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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} |