From 62bcc203f7aad3320547d5732161faedcc082613 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias 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