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| -rw-r--r-- | frsb_kac-rice.tex | 5 |
1 files changed, 3 insertions, 2 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index ee3c138..9e9d40d 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -863,7 +863,7 @@ for different energies and typical vs minima. \section{Interpretation} \label{sec:interpretation} -Let $\langle A\rangle$ be the average of $A$ over stationary points with given $E$ and $\mu$, i.e., +Let $\langle A\rangle$ be the average of $A$ over stationary points with given $E$ and $\mu^*$, i.e., \begin{equation} \langle A\rangle =\frac1{\mathcal N}\sum_{\sigma}A(s_\sigma) @@ -872,7 +872,8 @@ Let $\langle A\rangle$ be the average of $A$ over stationary points with given $ \end{equation} with \begin{equation} - d\nu(s)=ds\,\delta(NE-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)| + d\nu(s)=ds\,d\mu\,\delta\big(\tfrac12(\|s\|^2-N)\big)\,\delta\big(\nabla H(s,\mu)\big)\,\big|\det\operatorname{Hess}H(s,\mu)\big| + \delta\big(NE-H(s)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s,\mu)\big) \end{equation} the Kac--Rice measure. The fields $C$, $R$, and $D$ defined in \eqref{eq:fields} can be related to certain averages of this type. |