Started expanding on FRSB supersymmetric solution.

1 files changed, 14 insertions, 0 deletions

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}