From 1a4aed1e217a6f851630627a1ad8d6882eeee76a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 13 May 2023 19:16:55 +0200
Subject: More writing.

---
 2-point.tex | 77 +++++++++++++++++++++++++++++++++++++++++++------------------
 1 file changed, 55 insertions(+), 22 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index c5e404e..d4bff09 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -77,6 +77,9 @@ The gradient and Hessian at a stationary point are then
 \end{align}
 where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivative with respect to $\mathbf s$.
 
+
+\section{Complexity}
+
 We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
 \begin{equation}
   d\nu_H(\mathbf s,\omega)
@@ -89,12 +92,9 @@ measure conditioned on the energy density $E$ and stability $\mu$,
 \begin{equation}
   d\nu_H(\mathbf s,\omega\mid E,\mu)
   =d\nu_H(\mathbf s,\omega)\,
-  \delta\big(H(\mathbf s)-NE\big)\,
+  \delta\big(NE-H(\mathbf s)\big)\,
   \delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\omega)\big)
 \end{equation}
-
-\section{Complexity}
-
 We want the typical number of stationary points with energy density
 $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference
 stationary point of energy density $E_0$ and stability $\mu_0$.
@@ -169,7 +169,7 @@ resulting from the average of the Hessians, the remaining part of the integrand
 has the form
 \begin{equation}
   e^{
-    -Nm\hat\beta_0E_0-Nn\hat\beta_1E_1
+    Nm\hat\beta_0E_0+Nn\hat\beta_1E_1
     -\sum_a^m\left[(\pmb\sigma_a\cdot\hat{\pmb\sigma}_a)\mu_0
       -\frac12\hat\mu_0(N-\pmb\sigma_a\cdot\pmb\sigma_a)
     \right]
@@ -184,11 +184,11 @@ where we have introduced the linear operator
 \begin{equation}
   \mathcal O(\mathbf t)
   =\sum_a^m\delta(\mathbf t-\pmb\sigma_a)\left(
-    i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}+\hat\beta_0
+    i\hat{\pmb\sigma}_a\cdot\partial_{\mathbf t}-\hat\beta_0
     \right)
     +
     \sum_a^n\delta(\mathbf t-\mathbf s_a)\left(
-    i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}+\hat\beta_1
+    i\hat{\mathbf s}_a\cdot\partial_{\mathbf t}-\hat\beta_1
     \right)
 \end{equation}
 We have written the $H$-dependant terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
@@ -242,10 +242,10 @@ where
 \begin{equation}
   \begin{aligned}
     &\mathcal S_0(\mathcal Q_{00})
-    =-\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\
+    =\hat\beta_0E_0-r^{00}_\mathrm d\mu_0-\frac12\hat\mu_0(1-c^{00}_\mathrm d)+\mathcal D(\mu_0)\\
     &\quad+\frac1m\bigg\{
       \frac12\sum_{ab}^m\left[
-        \hat\beta_1^2f(C^{00}_{ab})-(2\hat\beta_1R^{00}_{ab}+D^{00}_{ab})f'(C^{00}_{ab})+(R_{ab}^{00})^2f''(C_{ab}^{00})
+        \hat\beta_1^2f(C^{00}_{ab})+(2\hat\beta_1R^{00}_{ab}-D^{00}_{ab})f'(C^{00}_{ab})+(R_{ab}^{00})^2f''(C_{ab}^{00})
         \right]+\frac12\log\det\begin{bmatrix}C^{00}&R^{00}\\R^{00}&D^{00}\end{bmatrix}
     \bigg\}
   \end{aligned}
@@ -254,14 +254,14 @@ is the action for the ordinary, one-point complexity, and remainder is given by
 \begin{equation}
   \begin{aligned}
     &\mathcal S(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01})
-    =-\hat\beta_1E_1-r^{11}_\mathrm d\mu_1-\frac12\hat\mu_1(1-c^{11}_\mathrm d)+\mathcal D(\mu_1) \\
+    =\hat\beta_1E_1-r^{11}_\mathrm d\mu_1-\frac12\hat\mu_1(1-c^{11}_\mathrm d)+\mathcal D(\mu_1) \\
     &\quad+\frac1n\sum_b^n\left\{-\frac12\hat\mu_{12}(q-C^{01}_{1b})+\sum_a^m\left[
-      \hat\beta_0\hat\beta_1f(C^{01}_{ab})-(\hat\beta_0R^{01}_{ab}+\hat\beta_1R^{10}_{ab}+D^{01}_{ab})f'(C^{01}_{ab})+R^{01}_{ab}R^{10}_{ab}f''(C^{01}_{ab})
+      \hat\beta_0\hat\beta_1f(C^{01}_{ab})+(\hat\beta_0R^{01}_{ab}+\hat\beta_1R^{10}_{ab}-D^{01}_{ab})f'(C^{01}_{ab})+R^{01}_{ab}R^{10}_{ab}f''(C^{01}_{ab})
   \right]\right\}
     \\
     &\quad+\frac1n\bigg\{
       \frac12\sum_{ab}^n\left[
-        \hat\beta_1^2f(C^{11}_{ab})-(2\hat\beta_1R^{11}_{ab}+D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab})
+        \hat\beta_1^2f(C^{11}_{ab})+(2\hat\beta_1R^{11}_{ab}-D^{11}_{ab})f'(C^{11}_{ab})+(R^{11}_{ab})^2f''(C^{11}_{ab})
       \right]\\
     &\quad+\frac12\log\det\left(
         \begin{bmatrix}
@@ -380,7 +380,7 @@ For these models, the saddle point parameters for the reference stationary
 point are well known, and take the values.
 \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}\\
+  &=-\frac{(\epsilon_0+\mu_0)f'(1)+\epsilon_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\
   r_\mathrm d^{00}
   &=\frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\
   d_\mathrm d^{00}
@@ -390,23 +390,56 @@ point are well known, and take the values.
     \right)^2
 \end{align}
 
