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| -rw-r--r-- | 2-point.tex | 193 |
1 files changed, 130 insertions, 63 deletions
diff --git a/2-point.tex b/2-point.tex index b187e96..ca84552 100644 --- a/2-point.tex +++ b/2-point.tex @@ -116,55 +116,153 @@ stationary point of energy density $E_1$ and stability $\mu_1$. \end{aligned} \end{equation} -\begin{align*} - &\Sigma_{12} - =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int e^{Nm\mathcal S_0(\hat\beta_0,C^{00},R^{00},D^{00})+Nn\mathcal S_1(\hat\beta_0,\hat\beta_1,C^{00},C^{01},C^{11},R^{00},R^{01},R^{10},R^{11},D^{00},D^{01},D^{11})} -\end{align*} +\begin{align} + \mathcal Q_{00}=\begin{bmatrix} + \hat\beta_0\\\hat\mu_0\\C^{00}\\R^{00}\\D^{00} + \end{bmatrix} + && + \mathcal Q_{11}=\begin{bmatrix} + \hat\beta_1\\\hat\mu_1\\C^{11}\\R^{11}\\D^{11} + \end{bmatrix} + && + \mathcal Q_{01}=\begin{bmatrix} + \hat\mu_{01}\\C^{01}\\R^{01}\\R_{10}\\D^{01} + \end{bmatrix} +\end{align} +\begin{equation} + \Sigma_{01} + =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\int d\mathcal Q_{00}\,d\mathcal Q_{11}\,d\mathcal Q_{01}\,e^{Nm\mathcal S_0(\mathcal Q_{00})+Nn\mathcal S_1(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01})} +\end{equation} +\begin{equation} + \begin{aligned} + &\mathcal S_0(\mathcal Q_{00}) + =-\hat\beta_0E_0-r^{00}_d\mu_0-\frac12\hat\mu_0(1-c^{00}_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}) + \right]+\frac12\log\det\begin{bmatrix}C^{00}&R^{00}\\R^{00}&D^{00}\end{bmatrix} + \bigg\} + \end{aligned} +\end{equation} + +\begin{equation} + \begin{aligned} + &\mathcal S(\mathcal Q_{00},\mathcal Q_{11},\mathcal Q_{01}) + =-\hat\beta_1E_1-\mu_1r^{11}_d-\frac12\hat\mu_1(1-c^{11}_d) \\ + &\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}) + \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}) + \right]\\ + &\quad+\frac12\log\det\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) + \bigg\} + \end{aligned} +\end{equation} \begin{align} C^{01} + = + \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} + && + R^{01} =\begin{bmatrix} - q&\cdots&q\\ - q'&\cdots&q'\\ + r_{01}&\cdots&r_{01}\\ + 0&\cdots&0\\ \vdots&\ddots&\vdots\\ - q'&\cdots&q' + 0&\cdots&0 \end{bmatrix} && - R^{01} + R^{10} =\begin{bmatrix} - r_{01}&\cdots&r_{01}\\ - r_{01}'&\cdots&r_{01}'\\ + r_{10}&\cdots&r_{10}\\ + 0&\cdots&0\\ + \vdots&\ddots&\vdots\\ + 0&\cdots&0 + \end{bmatrix} + && + D^{01} + =\begin{bmatrix} + d_{01}&\cdots&d_{01}\\ + 0&\cdots&0\\ \vdots&\ddots&\vdots\\ - r_{01}'&\cdots&r_{01}' + 0&\cdots&0 \end{bmatrix} \end{align} -\begin{align*} - &\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\{ - \frac12\sum_{ab}\left[ - \hat\beta_1^2f(C_{ab})+(2\hat\beta_1R_{ab}-D_{ab})f'(C_{ab})+R_{ab}^2f''(C_{ab}) - \right] - \\& - +\frac12\hat\mu_1\operatorname{Tr}C-\mu_1\operatorname{Tr}R - +\frac12\log\det\begin{bmatrix} - 1&ir_{0}&q&ir_{01}\\ - ir_{0}&d_{0}&ir_{10}&d_{01}\\ - q&ir_{10}&C&iR\\ - ir_{01}&d_{01}&iR&D - \end{bmatrix} - \bigg\} -\end{align*} +The inverse of block hierarchical matrix is still a block hierarchical matrix, since (dropping the superscripts for clarity) +\begin{equation} + \begin{bmatrix} + C^{00}&iR^{00}\\iR^{00}&D^{00} + \end{bmatrix}^{-1} + = + \begin{bmatrix} + (C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00} & -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} \\ + -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} & (C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00} + \end{bmatrix} +\end{equation} +Because of the structure of the 01 matrices, the volume element will depend only on the diagonal if this matrix. If we write +\begin{align} + \tilde c_d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}]_{11} \\ + \tilde r_d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00}]_{11} \\ + \tilde d_d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00}]_{11} +\end{align} + +In the replica symmetric case, +\begin{align} + \tilde c_d^{00}=\frac1{(r^{00}_d)^2+d^{00}_d} && + \tilde r_d^{00}=\frac{r^{00}_d}{(r^{00}_d)^2+d^{00}_d} && + \tilde d_d^{00}=\frac{d^{00}_d}{(r^{00}_d)^2+d^{00}_d} +\end{align} + +\begin{equation} + \begin{bmatrix} + q^2\tilde d_d^{00}+2qr_{10}\tilde r^{00}_d-r_{10}^2\tilde d^{00}_d + & + i\left[d_{01}(r_{10}\tilde c^{00}_d-q\tilde r^{00}_d)+r_{01}(r_{10}\tilde r^{00}_d+q\tilde d^{00}_d)\right] + \\ + i\left[d_{01}(r_{10}\tilde c^{00}_d-q\tilde r^{00}_d)+r_{01}(r_{10}\tilde r^{00}_d+q\tilde d^{00}_d)\right] + & + d_{01}^2\tilde c^{00}_d+2r_{01}d_{01}\tilde r^{00}_d-r_{01}^2\tilde d^{00}_d + \end{bmatrix} +\end{equation} +where each block is a constant $n\times n$ matrix. +In the twin limits of $m$ and $n$ to zero, the saddle point conditions for the variables involving only the reference critical point (those in $\mathcal Q_{00}$) reduce to the ordinary, 1-point conditions. With a replica-symmetric ansatz, these conditions are \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}\\ - r_0 + r_d^{00} &=\frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\ - d_0 + d_d^{00} &=\frac1{f'(1)} -\left( \frac{\mu_0f(1)+\epsilon_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} @@ -172,37 +270,6 @@ stationary point of energy density $E_1$ and stability $\mu_1$. \end{align} \begin{align*} - \det\begin{bmatrix} - 1&ir_{0}&q&ir_{01}\\ - ir_{0}&d_{0}&ir_{10}&d_{01}\\ - q&ir_{10}&C&iR\\ - ir_{01}&d_{01}&iR&D - \end{bmatrix} - &= - \det\begin{bmatrix} - 1&ir_{0}\\ - ir_{0}&d_{0} - \end{bmatrix} - \det\left( - \begin{bmatrix} - C&iR\\ - iR&D - \end{bmatrix} - - - \begin{bmatrix} - q&ir_{10}\\ir_{01}&d_{01} - \end{bmatrix} - \begin{bmatrix} - 1&ir_{0}\\ - ir_{0}&d_{0} - \end{bmatrix}^{-1} - \begin{bmatrix} - q&ir_{01}\\ir_{10}&d_{01} - \end{bmatrix} - \right) -\end{align*} - -\begin{align*} & \begin{bmatrix} \tilde c&\tilde r\\\tilde r&\tilde d |