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authorJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-08 18:44:00 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-08 18:44:00 +0200
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Some writing.
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+++ 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\{