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(-)
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}
--
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