From 49f94f754322526aa271ab5c27219d01a2665bba Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 9 May 2023 17:48:02 +0200
Subject: Making the saddle point analysis more consistent.

---
 2-point.tex | 193 ++++++++++++++++++++++++++++++++++++++++--------------------
 1 file changed, 130 insertions(+), 63 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index b187e96..ca84552 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -116,92 +116,159 @@ 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}
     \right)^2
 \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}
-- 
cgit v1.2.3-54-g00ecf

