From 1cc4a4a0698e13d67669f0963143bb6e4b728932 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 4 Jun 2022 22:41:35 +0200
Subject: Added some.

---
 frsb_kac-rice.tex | 26 ++++++++++++++++++++++----
 1 file changed, 22 insertions(+), 4 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index ace1b28..7bf0c66 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -78,7 +78,25 @@ minima of two total energies such that one of the systems is frozen have nonzero
 
 \section{The model}
 
-Here we consider, for definiteness, the `toy' model introduced by M\'ezard and Parisi
+Here we consider, for definiteness, the `toy' model introduced by M\'ezard and Parisi (????? what)
+
+The mixed $p$-spin model
+\begin{equation}
+  H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
+\end{equation}
+for $\overline{J^2}=p!/2N^{p-1}$. Then
+\begin{equation}
+  \overline{H(s_1)H(s_2)}=Nf\left(\frac{s_1\cdot s_2}N\right)
+\end{equation}
+for
+\begin{equation}
+  f(q)=\frac12\sum_pa_pq^p
+\end{equation}
+Can be thought of as a model of generic gaussian functions on the sphere.
+To constrain the model to the sphere, we use a Lagrange multiplier $\mu$, with the total energy being
+\begin{equation}
+  H(s)+\frac\mu2(N-s\cdot s)
+\end{equation}
 
 
 
@@ -267,21 +285,21 @@ when $\mu=\mu_m$, the critical points are marginal minima.
     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
+        \frac12\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
+        \frac N2\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[
-        N\sum_{ab}^n
+        \frac N2\sum_{ab}^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)
-- 
cgit v1.2.3-70-g09d2

