From d69700b2249c2aaa71f7990af227696b748adb70 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 May 2023 08:41:30 +0200
Subject: Initial commit.

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+\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}
-- 
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