From 61bbfd74d7acfbe83ac65887e332b655e5329bed Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 May 2023 14:12:17 +0200
Subject: More work, adding double replica ansatz.

---
 2-point.tex | 27 +++++++++++++++++++++++++--
 1 file 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)}
@@ -116,6 +116,29 @@ 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}
+  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}
-- 
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