From a242819dab57a8fbfda184e0f6e448b6e6375717 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 21 Oct 2022 15:47:42 +0200
Subject: Rearranged some exposition about the algorithmic threshold.

---
 frsb_kac-rice_letter.tex | 37 ++++++++++++++++++-------------------
 1 file changed, 18 insertions(+), 19 deletions(-)

diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 1d8785c..c3628f9 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -145,7 +145,22 @@ landscape. This behavior of the complexity can be used to explain a rich
 variety of phenomena in the equilibrium and dynamics of the pure models: the
 threshold energy $E_\mathrm{th}$ corresponds to the average energy at the
 dynamic transition temperature, and the asymptotic energy reached by slow aging
-dynamics, and to the algorithmic limit $E_\mathrm{alg}$.
+dynamics.
+
+In the pure models, $E_\mathrm{th}$ also corresponds to the \emph{algorithmic
+threshold} $E_\mathrm{alg}$, defined by the lowest energy reached by local
+algorithms like approximate message passing \cite{ElAlaoui_2020_Algorithmic,
+ElAlaoui_2021_Optimization}. In the spherical models, this has been proven to
+be
+\begin{equation}
+  E_{\mathrm{alg}}=-\int_0^1dq\,\sqrt{f''(q)}
+\end{equation}
+For full RSB systems, $E_\mathrm{alg}=E_0$ and the algorithm can reach the
+ground state energy. For the pure $p$-spin models,
+$E_\mathrm{alg}=E_\mathrm{th}$, where $E_\mathrm{th}$ is the energy at which
+marginal minima are the most common stationary points. Something about the
+topology of the energy function might be relevant to where this algorithmic
+threshold lies.
 
 Things become much less clear in even the simplest mixed models. For instance,
 one mixed model known to have a replica symmetric complexity was shown to
@@ -247,24 +262,8 @@ This should correspond to 1RSB in the complexity. We take
 established to have a 2RSB ground state \cite{Crisanti_2011_Statistical}.
 With this covariance, the model sees a replica symmetric to 1RSB transition at
 $\beta_1=1.70615\ldots$ and a 1RSB to 2RSB transition at
-$\beta_2=6.02198\ldots$. At these transitions, the average energies in equilibrium are
-$\langle E\rangle_1=-0.906391\ldots$ and $\langle E\rangle_2=-1.19553\ldots$,
-respectively, and the ground state energy is $E_0=-1.287\,605\,530\ldots$.
-Besides these typical equilibrium energies, an energy of special interest for
-looking at the landscape topology is the \emph{algorithmic threshold}
-$E_\mathrm{alg}$, defined by the lowest energy reached by local algorithms like
-approximate message passing \cite{ElAlaoui_2020_Algorithmic,
-ElAlaoui_2021_Optimization}. In the spherical models, this has been proven to
-be
-\begin{equation}
-  E_{\mathrm{alg}}=-\int_0^1dq\,\sqrt{f''(q)}
-\end{equation}
-For full RSB systems, $E_\mathrm{alg}=E_0$ and the algorithm can reach the
-ground state energy. For the pure $p$-spin models,
-$E_\mathrm{alg}=E_\mathrm{th}$, where $E_\mathrm{th}$ is the energy at which
-marginal minima are the most common stationary points. Something about the
-topology of the energy function might be relevant to where this algorithmic threshold
-lies. For the $3+16$ model at hand, $E_\mathrm{alg}=-1.275\,140\,128\ldots$.
+$\beta_2=6.02198\ldots$. The typical equilibrium energies at these phase
+transitions are listed in Table~\ref{tab:energies}.
 
 \begin{table}
   \begin{tabular}{l|cc}
-- 
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