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authorJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-09 14:12:17 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2023-05-09 14:12:17 +0200
commit61bbfd74d7acfbe83ac65887e332b655e5329bed (patch)
treef6e7be4d3a2737c5a9b3480b44f4e6eb10d796e7
parent47bf153ea3077a245205e29d1ccfb360fb1ec272 (diff)
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More work, adding double replica ansatz.
-rw-r--r--2-point.tex27
1 files changed, 25 insertions, 2 deletions
diff --git a/2-point.tex b/2-point.tex
index 6e208cd..b187e96 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -87,9 +87,9 @@ stationary point of energy density $E_1$ and stability $\mu_1$.
\end{align*}
\begin{align*}
\Sigma_{12}
- &=\frac1N\lim_{n\to0}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\,
+ &=\frac1N\lim_{n\to0}\lim_{m\to-1}\overline{\int d\nu_H(\mathbf s_0\mid E_0,\mu_0)\left(\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)\right)^m\,
\frac\partial{\partial n}\bigg(\int d\nu_H(\mathbf s_1\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_1)\bigg)^n}\\
- &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\overline{\int\frac{d\nu_H(\mathbf s_0\mid E_0,\mu_0)}{\int d\nu_H(\mathbf s_0'\mid E_0,\mu_0)}\int\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\mathbf s_0\cdot \mathbf s_a)}
+ &=\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)}
\end{align*}
\begin{equation}
\overline{\big|\det\operatorname{Hess}H(s)\big|\,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(s)\big)}
@@ -118,6 +118,29 @@ stationary point of energy density $E_1$ and stability $\mu_1$.
\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}
+ C^{01}
+ =\begin{bmatrix}
+ q&\cdots&q\\
+ q'&\cdots&q'\\
+ \vdots&\ddots&\vdots\\
+ q'&\cdots&q'
+ \end{bmatrix}
+ &&
+ R^{01}
+ =\begin{bmatrix}
+ r_{01}&\cdots&r_{01}\\
+ r_{01}'&\cdots&r_{01}'\\
+ \vdots&\ddots&\vdots\\
+ r_{01}'&\cdots&r_{01}'
+ \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)
\\&