diff options
Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 85 |
1 files changed, 84 insertions, 1 deletions
@@ -1642,11 +1642,94 @@ peaked around $\varphi=45^\circ$, while above the threshold it is peaked strongly near the minimum allowed $\varphi$. At the threshold, the function becomes extremely flat. +\begin{figure} + \includegraphics{figs/neighbor_thres.pdf} + \caption{ + The scaled two-replica complexity $\Upsilon$ as a function of angle + $\varphi$ with $\epsilon_2=\epsilon_1$, $\Delta=2^{-7}$, and various + $\epsilon_1$. At the threshold, the function undergoes a geometric + transition and becomes sharper with decreasing $\Delta$. + } \label{fig:neighbor.complexity.passing.threshold} +\end{figure} + One can examine the scaling of these curves as $\Delta$ goes to zero. Both above and below the threshold, one finds a quickly-converging limit of $(\Sigma(\epsilon_1,\epsilon_1,\varphi,\Delta)/\Sigma_2(\epsilon_1)-1)/\Delta$. Above the threshold, these curves converge to a function whose peak is always precisely at $45^\circ$, while below they converge to a function with a peak that grows linearly with $\Delta^{-1}$. At the threshold, the scaling is different, and the function approaches a flat function extremely rapidly, as $\Delta^3$. +\begin{figure} + \includegraphics{figs/neighbor_limit_thres_above.pdf} + \hspace{-1em} + \includegraphics{figs/neighbor_limit_thres_at.pdf} + \hspace{-1em} + \includegraphics{figs/neighbor_limit_thres_below.pdf} + \hfill + \includegraphics{figs/neighbor_limit_thres_legend.pdf} + \caption{ + The scaled two-replica complexity $\Upsilon$ as a function of angle + $\varphi$ for various $\Delta$, $\epsilon_2=\epsilon_1$, and \textbf{Left:} + $\epsilon_1=\epsilon_\mathrm{th}+0.001$ \textbf{Center:} + $\epsilon_1=\epsilon_\mathrm{th}$ \textbf{Right:} + $\epsilon=\epsilon_\mathrm{th}-0.001$. All lines have been normalized by + the complexity $\Sigma$ of the real 3-spin model at the same energy, or + (where relevant) by the complexity $\Sigma_2$ of rank-two saddles of the + real 3-spin model. + } +\end{figure} + Thus, there is an abrupt geometric transition in the population of nearest neighbors as the threshold is crossed: above they are broadly distributed at -all angles, while below they are highly concentrated around $90^\circ$. From this analysis it appears that the complexity of the nearest neighbors, at zero distance, behaves as that of the index-2 saddles at all angles, which would imply that the nearest neighbors vanish at the same point as the index-2 saddles. However, this is not the case: we have only shown that this is how the neighbors at \emph{identical energy} scale, which is correct above the threshold, but no longer underneath. +all angles, while below they are highly concentrated around $90^\circ$. From +this analysis it appears that the complexity of the nearest neighbors, at zero +distance, behaves as that of the index-2 saddles at all angles, which would +imply that the nearest neighbors vanish at the same point as the index-2 +saddles. However, this is not the case: we have only shown that this is how the +neighbors at \emph{identical energy} scale, which is correct above the +threshold, but no longer underneath. + +If an energy is taken under the threshold and the two-replica complexity +maximized with respect to both $\epsilon_2$ and $\varphi$, one finds that as +$\Delta\to0$, $\epsilon_2\to\epsilon_1$, as must be the case the find a +positive complexity at zero distance, but the maximum is never at +$\epsilon_2=\epsilon_1$, but rather at a small distance $\Delta\epsilon$ that +decreases with decreasing $\Delta$ like $\Delta^2$. When the complexity is +maximized in both parameters, one finds that, in the limit as $\Delta\to0$, the +peak is at $90^\circ$ but has a height equal to $\Sigma_1$, the complexity of +rank-1 saddles. + +\begin{figure} + \includegraphics{figs/neighbor_energy_limit.pdf} + \caption{ + The two-replica complexity $\Upsilon$ scaled by $\Sigma_1$ as a function of + angle $\varphi$ for various $\Delta$ at $\epsilon_1=\mathcal E_2$, the + point of zero complexity for rank-two saddles in the real problem. + \textbf{Solid lines:} The complexity evaluated at the value of $\epsilon_2$ + which leads to the largest maximum value. As $\Delta$ varies this varies + like $\epsilon_2-\epsilon_1\propto\Delta^2$. \textbf{Dashed lines:} The + complexity evaluated at $\epsilon_2=\epsilon_1$. + } +\end{figure} + +Below $\mathcal E_1$, where the rank-1 saddle complexity vanishes, the complexity of stationary points of any type at zero distance is negative. To find what the nearest population looks like, one must find the minimum $\Delta$ at which the complexity is nonnegative, or +\begin{equation} + \Delta_\textrm{min}=\operatorname{argmin}_\Delta\left(0\leq\max_{\epsilon_2, \varphi}\Upsilon(\epsilon_1,\epsilon_2,\Delta,\varphi)\right) +\end{equation} +The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produces it is plotted in Fig.~\ref{fig:nearest.properties}. As the energy is brought below $\mathcal E_1$, $\epsilon_2-\epsilon_1\propto -|\epsilon_1-\mathcal E_1|^2$, $\varphi-90^\circ\propto-|\epsilon_1-\mathcal E_1|^{1/2}$, and $\Delta_\textrm{min}\propto|\epsilon_1-\mathcal E_1|$. The fact that the population of nearest neighbors has a energy lower than the stationary point gives some hope for the success of continuation involving these points: since Stokes points only lead to a change in weight when they involve upward flow from a point that already has weight, neighbors that have a lower energy won't be eligible to be involved in a Stokes line that causes a change of weight until the phase of $\beta$ has rotated almost $180^\circ$. + +\begin{figure} + \includegraphics{figs/neighbor_plot.pdf} + + \caption{ + The properties of the nearest neighbor saddles as a function of energy + $\epsilon$. Above the threshold energy $\mathcal E_\mathrm{th}$, stationary + points are found at arbitrarily close distance and at all angles $\varphi$ + in the complex plane. Below $\mathcal E_\mathrm{th}$ but above $\mathcal + E_2$, stationary points are still found at arbitrarily close distance and + all angles, but there are exponentially more found at $90^\circ$ than at + any other angle. Below $\mathcal E_2$ but above $\mathcal E_1$, stationary + points are found at arbitrarily close distance but only at $90^\circ$. + Below $\mathcal E_1$, neighboring stationary points are separated by a + minimum squared distance $\Delta_\textrm{min}$, and the angle they are + found at drifts. + } \label{fig:nearest.properties} +\end{figure} \subsection{Pure {\it p}-spin: is analytic continuation possible?} |