Restored paragraph on difference between this work and Müller et al.

1 files changed, 19 insertions, 0 deletions

diff --git a/marginal.tex b/marginal.tex
index 85743c3..3902aa8 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1705,6 +1705,25 @@ self-similarity and stochastic stability of minima have
 recently been suggested as a route to understanding this problem, but this approach is
 still in its infancy \cite{Urbani_2024_Statistical}.
 
+The title of our paper and that of \citeauthor{Muller_2006_Marginal} suggest
+they address the same topic, but this is not the case
+\cite{Muller_2006_Marginal}. That work differs in three important and
+fundamental ways. First, it describes minima of the TAP free energy and
+involves peculiarities specific to the TAP. Second, it describes dominant
+minima which happen to be marginal, not a condition for finding subdominant marginal minima. Finally, it
+focuses on minima with a single soft direction (which are the typical minima of
+the low temperature Sherrington--Kirkpatrick TAP free energy), while we aim to
+avoid such minima in favor of ones that have a pseudogap (which we argue are relevant
+to out-of-equilibrium dynamics). The fact that the typical minima studied by
+\citeauthor{Muller_2006_Marginal} are not marginal in this latter sense may
+provide an intuitive explanation for the seeming discrepancy between the proof
+that the low-energy Sherrington--Kirkpatrick model cannot be sampled
+\cite{ElAlaoui_2022_Sampling} and the proof that a message passing algorithm
+can find near-ground states \cite{Montanari_2021_Optimization}: the algorithm
+finds the atypical low-lying states that are marginal in the sense considered
+here but cannot find the typical ones that are marginal in the sense of
+\citeauthor{Muller_2006_Marginal}.
+
 \begin{acknowledgements}
   JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
 \end{acknowledgements}