From e05be8a3242c4b146bb4878c62a948a0f0cfaa79 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 19 May 2023 08:07:58 +0200
Subject: Started adding C matrix elements to paper.

---
 2-point.tex | 55 ++++++++++++++++++++++++++++++++++++++++++++++++++++---
 1 file changed, 52 insertions(+), 3 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index e20792e..2e4df01 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -346,8 +346,11 @@ zero, the parameters $\mathcal Q_{00}$ can be evaluated at a saddle point of
 $\mathcal S_0$ alone. This means that these parameters will take the same value
 they take when the ordinary, 1-point complexity is calculated.
 
-The $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$, and $D^{01}$ are
-expected to have the following form at the saddle point:
+In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
+and $D^{01}$ to have constant \emph{rows} of length $n$, with blocks of rows
+corresponding to the \textsc{rsb} structure of the single-point complexity. For
+the scope of this paper, where we restrict ourselves to replica symmetric
+complexities, they have the following form at the saddle point:
 \begin{align}
   C^{01}=
   \begin{subarray}{l}
@@ -625,7 +628,7 @@ which the Hessian is evaluated.
   \caption{
     A sketch of the vectors involved in the calculation of the isolated
     eigenvalue. All replicas $\mathbf x$ sit in an $N-2$ sphere corresponding
-    with the tangent plane of the first $\mathbf s$ replica (not to scale). All of the
+    with the tangent plane (not to scale) of the first $\mathbf s$ replica. All of the
     $\mathbf s$ replicas lie on the sphere, constrained to be at fixed overlap
     $q$ with the first of the $\pmb\sigma$ replicas, the reference
     configuration. All of the $\pmb\sigma$ replicas lie on the sphere.
@@ -946,6 +949,52 @@ for
     0&0&0&0&0
   \end{bmatrix}
 \]
+\begin{align}
+  &
+  C_{11}=d^{00}_\mathrm df'(1)
+  \quad
+  C_{12}=r^{00}_\mathrm df'(1)
+  \quad
+  C_{22}=-f'(1)
+  \\
+  &
+  C_{13}
+  =\frac1{1-q_0}\left(
+    (r^{11}_d-r^{11}_0)\left(r^{01}-q\frac{r^{11}_d-r^{11}_0}{1-q_0}\right)(f'(1)-f'(q_0))+qf'(1)d^{00}_d+r^{00}_d(r^{10}f'(1)+(r^{11}_d-r^{11}_0)f'(q))
+    \right)
+  \\
+  &
+  C_{15}=r^{00}_df'(q)+\left(r^{01}-q\frac{r^{11}_d-r^{11}_0}{1-q_0}\right)(f'(1)-f'(q_0))
+  \quad
+  C_{14}=-C_{15}
+  \\
+  &
+  C_{23}=\frac1{1-q_0}\left((qr^{00}_d-r^{10})f'(1)-(r^{11}_d-r^{11}_0)f'(q)\right)
+  \quad
+  C_{24}=f'(q)
+  \quad
+  C_{25}=-C_{24}
+  \\
+  &
+  C_{33}
+  =-\frac1{1-q_0}\left(
+    \frac{1-q^2}{1-q_0}+\frac{r^{11}_d-r^{11}_0}{1-q_0}\left[
+      (r^{11}_d-r^{11}_0)f'(1)
+      -2\left(
+        qr^{01}-r^{11}_0+\frac{1-q^2}{1-q_0}(r^{11}_d-r^{11}_0)
+        \right)(f'(1)-f'(q_0))
+        -2(qr^{00}-r^{10})f'(q)
+    \right]
+    +\frac{(r^{10}-qr^{00}_d)^2}{1-q_0}f'(1)
+    \right)
+    \\
+  &
+    C_{44}=f'(1)-2f'(q_0)
+    \quad
+    C_{45}=f'(q_0)
+    \quad
+    C_{55}=-f'(1)
+\end{align}
 Use $X$ for the big vector. Then
 \[
   0=-\beta^2f''(1)a_0+\frac{a_0-X^TCX}{(1-a_0)^2}
-- 
cgit v1.2.3-54-g00ecf

