diff options
Diffstat (limited to '2-point.tex')
| -rw-r--r-- | 2-point.tex | 261 |
1 files changed, 261 insertions, 0 deletions
diff --git a/2-point.tex b/2-point.tex new file mode 100644 index 0000000..32d88a8 --- /dev/null +++ b/2-point.tex @@ -0,0 +1,261 @@ +\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{ + The character 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} + +We introduce the Kac--Rice measure +\begin{equation} + d\nu_H(s)=ds\,\delta\big(\nabla H(s)\big)\,\big|\det\operatorname{Hess}H(s)\big| +\end{equation} +which counts stationary points of the function $H$. More interesting is the measure conditioned on a given energy density $E$ and stability $\mu$, +\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) +\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 $\mu_1$. +\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)} +\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)} +\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}-\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] + \\& + +\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} |