From 2163176d888323c18805ad8a5d1766fe3df556ac Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 5 Jun 2022 00:04:55 +0200
Subject: Cleaned up derivation equations.

---
 frsb_kac-rice.tex | 70 ++++++++++++++++++++++++++-----------------------------
 1 file changed, 33 insertions(+), 37 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 8881f10..a8f7188 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -259,53 +259,49 @@ index $\frac12N(1-\mu/\mu_m)$. When $\mu<-\mu_m$, the critical
 points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
 when $\mu=\mu_m$, the critical points are marginal minima.
 
+\begin{equation}
+  \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)
+  =\int \frac{\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi}
+    e^{\hat\epsilon(N\epsilon-H(s_a))+\hat s_a\cdot(\partial H(s_a)-\mu s_a)}
+\end{equation}
+
 \begin{equation}
   \begin{aligned}
-    \overline{\Sigma(\epsilon, \mu)}
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
-    e^{nN\mathcal D(\mu)}
-    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}\overline{
-      \exp\left[
-        \sum_a^n
-        (\hat s_a\partial_a-\hat\epsilon)H(s_a)
-      \right]
+    \overline{
+      \exp\left\{
+        \sum_a^n(\hat s_a\cdot\partial_a-\hat\epsilon)H(s_a)
+      \right\}
     }
-    \\
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
-    e^{nN\mathcal D(\mu)}
-    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
-      \exp\left[
+    &=\exp\left\{
         \frac12\sum_{ab}^n
-        (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)\overline{H(s_a)H(s_b)}
-      \right] \\
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
-    e^{nN\mathcal D(\mu)}
-    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
-      \exp\left[
+        (\hat s_a\cdot\partial_a-\hat\epsilon)
+        (\hat s_b\cdot\partial_b-\hat\epsilon)
+        \overline{H(s_a)H(s_b)}
+      \right\} \\
+    &=\exp\left\{
         \frac N2\sum_{ab}^n
-        (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)f(s_as_b/N)
-      \right] \\
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
-    e^{nN\mathcal D(\mu)}
-    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
-      \exp\left[
+        (\hat s_a\cdot\partial_a-\hat\epsilon)
+        (\hat s_b\cdot\partial_b-\hat\epsilon)
+        f\left(\frac{s_a\cdot s_b}N\right)
+      \right\} \\
+    &\hspace{-13em}\exp\left\{
         \frac N2\sum_{ab}^n
-        (
-        \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\frac{\hat s_as_b}Nf'(s_as_b/N)+\frac{\hat s_a\hat s_b}Nf'(s_as_b/N)
-        +\left(\frac{\hat s_as_b}N\right)^2f''(s_as_b/N)
-        )
-      \right]
+        \left[
+          \hat\epsilon^2f\left(\frac{s_a\cdot s_b}N\right)
+          -2\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
+          +\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
+          +\left(\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right)
+        \right]
+      \right\}
   \end{aligned}
 \end{equation}
 
 The parameters:
-\begin{equation}
-  \begin{aligned}
-    Q_{ab}=\frac1Ns_a\cdot s_b \\
-    R_{ab}=\frac1N\hat s_a\cdot s_b \\
-    D_{ab}=\frac1N\hat s_a\cdot\hat s_b
-  \end{aligned}
-\end{equation}
+\begin{align}
+  Q_{ab}=\frac1Ns_a\cdot s_b &&
+  R_{ab}=\frac1N\hat s_a\cdot s_b &&
+  D_{ab}=\frac1N\hat s_a\cdot\hat s_b
+\end{align}
 
 \begin{equation}
   S
-- 
cgit v1.2.3-70-g09d2

