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Diffstat (limited to 'marginal.tex')
| -rw-r--r-- | marginal.tex | 21 |
1 files changed, 12 insertions, 9 deletions
diff --git a/marginal.tex b/marginal.tex index f167ec7..8414310 100644 --- a/marginal.tex +++ b/marginal.tex @@ -75,13 +75,18 @@ compared to stiff minima or saddle points. This ubiquity of behavior suggests that the distribution of marginal minima can be used to bound out-of-equilibrium dynamical behavior. -Despite their importance in a wide variety of in and out of equilibrium settings \cite{Muller_2015_Marginal, Anderson_1984_Lectures, Sommers_1984_Distribution, Parisi_1995-01_On, Horner_2007_Time, Pankov_2006_Low-temperature, Erba_2024_Quenches, Efros_1985_Coulomb, Shklovskii_2024_Half}, it is not straightforward to condition on the marginality of minima using the -traditional methods for analyzing the distribution of minima in rugged -landscapes. Using the method of a Legendre transformation of the Parisi -parameter corresponding to a set of real replicas, one can force the result to -correspond with marginal minima by tuning the value of that parameter \cite{Monasson_1995_Structural}. However, this -results in only a characterization of the threshold energy and cannot characterize marginal minima at -other energies where they are a minority. +Despite their importance in a wide variety of in and out of equilibrium +settings \cite{Muller_2015_Marginal, Anderson_1984_Lectures, +Sommers_1984_Distribution, Parisi_1995-01_On, Horner_2007_Time, +Pankov_2006_Low-temperature, Erba_2024_Quenches, Efros_1985_Coulomb, +Shklovskii_2024_Half, Folena_2022_Marginal}, it is not straightforward to condition on the +marginality of minima using the traditional methods for analyzing the +distribution of minima in rugged landscapes. Using the method of a Legendre +transformation of the Parisi parameter corresponding to a set of real replicas, +one can force the result to correspond with marginal minima by tuning the value +of that parameter \cite{Monasson_1995_Structural}. However, this results in +only a characterization of the threshold energy and cannot characterize +marginal minima at other energies where they are a minority. The alternative approach, used to great success in the spherical spin glasses, is to start by making a detailed understanding of the Hessian matrix at stationary @@ -517,7 +522,6 @@ $E$, Hessian trace $\mu$, and smallest eigenvalue $\lambda^*$ as \delta\big(N\lambda^*-\mathbf s^T\operatorname{Hess}H(\mathbf x,\pmb\omega)\mathbf s\big) \end{aligned} \end{equation} -\end{widetext} where the additional $\delta$-functions \begin{equation} \delta(\mathbf s^T\partial\mathbf g(\mathbf x)) @@ -538,7 +542,6 @@ In practice, this can be computed by introducing replicas to treat the logarithm ($\log x=\lim_{n\to0}\frac\partial{\partial n}x^n$) and introducing another set of replicas to treat each of the normalizations in the numerator ($x^{-1}=\lim_{m\to-1}x^m$). This leads to the expression -\begin{widetext} \begin{equation} \label{eq:min.complexity.expanded} \begin{aligned} \Sigma_{\lambda^*}(E,\mu) |