From 62bcc203f7aad3320547d5732161faedcc082613 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 8 Jul 2022 11:37:35 +0200
Subject: Started expanding on FRSB supersymmetric solution.

---
 frsb_kac-rice.tex | 14 ++++++++++++++
 1 file changed, 14 insertions(+)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 5a69d6d..c65c55d 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -672,6 +672,20 @@ This has several implications. First, other than the ground state, there are
 As we will see, stable minima are numerous at energies above the ground state,
 but these vanish at the ground state.
 
+Evaluated at $\mu^*_\mathrm{ss}=r_d^{-1}+f''(1)r_d$, the complexity further simplies to
+\begin{equation} \label{eq:functional.action.ss}
+  \Sigma(E,\mu^*_\mathrm{ss})
+    =
+    \hat\beta E+\frac12\left(
+      \hat\beta f'(1)r_d-f''(1)r_d^2+\frac1{f''(1)r_d^2}
+    \right)
+    +\log(f''(1)r_d^2)
+      +\frac12\int_0^1dq\,\left(
+        \hat\beta^2f''(q)\chi(q)+\frac1{\chi(q)+r_d/\hat\beta}
+      \right)
+\end{equation}
+At the ground state, the solution $\chi$ is smooth for all values of $q$. In order to satisfy the boundary conditions, we must have $r_d=f''(1)^{-1/2}$ and $\hat\beta=\frac12f'''(1)/f''(1)^{3/2}$
+
 \subsection{Expansion near the transition}
 \label{subsec:expansion}
 
-- 
cgit v1.2.3-70-g09d2

