From 5fad5448bdc0f70b67888fe65a5dcc958d368698 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 8 Jul 2022 17:48:33 +0200
Subject: Starting to tidy up Jorge's section.

---
 frsb_kac-rice.tex | 23 ++++++++++++++++++-----
 1 file changed, 18 insertions(+), 5 deletions(-)

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index cbb33c3..5dff8b0 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -1055,12 +1055,25 @@ points.
 
 \subsection{A concrete example}
 
-Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and $H_{p_2}({\mathbf \sigma})$ of sizes  $N$, and couple them weakly with $\varepsilon \;
-{\mathbf \sigma} \cdot {\mathbf s}$.
-The complexities are
+One can construct a schematic 2RSB model from two 1RSB models.
+Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and
+$H_{p_2}({\mathbf \sigma})$ of sizes  $N$, and couple them weakly with
+$\varepsilon \;
+{\mathbf \sigma} \cdot {\mathbf s}$. The landscape of the pure models is much
+simpler than that of the mixed because, in these models, fixing the stability
+$\mu$ is equivalent to fixing the energy: $\mu=pE$. This implies that at each
+energy level there is only one type of stationary point. Therefore, for the
+pure models our formulas for the complexity and its Legendre transforms are
+functions of one variable only, $E$, and each instance of $\mu^*$ inside mus be
+replaced with $pE$.
+
+In the joint model, we wish to fix the total energy, not the energies of the
+individual two models. Therefore, we insert a $\delta$-function containing
+$(E_1+E_2)-E$ and integrate over $E_1$ and $E_2$. This results in a joint
+complexity (and Legendre transform)
 \begin{eqnarray}
-    e^{N\Sigma(e)}&=&\int de_1 de_2 d\lambda \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\
-    e^{-G(\hat \beta)}&=&\int de de_1 de_2 d\lambda\; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}
+    e^{N\Sigma(E)}&=&\int dE_1\, dE_2\, d\lambda \, e^{N[ \Sigma_1(E_1) + \Sigma_2(E_2) + O(\varepsilon) -\lambda N [(E_1+E_2)-E]}\nonumber \\
+    e^{NG(\hat \beta)}&=&\int dE\, dE_1\, dE_2\, d\lambda\, e^{N[-\hat \beta E+ \Sigma_1(E_1) + \Sigma_2(E_2) + O(\varepsilon) -\lambda N [(E_1+E_2)-E]}
 \end{eqnarray}
 The maximum is given by $\Sigma_1'=\Sigma_2'=\hat \beta$, provided it occurs in the phase
 in which both $\Sigma_1$ and $\Sigma_2$ are non-zero. The two systems are `thermalized',
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