diff options
Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 238 |
1 files changed, 107 insertions, 131 deletions
diff --git a/topology.tex b/topology.tex index 65cede5..08c3a0b 100644 --- a/topology.tex +++ b/topology.tex @@ -398,38 +398,128 @@ principle behind the calculation of the effective action, or it could be the result of a negative Euler characteristic. To address this problem, we compute also the average of the square of the Euler -characteristic, $\overline{\chi(\Omega)^2}$. This has the benefit of always -being positive, so that the saddle-point approach to the calculation at large -$N$ does not produce complex values even when $\overline{\chi(\Omega)}$ is -negative. We find three saddle points that could contribute to the value of +characteristic, $\overline{\chi(\Omega)^2}$, with details in +Appendix~\ref{sec:rms}. This has the benefit of always being positive, so that +the saddle-point approach to the calculation at large $N$ does not produce +complex values even when $\overline{\chi(\Omega)}$ is negative. We find three +saddle points that could contribute to the value of $\overline{\chi(\Omega)^2}$: two at $\pm m^*$ where -$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$, and one at $m=0$ where +$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$, +and one at $m=0$ where \begin{equation} \frac1N\log\overline{\chi(\Omega)^2}=2\operatorname{Re}\mathcal S_\chi(0) \end{equation} +which is consistent with +$\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$. Such a +correspondence, which indicates that the `annealed' calculation here is also +representative of typical realizations of the constraints, is not always true. +With average squared Euler characteristic we find instabilities of the solution +described here to replica symmetry breaking (\textsc{rsb}). We do not explore +these \textsc{rsb} solutions here, except in the context of $M=1$ in +Section~\ref{sec:ssg}. However, in the following Figures \ref{fig:phases} and +\ref{fig:crossover} we depict the unstable region with shading. -We therefore have four possible topological regimes: +\begin{figure} + \includegraphics[width=0.196\textwidth]{figs/connected.pdf} + \includegraphics[width=0.196\textwidth]{figs/middle.pdf} + \includegraphics[width=0.196\textwidth]{figs/complex.pdf} + \includegraphics[width=0.196\textwidth]{figs/shattered.pdf} + \includegraphics[width=0.196\textwidth]{figs/gone.pdf} + + \hspace{1.5em} + \textbf{Regime I} + \hfill + \textbf{Regime II} + \hfill + \textbf{Regime III} + \hfill + \textbf{Regime IV} + \hfill + \textbf{Regime V} + \hspace{1.5em} + + \caption{ + \textbf{Cartoons of the solution manifold in five topological regimes.} + The solution manifold is shown as a shaded region with a black boundary, + and the height axis $\mathbf x_0$ is a black arrow. In Regime I, the + statistics of the Euler characteristic is consistent with a manifold with a + single simply-connected component. In Regime II, holes occupy the equator + but its most polar regions are + topologically simple. In Regime III, holes dominate and the edge of the + manifold is not necessarily simple. In Regime IV, disconnected components + dominate. In Regime V, the manifold is empty. + } \label{fig:cartoons} +\end{figure} + +We therefore have five possible topological regimes: \begin{itemize} - \item \textbf{Simple connected: - \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}=0$, - $\frac1N\log\overline{\chi(\Omega)}=0$}.} This regime occurs when $m_*^2>0$ + \item \textbf{Regime I: + \boldmath{$\overline{\chi(\Omega)}=2$}.} + + $\frac1N\log\overline{\chi(\Omega)^2}=0$, + $\frac1N\log\overline{\chi(\Omega)}=0$ + This regime occurs when $m_*^2>0$ and $\operatorname{Re}\mathcal S_\chi(0)<0$. Here, $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$, strongly indicating a topology homeomorphic to the $S^{N-M-1}$ sphere. - \item \textbf{Complex connected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}<0$}.} + + \item \textbf{Regime II: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, isolated contribution at \boldmath{$m=\pm m^*$}.} + + Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}<0$. This regime occurs when $m_\text{min}^2>0$ and $\operatorname{Re}\mathcal S_\chi(0)>0$. Here the average Euler characteristic is large and negative. While the topology of the manifold is not necessarily connected in this - regime, holes are more numerous than components. - \item \textbf{Disconnected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}>0$}.