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authorJaron Kent-Dobias <jaron@kent-dobias.com>2022-07-13 13:18:32 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2022-07-13 13:18:32 +0200
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Tiny disclaimer.
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@@ -536,11 +536,12 @@ matrix products and Hadamard products. In particular, the determinant of the blo
\begin{equation}
\ln\det\begin{bmatrix}C&iR\\iR&D\end{bmatrix}=\ln\det(CD+R^2)
\end{equation}
-This is straightforward to write down at $k$RSB, since the product and sum of
-the hierarchical matrices is still a hierarchical matrix. The algebra of
-hierarchical matrices is reviewed in \S\ref{sec:dict}. Using the product formula
-\eqref{eq:replica.prod}, one can write down the hierarchical matrix $CD+R^2$,
-and then compute the $\ln\det$ using the formula \eqref{eq:replica.logdet}.
+This is straightforward (if strenous) to write down at $k$RSB, since the
+product and sum of the hierarchical matrices is still a hierarchical matrix.
+The algebra of hierarchical matrices is reviewed in \S\ref{sec:dict}. Using the
+product formula \eqref{eq:replica.prod}, one can write down the hierarchical
+matrix $CD+R^2$, and then compute the $\ln\det$ using the formula
+\eqref{eq:replica.logdet}.
The extremal conditions are given by differentiating the complexity with
respect to its parameters, yielding