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| -rw-r--r-- | frsb_kac-rice.tex | 11 |
1 files changed, 6 insertions, 5 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index ec3a933..10552b6 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -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 |