} - This regime occurs when $m_\text{min}^2<0$ and $\mathcal S(0)>0$. Here the + regime, holes are more numerous than components. In addition, here + $m_*^2>m_\text{min}^2$, meaning that the solutions at $m=\pm m_*$ are still + present. This indicates that the manifold has topologically simple + boundaries with some separation from the sea of holes. + + \item \textbf{Regime III: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, no contribution at \boldmath{$m=\pm m^*$}.} + + The same as Regime II, but with $m_*^2<m_\text{min}^2$. The solutions at + $m=\pm m_*$ no longer exist, and nontrivial contributions to the Euler + characteristic are made all the way to the solution manifold's boundary. + + \item \textbf{Regime IV: \boldmath{$\overline{\chi(\Omega)}$} very large and positive.} + + Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}>0$. + This regime occurs when $m_\text{min}^2<0$ and $\mathcal S_\chi(0)>0$. Here the average Euler characteristic is large and positive. Large connected - components of the manifold may or may not exist, but disconnected + components of the manifold may or may not exist, but small disconnected components outnumber holes. - \item \textbf{\textsc{Unsat}: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}<0$}.} - There is typically not a manifold at all, indicating that the equations cannot be satisfied. + + \item \textbf{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.} + + Here $\frac1N\log\overline{\chi(\Omega)}<0$, indicating that the average + Euler characteristic shrinks exponentially with $N$. Under most conditions + we conclude this is the \textsc{unsat} regime where no manifold exists, but + there may be circumstances where part of this regime is characterized by + nonempty solution manifolds that are overwhelmingly likely to have Euler + characteristic zero. \end{itemize} +\begin{figure} + \includegraphics{figs/phases_1.pdf} + \hspace{-3em} + \includegraphics{figs/phases_2.pdf} + \hspace{-3em} + \includegraphics{figs/phases_3.pdf} + + \caption{ + \textbf{Topological phase diagram.} + Topological phases of the model for three different homogeneous covariance + functions. The onset transition $V_\text{on}$, shattering transition + $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$ + are indicated when they exist. In the limit of $\alpha\to0$, the behavior + of level sets of the spherical spin glasses are recovered: the final plot + shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy + $E_\text{th}$ correspond with the limits of the satisfiability and + shattering transitions, respectively. Note that for mixed models with + inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of + $V_\text{sh}$. + } \label{fig:phases} +\end{figure} + However, when the magnitude of $V_0$ is sufficiently large, with \begin{equation} V_0^2>V_\text{on}^2\equiv\frac{1-\alpha+\sqrt{1-\alpha}}\alpha f(1) @@ -471,28 +561,6 @@ $V_\text{\textsc{sat}}=V_{\text{\textsc{sat}}\ast}$, and indeed this is the case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$. \begin{figure} - \includegraphics{figs/phases_1.pdf} - \hspace{-3em} - \includegraphics{figs/phases_2.pdf} - \hspace{-3em} - \includegraphics{figs/phases_3.pdf} - - \caption{ - \textbf{Topological phase diagram.} - Topological phases of the model for three different homogeneous covariance - functions. The onset transition $V_\text{on}$, shattering transition - $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$ - are indicated when they exist. In the limit of $\alpha\to0$, the behavior - of level sets of the spherical spin glasses are recovered: the final plot - shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy - $E_\text{th}$ correspond with the limits of the satisfiability and - shattering transitions, respectively. Note that for mixed models with - inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of - $V_\text{sh}$. - } \label{fig:phases} -\end{figure} - -\begin{figure} \includegraphics{figs/phases_12_1.pdf} \hspace{-2.85em} \includegraphics{figs/phases_12_2.pdf} @@ -506,7 +574,7 @@ case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$. Topological phases for models with a covariance function $f(q)=(1-\lambda)q+\lambda\frac12q^2$ for several values of $\lambda$, interpolating between homogeneous linear ($\lambda=0$) and quadratic ($\lambda=1$) constraints. - } + } \label{fig:crossover} \end{figure} The phase diagram implied by these transitions is shown in @@ -589,98 +657,6 @@ Thus we find the average Euler characteristic in this simple example is 2 despite the fact that the possible manifolds resulting from the constraints have characteristics of either 0 or 4. -\begin{figure} - \includegraphics{figs/regime_1.pdf} - \hspace{-3.5em} - \includegraphics{figs/regime_2.pdf} - \hspace{-3.5em} - \includegraphics{figs/regime_3.pdf} - \hspace{-3.5em} - \includegraphics{figs/regime_4.pdf} - - \caption{ - \textbf{Behavior of the action in four regimes.