From 47bf153ea3077a245205e29d1ccfb360fb1ec272 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 May 2023 18:44:00 +0200
Subject: Some writing.

---
 2-point.tex | 65 +++++++++++++++++++++++++++++++++++++++++++------------------
 1 file changed, 46 insertions(+), 19 deletions(-)

(limited to '2-point.tex')

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\{
-- 
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