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-rw-r--r--2-point.tex55
1 files changed, 52 insertions, 3 deletions
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}