} The effective action $\mathcal S_\chi$ as a function of overlap $m$ with - the height axis for our model with $f(q)=\frac12q^3$ and $\alpha=\frac12$ - at three different target values $V_0$. \textbf{Left: the connected - regime.} The action is maximized with $\mathcal S_\chi(m^*)=0$, and no - stationary points are found with overlap less than $m_\text{min}$ with a - random point on the sphere. \textbf{Center: the onset regime.} The action - is maximized with $\mathcal S_\chi(m_\text{min})>0$, and is positive up - to $m_\text{max}$. No stationary points are found with overlap less than $m_\text{min}$. \textbf{Right: the shattered regime.} The action is - maximized with $\mathcal S_\chi(0)>0$, and is positive up to - $m_\text{max}$. - } -\end{figure} - -With these caveats in mind, we can combine the information from the previous -calculations to reason about what the topology and geometry of $\Omega$ should -be. We know what the average Euler characteristic is, and we know that it -corresponds to the average number of stationary points. We also know these two -averages correspond with each other when restricted to arbitrary latitudes $m$ -associated with an arbitrary height axis $\mathbf x_0$, and we know their value -at each latitude. From this information, we draw the following inferences about -the form of the solution manifold in our four possible regimes. - -\paragraph{The connected regime: \boldmath{$V_0^2<V_\text{on}^2$}.} - -In our calculation above, $\overline{\chi(\Omega)}=2$ could mean a fine-tuned -average like this, or it could indicate the presence of manifold homeomorphic -to $S^{N-M-1}$ for even $N-M-1$. In either case it strongly indicates a single -connected component, whose few minima and maxima are usually found at the -latitude $m^*$. Randomly chosen points on the sphere have a typical nearest -overlap $m^*$ with the solution manifold, but can never have a smaller overlap than -$m_\text{min}$, indicating that the manifold is extensive. - -\paragraph{The onset regime: \boldmath{$V_\text{on}^2<V_0^2<V_\text{sh}^2$}.} - -In this regime $\log\overline{\chi(\Omega)}=O(N)$ but a minimum overlap -$m_\text{min}>0$ still exists. The minimum overlap indicates that the solution -manifold is exclusively made up of extensive components, because the existence -of small components would lead to stationary points near the equator with -respect to a randomly chosen axis $\mathbf x_0$. The solution manifold is -homeomorphic to a topological space with large Euler characteristic like the -product or disjoint union of many spheres. In the former case we would have one -topologically nontrivial connected component, while in the latter we would have -many simple disconnected components; the reality could be a combination of the -two. In the framework of this calculation, it is not possible to distinguish between -these scenarios. In any case, the minima and maxima of the height on the -solution manifold are typically found at latitude $m_\text{min}$ but are found -in exponential number up to the latitude $m_\text{max}$. - -\paragraph{The shattered regime: \boldmath{$V_\text{sh}^2<V_0^2<V_\text{\textsc{sat}}^2$}.} - -Here $\log\overline{\chi(\Omega)}=O(N)$ and the primary contribution to the -number of stationary points and to the Euler characteristic comes from the -equator. This indicates the presence of a large number of small disconnected -components to the solution manifold. While most minima and maxima of the height -are located at the equator $m=0$, they are found in exponential number up to -the latitude $m_\text{max}$. - -\paragraph{The \textsc{unsat} regime: \boldmath{$V_\text{\textsc{sat}}^2<V_0^2$}.} - -In this regime $\log\overline{\chi(\Omega)}<0$, indicating that in most -realizations of the functions $V_k$ the set $\Omega$ is empty. - -\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{ - \textbf{Cartoon of the solution manifold.} - A low-dimensional sketch of the solution manifold in black, with stationary - points of the height function as red points. - 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 large connected components with nontrivial topology. Above the shattering transition, or - $V_\text{sh}<V_0<V_\text{\textsc{sat}}$, small disconnected pieces appear. Finally, above the satisfiability transition - $V_\text{\textsc{sat}}$ the manifold vanishes. - } \label{fig:cartoons} -\end{figure} \paragraph{} Fig.~\ref{fig:cartoons} shows a low-dimensional facsimile of what these |