mathcal D(\mu_1)+\hat\beta_1E_1-\frac12\hat\mu_1-\mu_0r_{00} +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r_{01}+\hat\beta_1r_{10}-d_{01})f'(q)+r_{01}r_{10}f''(q) \\& +\lim_{n\to0}\frac1n\bigg\{ |