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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} |