From c097791646adc23704b0230134574e25b78e6794 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 12 Jul 2022 14:31:42 +0200
Subject: Wrote some conclusion.

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 frsb_kac-rice.tex | 41 +++++++++++++++++++++++++++++++++--------
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@@ -1395,14 +1395,39 @@ resulting magnetization of the stationary points.
 
 
 \section{Conclusion}
-We have constructed a  replica solution for the general  problem of finding saddles of random mean-field landscapes, including systems
-with many steps of RSB. 
-For systems with full RSB, we find that minima are, at all energy densities above the ground state, exponentially subdominant with respect to saddles.
-The solution contains valuable geometric information that has yet to be
-extracted in all detail. 
-
-A first and very important application of the method here is to perform the calculation for high dimensional spheres, where it would give us
-a clear understanding of what happens in a low-temperature realistic jamming dynamics \cite{Maimbourg_2016_Solution}. 
+
+We have constructed a  replica solution for the general  problem of finding
+saddles of random mean-field landscapes, including systems with many steps of
+RSB.  For systems with full RSB, we find that minima are, at all energy
+densities above the ground state, exponentially subdominant with respect to
+saddles.  The solution contains valuable geometric information that has yet to
+be extracted in all detail.
+
+A first and very important application of the method here is to perform the
+calculation for high dimensional spheres, where it would give us a clear
+understanding of what happens in a low-temperature realistic jamming dynamics
+\cite{Maimbourg_2016_Solution}. More simply, examining the landscape of a
+spherical model with a glass to glass transition from 1RSB to RS, like the
+$2+4$ model when $a_4$ is larger than we have taken it in our example, might
+give insight into the cases of interest for Gardner physics
+\cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. In any case, our
+analysis of typical 1RSB and FRSB landscapes indicates that the highest energy
+signature of RSB phases at lower energies is in the overlap structure of the
+high-index saddle points. Though measuring the statistics of saddle points is
+difficult to imagine for experiments, this insight could find application in
+simulations of glass formers, where saddle-finding methods are possible.
+
+A second application is to evaluate in more detail the landscape of these RSB
+systems. In particular, examining the complexity of stationary points with
+non-extensive indices (like rank-one saddles), the complexity of pairs of
+stationary points at fixed overlap, or the complexity of barriers
+\cite{Auffinger_2012_Random, Ros_2019_Complexity}. These other properties of
+the landscape might shed light on the relationship between landscape RSB and
+dynamical features, like the algorithmic energy $E_\mathrm{alg}$. For our 1RSB
+example, because $E_\mathrm{alg}$ is just below the energy where
+dominant saddles transition to a RSB complexity, we speculate that
+$E_\mathrm{alg}$ may be related to the statistics of minima connected to the
+saddles at this transition point.
 
 \paragraph{Acknowledgements}
 The authors would like to thank Valentina Ros for helpful discussions.
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