From ae596d037ce69d92068c5886068a1417fcb3a102 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 6 Jul 2022 18:51:00 +0200
Subject: A couple small fixes.
---
frsb_kac-rice.tex | 5 +++--
1 file 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.
--
cgit v1.2.3-70-g09d2