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authorJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-10 19:00:03 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-10 19:00:03 +0200
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Lots of work.
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@@ -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}