From cf33eb863768470e498374eb325f648192c50ac1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 5 Jun 2022 08:46:02 +0200
Subject: New convention needed a minus sign in the energy.

---
 frsb_kac-rice.tex | 13 +++++++++----
 1 file changed, 9 insertions(+), 4 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 3c6276d..6013f2f 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -119,7 +119,7 @@ when $\mu=\mu_m$, the critical points are marginal minima.
     =\mathcal D(\mu)
     +\operatorname*{extremum}_{\substack{R_d,D_d,\hat\epsilon\in\mathbb R\\\chi\in\Lambda}}
     \left\{
-      \hat\epsilon\epsilon-\mu R_d
+      -\hat\epsilon\epsilon-\mu R_d
       +\frac12(2\hat\epsilon R_d-D_d)f'(1)+\frac12R_d^2f''(1)
       +\log R_d \right.\\\left.
       +\frac12\int_0^1dq\,\left(
@@ -148,9 +148,14 @@ result is that, if the equilibrium state in the vicinity of zero temperature is
 given by a $k$-RSB ansatz, then the complexity is given by a $(k-1)$-RSB
 ansatz. Moreover, there is an exact correspondence between the parameters of
 the equilibrium saddle point in the limit of zero temperature and those of the
-complexity saddle saddle at the ground state. If the equilibrium is given by $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the complexity in the ground state are
+complexity saddle at the ground state. If the equilibrium is given by
+$x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\tilde
+x_1,\ldots,\tilde x_{k-1}$ and $\tilde q_1,\ldots,\tilde q_{k-1}$ for the
+complexity in the ground state are
 \begin{align}
-  \tilde x_i=\frac1{\hat\epsilon}\lim_{\beta\to\infty}\beta x_i
+  \hat\epsilon=\lim_{\beta\to\infty}\beta x_k
+  &&
+  \tilde x_i=\lim_{\beta\to\infty}\frac{x_i}{x_k}
   &&
   \tilde q_i=\lim_{\beta\to\infty}q_i
   &&
@@ -366,7 +371,7 @@ The parameters:
 
 \begin{equation}
   S
-  =\mathcal D(\mu)+\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left(
+  =\mathcal D(\mu)-\hat\epsilon\epsilon+\lim_{n\to0}\frac1n\left(
     -\mu\sum_a^nR_{aa}
     +\frac12\sum_{ab}\left[
       \hat\epsilon^2f(Q_{ab})+2\hat\epsilon R_{ab}f'(Q_{ab})
-- 
cgit v1.2.3-70-g09d2

