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diff --git a/topology.tex b/topology.tex index 4c14207..102dc6b 100644 --- a/topology.tex +++ b/topology.tex @@ -141,6 +141,29 @@ putting strong constraints on the resulting topology and geometry. \subsection{Topology of solutions to many equations and the satisfiability transition} \begin{figure} + \includegraphics{figs/action_1.pdf} + \hspace{-3.5em} + \includegraphics{figs/action_3.pdf} + + \caption{ + The effective action governing the as a function of the overlap + $m=\frac1N\mathbf x\cdot\mathbf x_0$ with the height direction for two + different homogeneous polynomial functions and a variety of $V_0$. In both + plots $\alpha=\frac12$. \textbf{Left:} With linear functions there are two + regimes. For small $V_0$, there are maxima at $m=\pm m^*$ where the action + is zero, while after the satisfiability transition at $V_0=V_\text{\textsc{sat}}=1$, $m^*$ + goes to zero and the action becomes negative. \textbf{Left:} With nonlinear + functions there are four regimes. For small $V_0$ the behavior is the same + as in the linear case, with zero action. After an onset transition at + $V_0=V_\text{on}\simeq1.099$ the maxima are at the edge of validity of the + action and the action is positive. At a shattering transition at + $V_0=V_\text{sh}\simeq1.394$, $m^*$ goes to zero and the action is positive. + Finally, at the satisfiability transition + $V_0=V_\text{\textsc{sat}}\simeq1.440$ the action becomes negative. + } +\end{figure} + +\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} |