From bb3f5cbd80482791340e9f9bb3d40767fc249a0a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 10 May 2023 19:00:03 +0200
Subject: Lots of work.

---
 2-point.tex | 200 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 200 insertions(+)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index ca84552..b4cb216 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -368,6 +368,206 @@ What about the average for the Hessian terms?
   }
 \]
 
+\section{Isolated eigenvalue}
+
+
+\begin{align*}
+  \beta F(\beta\mid\mathbf s)
+  &=-\frac1N\log\left(\int d\mathbf x\,\delta(\mathbf x\cdot\mathbf s)\delta(N-\mathbf x\cdot\mathbf x)\exp\left\{
+    -\beta\frac12\mathbf x^T\partial\partial H(\mathbf s)\mathbf x
+  \right\}\right) \\
+  &=-\lim_{\ell\to0}\frac1N\frac\partial{\partial\ell}\int\left[\prod_{\alpha=1}^\ell d\mathbf x_\alpha\,\delta(\mathbf x_\alpha^T\mathbf s)\delta(N-\mathbf x_\alpha^T\mathbf x_\alpha)\exp\left\{
+    -\beta\frac12\mathbf x^T_\alpha\partial\partial H(\mathbf s)\mathbf x_\alpha
+  \right\}\right]
+\end{align*}
+\begin{align*}
+  F(\beta\mid E_1,\mu_1,q,\pmb\sigma)
+  &=\int\frac{d\nu(\mathbf s\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s)}{\int d\nu(\mathbf s'\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s')}F(\beta\mid\mathbf s) \\
+  &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F(\beta\mid\mathbf s_1)
+\end{align*}
+\[
+  \begin{aligned}
+    F(\beta\mid\epsilon_1,\mu_1,\epsilon_2,\mu_2,q)
+    &=\int\frac{d\nu(\pmb\sigma\mid E_0,\mu_0)}{\int d\nu(\pmb\sigma'\mid E_0,\mu_0)}\,F(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\
+    &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu(\pmb\sigma_a\mid E_0,\mu_0)\right]\,F(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
+  \end{aligned}
+\]
+
+\begin{align}
+  &\log\det
+    \begin{bmatrix}
+      C^{00}&iR^{00}&C^{01}&iR^{01}&X_1\\
+      iR^{00}&D^{00}&iR^{10}&D^{01}&\hat X_1\\
+      C^{01})^T&iR^{10})^T&C^{11}&iR^{11}&X_2\\
+      iR^{01})^T&D^{10})^T&iR^{11}&D^{11}&\hat X_2\\
+      X_1)^T&\hat X_1)^T&X_2)^T&\hat X_2)^T&A
+    \end{bmatrix}\\
+  &=\log\det\left(
+    A-
+    \begin{bmatrix}
+      X_1\\\hat X_1\\X_2\\\hat X_2
+    \end{bmatrix})^T
+    \begin{bmatrix}
+      C^{00}&iR^{00}&C^{01}&iR^{01}\\
+      iR^{00}&D^{00}&iR^{10}&D^{01}\\
+      (C^{01})^T&(iR^{10})^T&C^{11}&iR^{11}\\
+      (iR^{01})^T&(D^{10})^T&iR^{11}&D^{11}\\
+    \end{bmatrix}^{-1}
+    \begin{bmatrix}
+      X_1\\\hat X_1\\X_2\\\hat X_2
+    \end{bmatrix}
+  \right)
+\end{align}
+\begin{equation}
+  \begin{bmatrix}
+    A & B \\
+    C & D
+  \end{bmatrix}
+\end{equation}
+\begin{equation}
+    A=
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+      +
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}
+      D
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}^T
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+\end{equation}
+\begin{equation}
+    B=-
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}
+      D
+\end{equation}
+\begin{equation}
+    C=-
+      D
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}^T
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+\end{equation}
+\begin{equation}
+    D=
+    \left(
+      \begin{bmatrix}
+        C^{11}&iR^{11}\\iR^{11}&D^{11}
+      \end{bmatrix}
+      -
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}^T
+      \begin{bmatrix}
+        C^{00}&iR^{00}\\iR^{00}&D^{00}
+      \end{bmatrix}^{-1}
+      \begin{bmatrix}
+        C^{01}&iR^{01}\\
+        iR^{10}&D^{01}
+      \end{bmatrix}
+    \right)^{-1}
+\end{equation}
+
+\begin{equation}
+    \begin{bmatrix}
+      X_0\\\hat X_0
+    \end{bmatrix}^TA
+    \begin{bmatrix}
+      X_0\\\hat X_0
+    \end{bmatrix}
+    +
+    \begin{bmatrix}
+      X_1\\\hat X_1
+    \end{bmatrix}^TC
+    \begin{bmatrix}
+      X_0\\\hat X_0
+    \end{bmatrix}
+    +
+    \begin{bmatrix}
+      X_0\\\hat X_0
+    \end{bmatrix}^TB
+    \begin{bmatrix}
+      X_1\\\hat X_1
+    \end{bmatrix}
+    +
+    \begin{bmatrix}
+      X_1\\\hat X_1
+    \end{bmatrix}^TD
+    \begin{bmatrix}
+      X_1\\\hat X_1
+    \end{bmatrix}
+\end{equation}
+
+\begin{align}
+  X_0
+  =
+  \begin{subarray}{l}
+    \hphantom{[}\begin{array}{ccc}\leftarrow&m&\rightarrow\end{array}\hphantom{\Bigg]}\\
+  \left[
+    \begin{array}{ccc}
+      x_0&\cdots&x_0\\
+      \vdots&\ddots&\vdots\\
+      x_0&\cdots&x_0
+    \end{array}
+  \right]\begin{array}{c}
+    \uparrow\\\ell\\\downarrow
+  \end{array}\\
+  \vphantom{\begin{array}{c}n\end{array}}
+  \end{subarray}
+  &&
+  \hat X_0
+  =\begin{bmatrix}
+    \hat x_0&\cdots&\hat x_0\\
+    \vdots&\ddots&\vdots\\
+    \hat x_0&\cdots&\hat x_0
+  \end{bmatrix}
+  &&
+  X_1
+  =
+  \begin{subarray}{l}
+    \hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\
+  \left[
+    \begin{array}{ccc}
+      q&\cdots&q\\
+      0&\cdots&0\\
+      \vdots&\ddots&\vdots\\
+      0&\cdots&0
+    \end{array}
+  \right]\begin{array}{c}
+    \\\uparrow\\m-1\\\downarrow
+  \end{array}\\
+  \vphantom{\begin{array}{c}n\end{array}}
+  \end{subarray}
+  &&
+  D^{01}
+  =\begin{bmatrix}
+    d_{01}&\cdots&d_{01}\\
+    0&\cdots&0\\
+    \vdots&\ddots&\vdots\\
+    0&\cdots&0
+  \end{bmatrix}
+\end{align}
 
 \paragraph{Acknowledgements}
 
-- 
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