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| -rw-r--r-- | topology.tex | 46 |
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diff --git a/topology.tex b/topology.tex index f420411..4d2d660 100644 --- a/topology.tex +++ b/topology.tex @@ -135,6 +135,28 @@ putting strong constraints on the resulting topology and geometry. \subsection{Topology of solutions to many equations and the satisfiability transition} \begin{figure} + \includegraphics[width=0.245\textwidth]{figs/connected.pdf} + \includegraphics[width=0.245\textwidth]{figs/coexist.pdf} + \includegraphics[width=0.245\textwidth]{figs/shattered.pdf} + \includegraphics[width=0.245\textwidth]{figs/gone.pdf} + + \includegraphics{figs/bar.pdf} + + \caption{ + Cartoon of the topology of the solution manifold implied by our + calculation. The arrow shows the vector $\mathbf x_0$ defining the height + function. For $V_0<V_\text{on}$, the manifold has a single connected + component. Above the onset with $V_\text{on}<V_0<V_\text{sh}$, the manifold + has a large connected component around the equator, and many disconnected + pieces in a certain range of latitudes. Above the shattering transition, or + $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, the large connected + component vanishes and small disconnected pieces occupy the entire equatorial + region. Finally, above the satisfiability transition + $V_\text{\textsc{sat}}$ the manifold vanishes. + } \label{fig:cartoons} +\end{figure} + +\begin{figure} \includegraphics{figs/phases_1.pdf} \hspace{-3em} \includegraphics{figs/phases_2.pdf} @@ -155,30 +177,6 @@ putting strong constraints on the resulting topology and geometry. } \end{figure} -\begin{figure} - \includegraphics[width=0.24\textwidth]{figs/connected.pdf} - \hfill - \includegraphics[width=0.24\textwidth]{figs/coexist.pdf} - \hfill - \includegraphics[width=0.24\textwidth]{figs/shattered.pdf} - \hfill - \includegraphics[width=0.24\textwidth]{figs/gone.pdf} - - \includegraphics{figs/bar.pdf} - - \caption{ - Cartoon of the topology of the solution manifold implied by our - calculation. The arrow shows the vector $\mathbf x_0$ defining the height - function. The region of solutions is marked in black, and the critical points - of the height function restricted to this region are marked with a point. - For $\alpha<1$, there are few simply connected regions with most of the - minima and maxima contributing to the Euler characteristic concentrated at - the height $m^*$. For $\alpha\geq1$, there are many simply - connected regions and most of their minima and maxima are concentrated at - the equator. - } \label{fig:cartoons} -\end{figure} - \subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold} \begin{figure} |