From d389ec5117d64f4f1edc4a2ca6d381af7b0b4636 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 5 Jun 2022 10:01:38 +0200
Subject: Finally settled on a convention that gets me the signs I want.

---
 frsb_kac-rice.tex | 38 +++++++++++++++++++-------------------
 1 file changed, 19 insertions(+), 19 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 6013f2f..df30c38 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -119,9 +119,9 @@ when $\mu=\mu_m$, the critical points are marginal minima.
     =\mathcal D(\mu)
     +\operatorname*{extremum}_{\substack{R_d,D_d,\hat\epsilon\in\mathbb R\\\chi\in\Lambda}}
     \left\{
-      -\hat\epsilon\epsilon-\mu R_d
+      \hat\epsilon\epsilon+\mu R_d
       +\frac12(2\hat\epsilon R_d-D_d)f'(1)+\frac12R_d^2f''(1)
-      +\log R_d \right.\\\left.
+      +\frac12\log R_d^2 \right.\\\left.
       +\frac12\int_0^1dq\,\left(
         \hat\epsilon^2f''(q)\chi(q)+\frac1{\chi(q)+R_d^2/D_d}
       \right)
@@ -283,7 +283,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
 
 \begin{equation}
   \mathcal N(\epsilon, \mu)
-  =\int ds\,\delta(H(s)-N\epsilon)\delta(\partial H(s)-\mu s)|\det(\partial\partial H(s)-\mu I)|
+  =\int ds\,\delta(N\epsilon-H(s))\delta(\partial H(s)-\mu s)|\det(\partial\partial H(s)-\mu I)|
 \end{equation}
 \begin{equation}
   \Sigma(\epsilon,\mu)=\frac1N\log\mathcal N(\epsilon, \mu)
@@ -298,7 +298,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
   \begin{aligned}
     \Sigma(\epsilon, \mu)
     &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\mathcal N^n(\epsilon) \\
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)|
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)|\det(\partial\partial H(s_a)-\mu I)|
   \end{aligned}
 \end{equation}
 
@@ -307,7 +307,7 @@ the question of independence \cite{Bray_2007_Statistics}
 \begin{equation}
   \begin{aligned}
     \overline{\Sigma(\epsilon, \mu)}
-    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(H(s_a)-N\epsilon)\delta(\partial H(s_a)-\mu s_a)}
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\right)\,\overline{\prod_a^n \delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)}
     \times
     \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|}
   \end{aligned}
@@ -326,37 +326,37 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$.
 all saddles versus only minima
 
 \begin{equation}
-  \prod_a^n\delta(H(s_a)-N\epsilon)\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(H(s_a)-N\epsilon)+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)}
+  \prod_a^n\delta(N\epsilon-H(s_a))\delta(\partial H(s_a)-\mu s_a)
+  =\int \frac{d\hat\epsilon}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi}
+    e^{\hat\epsilon(N\epsilon-H(s_a))+i\hat s_a\cdot(\partial H(s_a)-\mu s_a)}
 \end{equation}
 
 \begin{equation}
   \begin{aligned}
     \overline{
       \exp\left\{
-        \sum_a^n(i\hat s_a\cdot\partial_a+\hat\epsilon)H(s_a)
+        \sum_a^n(i\hat s_a\cdot\partial_a-\hat\epsilon)H(s_a)
       \right\}
     }
     &=\exp\left\{
         \frac12\sum_{ab}^n
-        (i\hat s_a\cdot\partial_a+\hat\epsilon)
-        (i\hat s_b\cdot\partial_b+\hat\epsilon)
+        (i\hat s_a\cdot\partial_a-\hat\epsilon)
+        (i\hat s_b\cdot\partial_b-\hat\epsilon)
         \overline{H(s_a)H(s_b)}
       \right\} \\
     &=\exp\left\{
         \frac N2\sum_{ab}^n
-        (i\hat s_a\cdot\partial_a+\hat\epsilon)
-        (i\hat s_b\cdot\partial_b+\hat\epsilon)
+        (i\hat s_a\cdot\partial_a-\hat\epsilon)
+        (i\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
         \left[
           \hat\epsilon^2f\left(\frac{s_a\cdot s_b}N\right)
-          +2i\hat\epsilon\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
+          -2i\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)
+          +\left(i\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right)
         \right]
       \right\}
   \end{aligned}
@@ -365,19 +365,19 @@ all saddles versus only minima
 The parameters:
 \begin{align}
   Q_{ab}=\frac1Ns_a\cdot s_b &&
-  R_{ab}=i\frac1N\hat s_a\cdot s_b &&
+  R_{ab}=-i\frac1N\hat s_a\cdot s_b &&
   D_{ab}=\frac1N\hat s_a\cdot\hat s_b
 \end{align}
 
 \begin{equation}
   S
-  =\mathcal D(\mu)-\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left(
-    -\mu\sum_a^nR_{aa}
+  =\mathcal D(\mu)+\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left(
+    \mu\sum_a^nR_{aa}
     +\frac12\sum_{ab}\left[
       \hat\epsilon^2f(Q_{ab})+2\hat\epsilon R_{ab}f'(Q_{ab})
       -D_{ab}f'(Q_{ab})+R_{ab}^2f''(Q_{ab})
     \right]
-  +\frac12\log\det\begin{bmatrix}Q&-iR\\-iR&D\end{bmatrix}
+  +\frac12\log\det\begin{bmatrix}Q&iR\\iR&D\end{bmatrix}
   \right)
 \end{equation}
 
-- 
cgit v1.2.3-70-g09d2

