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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 130b7e9..2e0b971 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -664,7 +664,7 @@ is given by a Parisi matrix with parameters $x_1,\ldots,x_k$ and $q_1,\ldots,q_k$, then the parameters $\hat\beta$, $r_d$, $d_d$, $\tilde x_1,\ldots,\tilde x_{k-1}$, and $\tilde c_1,\ldots,\tilde c_{k-1}$ for the complexity in the ground state are -\begin{align} +\begin{align}\label{eq:equilibrium.complexity.map} \hat\beta=\lim_{\beta\to\infty}\beta x_k && \tilde x_i=\lim_{\beta\to\infty}\frac{x_i}{x_k} @@ -846,6 +846,22 @@ that this is the line of stability for the replica symmetric saddle. \section{General solution: examples} \label{sec:examples} +Though we have only written down an easily computable complexity along a +specific (and often uninteresting) line in energy and stability, this +computable (supersymmetric) solution gives a numeric foothold for computing the +complexity in the rest of that space. First, +\eqref{eq:ground.state.free.energy.cont} is \emph{maximized} with respect +to its parameters, since the equilibrium solution is equivalent to a +variational problem. Second, the mapping \eqref{eq:equilibrium.complexity.map} +is used to find the corresponding Kac--Rice saddle parameters in the ground +state. With these parameters in hand, small steps are then made in energy $E$ +or stability $\mu$, after which known values are used as the initial condition +for a saddle-finding problem. In this section, we use this basic numeric idea +to map out the complexity for two representative examples: a model with a +$2RSB$ equilibrium ground state and therefore $1RSB$ complexity in its +vicinity, and a model with a $FRSB$ equilibrium ground state, and therefore +$FRSB$ complexity as well. + \subsection{1RSB complexity} It is known that by choosing a covariance $f$ as the sum of polynomials with @@ -1002,17 +1018,22 @@ the phase boundary where $q_1$ goes to one. \begin{figure} \centering - \includegraphics{figs/24_func.pdf} + \includegraphics{figs/24_phases.pdf} + \caption{ + } \label{fig:frsb.phases} +\end{figure} + +\begin{figure} + \raggedright + \hspace{2em}\includegraphics{figs/24_opt_q(x).pdf} + \hspace{1em} + \includegraphics{figs/24_opt_xMax.pdf} + \\\vspace{1em} + \includegraphics{figs/24_opt_r(x).pdf} \hspace{1em} - \includegraphics{figs/24_qmax.pdf} + \includegraphics{figs/24_opt_d(x).pdf} \caption{ - \textbf{Left:} The spectrum $\chi$ of the replica matrix in the complexity - of dominant saddles for the $2+4$ model at several energies. - \textbf{Right:} The cutoff $q_{\mathrm{max}}$ for the nonlinear part of the - spectrum as a function of energy $E$ for both dominant saddles and marginal - minima. The colored vertical lines show the energies that correspond to the - curves on the left. } \label{fig:24.func} \end{figure} @@ -1031,7 +1052,7 @@ the phase boundary where $q_1$ goes to one. fixed energy $E$. Solid lines show the result of a FRSB ansatz and dashed lines that of a RS ansatz. All paired parameters coincide at the ground state energy, as expected. - } \label{fig:2rsb.comparison} + } \label{fig:frsb.comparison} \end{figure} @@ -1246,6 +1267,9 @@ 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}. +\paragraph{Acknowledgements} +J K-D and J K would like to thank Valentina Ros for helpful discussions. + \paragraph{Funding information} J K-D and J K are supported by the Simons Foundation Grant No. 454943. |