From 67f2a1fbc41a62c1418391e428da190f0ab307d7 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 4 Jun 2022 22:24:11 +0200
Subject: Some writing.

---
 frsb_kac-rice.tex | 25 ++++++++++++++++++++-----
 1 file changed, 20 insertions(+), 5 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 099b332..ace1b28 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -241,6 +241,16 @@ the question of independence \cite{Bray_2007_Statistics}
 \end{equation}
 for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$.
 
+all saddles versus only minima
+
+The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one
+can decide to find the complexity of saddles of a certain macroscopic index, or
+of minima with a certain harmonic stiffness. When $\mu$ is taken to be within
+the range $\pm2\sqrt{f''(1)}=\pm\mu_m$, the critical points are constrained to have
+index $\frac12N(1-\mu/\mu_m)$. When $\mu<-\mu_m$, the critical
+points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
+when $\mu=\mu_m$, the critical points are marginal minima.
+
 \begin{equation}
   \begin{aligned}
     \overline{\Sigma(\epsilon, \mu)}
@@ -258,23 +268,28 @@ for $\rho$ a semicircle distribution with radius $\sqrt{4f''(1)}$.
     \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
       \exp\left[
         \sum_{ab}^n
+        (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)\overline{H(s_a)H(s_b)}
+      \right] \\
+    &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
+    e^{nN\mathcal D(\mu)}
+    \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
+      \exp\left[
+        N\sum_{ab}^n
         (\hat s_a\partial_a-\hat\epsilon)(\hat s_b\partial_b-\hat\epsilon)f(s_as_b/N)
       \right] \\
     &=\frac1N\lim_{n\to0}\frac\partial{\partial n}
     e^{nN\mathcal D(\mu)}
     \int\left(\prod_a^nds_a\,d\hat s_a\right)\,d\hat\epsilon\,e^{nN\hat\epsilon\epsilon-\mu\sum_a^n\hat s_as_a}
       \exp\left[
-        \sum_{ab}^n
+        N\sum_{ab}^n
         (
-        \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\hat s_as_bf'(s_as_b/N)+\hat s_a\hat s_bf'(s_as_b/N)
-        +(\hat s_as_b)^2f''(s_as_b/N)
+        \hat\epsilon^2f(s_as_b/N)-2\hat\epsilon\frac{\hat s_as_b}Nf'(s_as_b/N)+\frac{\hat s_a\hat s_b}Nf'(s_as_b/N)
+        +\left(\frac{\hat s_as_b}N\right)^2f''(s_as_b/N)
         )
       \right]
   \end{aligned}
 \end{equation}
 
-all saddles versus only minima
-
 The parameters:
 \begin{equation}
   \begin{aligned}
-- 
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