From 8522f4e4b02d18c55380006852031d87da68684c Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 17 Jul 2023 14:53:20 +0200
Subject: Added discussion of possible differences in atypical calculation.

---
 2-point.tex | 36 ++++++++++++++++++++++++++++--------
 1 file changed, 28 insertions(+), 8 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 4ce640a..7a89651 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -174,7 +174,7 @@ find rich structure in this population, their properties pivot around the
 debunked threshold energy, and the apparent attractors of long-time dynamics
 are not distinguished by this measure. Moreover, we show that the usual picture of a
 marginal `manifold' of inherent states separated by subextensive barriers \cite{Kurchan_1996_Phase} is only true
-at the threshold energy, while at other energies marginal minima are far apart
+at the threshold energy, while at other energies typical marginal minima are far apart
 and separated by extensive barriers. Therefore, with respect to the problem of
 dynamics this paper merely deepens the outstanding issues.
 
@@ -333,7 +333,7 @@ model is plotted in Fig.~\ref{fig:complexities}.
 
 Our results are in the form of the two-point complexity, which is defined as
 the logarithm of the number of stationary points with energy $E_1$ and
-stability $\mu_1$ that lie at an overlap $q$ with a reference stationary point
+stability $\mu_1$ that lie at an overlap $q$ with a typical reference stationary point
 whose energy is $E_0$ and stability is $\mu_0$. When the complexity is
 positive, there are exponentially many stationary points with the given
 properties conditioned on the existence of the reference one. When it is zero,
@@ -512,12 +512,12 @@ energy, the energy at which most stationary points are marginal.
 \end{figure}
 
 This leads us to some general conclusions. First, at all energy densities
-except at the threshold energy, \emph{marginal minima are separated by
+except at the threshold energy, \emph{typical marginal minima are separated by
 extensive energy barriers}. Therefore, the picture of a marginal
 \emph{manifold} of many (even all) marginal states lying arbitrarily close and
 being connected by subextensive energy barriers can only describe the
-collection of marginal minima at the threshold energy. At energies both below and above,
-marginal minima are isolated from each other.
+collection of marginal minima at the threshold energy, or an atypical population of marginal minima. At energies both below and above the threshold energy,
+typical marginal minima are isolated from each other.
 
 We must put a small caveat here: in \emph{any} situation, this calculation
 admits order-one other marginal minima to lie a subextensive distance from the
@@ -531,14 +531,14 @@ cluster cannot describe aging, since the overlap with the initial condition
 will never change from one.
 
 This has implications for how quench dynamics should be interpreted. When
-marginal states are approached above the threshold energy, they must have been
+typical marginal states are approached above the threshold energy, they must have been
 via the neighborhood of saddles with an extensive index, not other marginal
-states. On the other hand, marginal states approached below the threshold
+states. On the other hand, typical marginal states approached below the threshold
 energy must  be reached after an extensive distance in configuration space
 without encountering any stationary point. The geometric conditions of the
 neighborhoods above and below are quite different, but the observed aging
 dynamics don't appear to qualitatively change \cite{Folena_2020_Rethinking,
-Folena_2021_Gradient}. Therefore, the conditions in the neighborhood of the
+Folena_2021_Gradient}. Therefore, if the marginal minima attracting dynamics are typical, the conditions in the neighborhood of the
 marginal minimum eventually reached at infinite time appear to be irrelevant
 for the nature of aging dynamics at any finite time.
 
@@ -556,6 +556,12 @@ spherical models, where \cite{Folena_2020_Rethinking} has shown aging dynamics
 asymptotically approaching marginal states that we have shown have $O(N)$
 saddles separating them, this lesson must be taken all the more seriously.
 
+On the other hand, it is possible that \emph{atypical} marginal minima are
+relevant for attracting the dynamics. Studying these points would require a
+different kind of computation, where the fixed reference point is abandoned and
+both points are treated on equal footing. Such a calculation is beyond the
+scope of this paper, but is clear fodder for future research.
+
 \section{Calculation of the two-point complexity}
 \label{sec:complexity}
 
@@ -1746,6 +1752,20 @@ and so \textsc{rsb} corrections can only decrease the complexity. Therefore,
 the overlap gaps, which correspond to regions of negative complexity, cannot be
 removed by a more detailed saddle point ansatz.
 
+Our calculation studied the neighborhood of typical reference points with the
+given energy and stability. However, it is possible that marginal minima with
+atypical neighborhoods actually attract the dynamics. To determine this, a
+different type of calculation is needed. As our calculation is akin to the
+quenched Franz--Parisi potential, study of atypical neighborhoods would entail
+something like the annealed Franz--Parisi approach, i.e.,
+\begin{equation}
+  \Sigma^*(E_0,\mu_0,E_1,\mu_1,q)=\frac1N\overline{\log\left(
+      \int d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)\,d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)
+  \right)}
+\end{equation}
+which puts the two points on equal footing. This calculation and exploration of
+the atypical neighborhoods it reveals is a clear future direction.
+
 The methods developed in this paper are straightforwardly (if not easily)
 generalized to landscapes with replica symmetry broken complexities
 \cite{Kent-Dobias_2023_How}. We suspect that many of the qualitative features
-- 
cgit v1.2.3-54-g00ecf

