From 37bcaf67a5c37a803675ba2026ca292ff6ef2aba Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 13 Jun 2023 10:30:29 +0200
Subject: Lots of work.

---
 when_annealed.tex | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index 91455a8..b5c9746 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -162,11 +162,11 @@ for the action $\mathcal S$ given by
         +(2\hat\beta r_\mathrm d-d_\mathrm d)f'(1)
         -\Delta x(2\hat\beta r_1-d_1)f'(q_1)
         +r_\mathrm d^2f''(1)-\Delta x\,r_1^2f''(q_1) \\
-        &+\log\Big(
+        &+\frac1x\log\Big(
           \big(r_\mathrm d-\Delta x\,r_1\big)^2+d_\mathrm d\big(1-\Delta x\,q_1\big)-\Delta x\,d_1\big(1-\Delta xq_1\big)
           \Big)
           -\frac{\Delta x}x\log\Big(
-            (r_\mathrm d-r_1)^2+d_\mathrm d\big(1-\Delta xq_1\big)
+            (r_\mathrm d-r_1)^2+(d_\mathrm d-d_1)(1-q_1)
           \Big)
       \bigg]
     \Bigg\}
@@ -224,7 +224,7 @@ bifurcating solution are known at this point, one can search for it by looking
 for a zero eigenvalue in the way described above. In the replica symmetric
 solution for points describing saddles, this line is
 \begin{equation} \label{eq:extremal.line}
-  \mu=-\frac1{z_f}\left(-2Ef'f''+\sqrt{-2f''u_f\big(E^2(f''-f')-\log\frac{f''}{f'}z_f\big)}\right)
+  \mu_0=-\frac1{z_f}\left(2Ef'f''+\sqrt{2f''u_f\bigg(\log\frac{f''}{f'}z_f-E^2(f''-f')\bigg)}\right)
 \end{equation}
 Let $M$ be the matrix of double partial derivatives of $\mathcal S$ with
 respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point
@@ -263,7 +263,7 @@ The expression inside the inner square root is proportional to
   \begin{aligned}
     G_f
     &=
-    2(f''-f')u_fw_f
+    -2(f''-f')u_fw_f
     -2\log^2\frac{f''}{f'}f'^2f''v_f
     \\
     &\qquad
@@ -288,7 +288,8 @@ between them. Therefore, $G_f>0$ is a necessary condition to see
     vanish, and enclosed inside they are found in exponential number. The red
     region (blown up in the inset) shows where the annealed complexity gives
     the wrong count and a {\oldstylenums1}\textsc{rsb} complexity in necessary.
-    The red points show where $\det M=0$. The gray shaded region highlights the
+    The red points show where $\det M=0$. The left point, which is only an
+    upper bound on the transition, coincides with it in this case. The gray shaded region highlights the
     minima, which are stationary points with $\mu>\mu_\mathrm m$.
   } \label{fig:complexity_35}
 \end{figure}
-- 
cgit v1.2.3-70-g09d2

