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| -rw-r--r-- | 2-point.tex | 55 |
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} |