\documentclass[fleqn,a4paper]{article} \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? \usepackage[T1]{fontenc} % vector fonts plz \usepackage{fullpage,amsmath,amssymb,latexsym,graphicx} \usepackage{newtxtext,newtxmath} % Times for PR \usepackage{appendix} \usepackage[dvipsnames]{xcolor} \usepackage[ colorlinks=true, urlcolor=MidnightBlue, citecolor=MidnightBlue, filecolor=MidnightBlue, linkcolor=MidnightBlue ]{hyperref} % ref and cite links with pretty colors \usepackage[ style=phys, eprint=true, maxnames = 100 ]{biblatex} \usepackage{anyfontsize,authblk} \addbibresource{2-point.bib} \begin{document} \title{ Arrangement of nearby minima and saddles in the mixed $p$-spin energy landscape } \author{Jaron Kent-Dobias} \affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma} \maketitle \begin{abstract} \end{abstract} \cite{Ros_2020_Distribution, Ros_2019_Complex, Ros_2019_Complexity} 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} f(q)=\frac12\sum_pa_pq^p \end{equation} 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(\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 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(\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}\lim_{m\to-1}\overline{\int d\nu_H(\mathbf s_0\mid E_0,\mu_0)\left(\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)\right)^m\, \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}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)} \end{align*} \begin{equation} \overline{\big|\det\operatorname{Hess}H(s)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big)} =e^{N\int d\lambda\,\rho(\lambda+\mu)\log|\lambda|}\delta(N\mu-s\cdot\partial H) \end{equation} \begin{equation} \rho(\lambda)=\begin{cases} \frac2{\pi}\sqrt{1-\big(\frac{\lambda}{\mu_\text m}\big)^2} & \lambda^2\leq\mu_\text m^2 \\ 0 & \text{otherwise} \end{cases} \end{equation} \begin{equation} \begin{aligned} \mathcal D(\mu) &=\int d\lambda\,\rho(\lambda+\mu)\ln|\lambda| \\ &=\begin{cases} \frac12+\log\left(\frac12\mu_\text m\right)+\frac\mu{\mu_\text m}\left(\frac\mu{\mu_\text m}-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) -\log\left(\frac{\mu}{\mu_\text m}-\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu>\mu_\text m \\ \frac12+\log\left(\frac12\mu_\text m\right)+\frac{\mu^2}{\mu_\text m^2} & -\mu_\text m\leq\mu\leq\mu_\text m \\ \frac12+\log\left(\frac12\mu_\text m\right)+\frac\mu{\mu_\text m}\left(\frac\mu{\mu_\text m}+\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) -\log\left(\frac{\mu}{\mu_\text m}+\sqrt{\big(\frac\mu{\mu_\text m}\big)^2-1}\right) & \mu<-\mu_\text m \end{cases} \end{aligned} \end{equation} \begin{align*} &\Sigma_{12} =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int e^{Nm\mathcal S_0(\hat\beta_0,C^{00},R^{00},D^{00})+Nn\mathcal S_1(\hat\beta_0,\hat\beta_1,C^{00},C^{01},C^{11},R^{00},R^{01},R^{10},R^{11},D^{00},D^{01},D^{11})} \end{align*} \begin{align} C^{01} =\begin{bmatrix} q&\cdots&q\\ q'&\cdots&q'\\ \vdots&\ddots&\vdots\\ q'&\cdots&q' \end{bmatrix} && R^{01} =\begin{bmatrix} r_{01}&\cdots&r_{01}\\ r_{01}'&\cdots&r_{01}'\\ \vdots&\ddots&\vdots\\ r_{01}'&\cdots&r_{01}' \end{bmatrix} \end{align} \begin{align*} &\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\{ \frac12\sum_{ab}\left[ \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-D_{ab})f'(C_{ab})+R_{ab}^2f''(C_{ab}) \right] \\& +\frac12\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R +\frac12\log\det\begin{bmatrix} 1&ir_{0}&q&ir_{01}\\ ir_{0}&d_{0}&ir_{10}&d_{01}\\ q&ir_{10}&C&iR\\ ir_{01}&d_{01}&iR&D \end{bmatrix} \bigg\} \end{align*} \begin{align} \hat\beta_0 &=-\frac{(\epsilon_0+\mu_0)f'(1)+\epsilon_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\ r_0 &=\frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\ d_0 &=\frac1{f'(1)} -\left( \frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \right)^2 \end{align} \begin{align*} \det\begin{bmatrix} 1&ir_{0}&q&ir_{01}\\ ir_{0}&d_{0}&ir_{10}&d_{01}\\ q&ir_{10}&C&iR\\ ir_{01}&d_{01}&iR&D \end{bmatrix} &= \det\begin{bmatrix} 1&ir_{0}\\ ir_{0}&d_{0} \end{bmatrix} \det\left( \begin{bmatrix} C&iR\\ iR&D \end{bmatrix} - \begin{bmatrix} q&ir_{10}\\ir_{01}&d_{01} \end{bmatrix} \begin{bmatrix} 1&ir_{0}\\ ir_{0}&d_{0} \end{bmatrix}^{-1} \begin{bmatrix} q&ir_{01}\\ir_{10}&d_{01} \end{bmatrix} \right) \end{align*} \begin{align*} & \begin{bmatrix} \tilde c&\tilde r\\\tilde r&\tilde d \end{bmatrix} = \begin{bmatrix} q&ir_{10}\\ir_{01}&d_{01} \end{bmatrix} \begin{bmatrix} 1&ir_{0}\\ ir_{0}&d_{0} \end{bmatrix}^{-1} \begin{bmatrix} q&ir_{01}\\ir_{10}&d_{01} \end{bmatrix}\\ &= \frac1{r_{0}^2+d_{0}}\begin{bmatrix} q^2d_{0}+2qr_{0}r_{10}-r_{10}^2 & i\left[d_{01}(r_{10}-r_0q)+r_{01}(r_0r_{10}+d_0q)\right] \\ i\left[d_{01}(r_{10}-r_0q)+r_{01}(r_0r_{10}+d_0q)\right] & d_{01}^2+2r_{0}r_{01}d_{01}-d_{0}r_{01}^2 \end{bmatrix} \end{align*} This matrix with modify the diagonal of the RS matrix for the second spin. Define $\tilde C=C-\tilde c$, $\tilde R=R-\tilde r$, $\tilde D=D-\tilde d$. Then \begin{align*} &\Sigma_{12} =\mathcal D(\mu_1)+\hat\beta_1E_1-\hat\mu_1 +\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\{ \frac12\sum_{ab}\left[ \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-D_{ab})f'(C_{ab})+R_{ab}^2f''(C_{ab}) \right] \\& +\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R +\frac12\log\det((C-\tilde c)(D-\tilde d)+(R-\tilde r)^2) \bigg\} \end{align*} These equations for $D^*$ are the same as those for the unpinned case, or \[ 0=-\frac12f'(C)+\frac12((C-\tilde c)(D-\tilde d)+(R-\tilde r)^2)^{-1}(C-\tilde c) \] Solving, we get \[ D=\tilde d+f'(C)^{-1}-(C-\tilde c)^{-1}(R-\tilde r)^2 \] \begin{align*} &\Sigma_{12} =\mathcal D(\mu_1)+\hat\beta_1E_1-\frac12\hat\mu_1 +\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\{ \frac12\sum_{ab}\left[ \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-(f'(C)^{-1}_{ab}-((C-\tilde c)^{-1}(R-\tilde r)^2)_{ab}-\tilde d))f'(C_{ab})+R_{ab}^2f''(C_{ab}) \right] \\& +\frac12\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R +\frac12\log\det(C-\tilde c)-\frac12\log\det f'(C) \bigg\} \end{align*} \begin{align*} 0&=C^*-f'(C)(C^*D^*+R^*R^*) \\ 0&=\big[\hat\beta_1f'(C)-\mu_1I+R\odot f''(C)\big](C-\tilde c)+f'(C)(R-\tilde r) \end{align*} \begin{align*} 0&=-f'(q)+\frac12(C^*D^*+R^*R^*)^{-1}_{ij}\left( C^*_{jk}\frac{D^*_{ki}}{d_{01}} + 2R^*_{jk}\frac{R^*_{ki}}{d_{01}} \right) \end{align*} \begin{equation} \hat\beta_2E_2-r_{22}^{(0)}\mu_2\frac12\left\{ \hat\beta_2^2\big(f(1)-f(q_{22}^{(0)})\big) +\left( r_{12}^2+2\hat\beta_2r_{22}-\frac{2q_{12}r_{12}(r_{22}-r_{22}^{(0)})}{1-q_{22}^{(0)}} \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \right\} \end{equation} What about the average for the Hessian terms? \[ \overline{ |\det\operatorname{Hess}H(s_0)|\delta(\mu_0-\operatorname{Tr}\operatorname{Hess}H(s_0))|\det\operatorname{Hess}H(s_a)|\delta(\mu_1-\operatorname{Tr}\operatorname{Hess}H(s_a)) } \] \paragraph{Acknowledgements} \paragraph{Funding information} \printbibliography \end{document}