+$(r^{00}_\mathrm d)^2+d^{00}_\mathrm d=1/f'(1)$
+
 Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, the diagonals of the inverse block matrix from above are simple expressions:
 \begin{align}
-  \tilde c_\mathrm d^{00}=\frac1{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d} &&
-  \tilde r_\mathrm d^{00}=\frac{r^{00}_\mathrm d}{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d} &&
-  \tilde d_\mathrm d^{00}=\frac{d^{00}_\mathrm d}{(r^{00}_\mathrm d)^2+d^{00}_\mathrm d}
+  \tilde c_\mathrm d^{00}=f'(1) &&
+  \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) &&
+  \tilde d_\mathrm d^{00}=d^{00}_\mathrm df'(1)
 \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\}
+  \begin{aligned}
+    &\Sigma_{12}=\mathcal D(\mu_1)-\frac12+\hat\beta_1E_1-r^{11}_\mathrm d\mu_1
+    +\hat\beta_1\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
+    +\hat\beta_0\hat\beta_1f(q)+(\hat\beta_0r^{01}+\hat\beta_1r^{10}+r^{00}_\mathrm d r^{01})f'(q)
+    \\
+    &+\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}(r^{10}-qr^{00}_\mathrm d)f'(q)+
+    \frac12\Bigg\{
+      \hat\beta_1^2\big(f(1)-f(q^{11}_{0})\big)
+      +(r^{11}_\mathrm d)^2f''(1)+2r^{01}r^{10}f''(q)-(r^{11}_0)^2f''(q^{11}_0)
+      \\
+    &+\left(
+      (r^{01})^2-\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\left(2qr^{01}-\frac{(1-q^2)r^{11}_0-(q^{11}_0-q^2)r^{11}_\mathrm d}{1-q^{11}_0}\right)
+      \right)\big(f'(1)-f'(q_{22}^{(0)})\big) \\
+    &+\frac{1-q^2}{1-q^{11}_0}+\frac{(r^{10}-qr^{00}_\mathrm d)^2}{1-q^{11}_0}f'(1)
+    -\frac1{f'(1)}\frac{f'(1)^2-f'(q)^2}{f'(1)-f'(q^{11}_0)}
+    +\frac{r^{11}_\mathrm d-r^{11}_0}{1-q^{11}_0}\big(r^{11}_\mathrm df'(1)-r^{11}_0f'(q^{11}_0)\big)
+    \\
+    &+\log\left(\frac{1-q_{11}^0}{f'(1)-f'(q_{11}^0)}\right)
+    \Bigg\}
+  \end{aligned}
 \end{equation}
 
+\subsection{Most common neighbors with given overlap}
+
+\begin{align}
+  \hat\beta_1=0 &&
+  \mu_1=2r^{11}_\mathrm df''(1)
+\end{align}
+
+\begin{equation}
+  \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(4f''(1)-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q)
+  +O\big((1-q)^2\big)
+\end{equation}
+\begin{equation}
+  E_0^*=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
+\end{equation}
+\begin{equation}
+  \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}(E_0-E_0^*)(1-q)+O\big((1-q)^2\big)
+\end{equation}
 
 \section{Isolated eigenvalue}
 
-- 
cgit v1.3

