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Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 45 |
1 files changed, 38 insertions, 7 deletions
@@ -1349,9 +1349,9 @@ $\gamma^\dagger\gamma$, $\eta^\dagger\gamma$, and $\eta^T\gamma$. The saddle point equations can be used to eliminate all but one of these, the `radius' like term $z^\dagger z$. When combined with the constraint, this term can be related directly to the magnitude of the imaginary part of $z$, since -$z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $Y=\frac1N(z^\dagger +$z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $(\operatorname{Im}s)^2=\frac1N(z^\dagger z-N)$, the specific measure of the distance into the complex plane from the -real sphere. The complexity can then be written +real sphere. The complexity can then be written in terms of $r=z^\dagger z/N$ as \begin{equation} \Sigma = @@ -1361,6 +1361,7 @@ real sphere. The complexity can then be written -\frac{(\operatorname{Re}\epsilon)^2}{R_+^2}-\frac{(\operatorname{Im}\epsilon)^2}{R_-^2} +I_p(\epsilon/\epsilon_\mathrm{th}) \end{equation} +where \begin{equation} R_\pm^2=\frac{p-1}2\frac{(r^{p-2}\pm1)\left[ r^{2(p-1)}\pm(p-1)r^{p-2}(r^2-1)-1 @@ -1368,7 +1369,7 @@ real sphere. The complexity can then be written 1+r^{2(p-2)}\left[p(p-2)(r^2-1)-1\right] } \end{equation} -$I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$. +and the function $I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$ and \begin{equation} \eqalign{ I_p(u) @@ -1384,14 +1385,12 @@ $I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$. \right) } \end{equation} -where the branch of the square roots are chosen such that the real part of the -root has the opposite sign as the real part of $u$, e.g., if +otherwise. The branch of the square roots are chosen such that the real part of +the root has the opposite sign as the real part of $u$, e.g., if $\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part is zero, then the sign is taken so that the imaginary part of the root has the opposite sign of the imaginary part of $u$. -\cite{Auffinger_2012_Random} - \begin{figure} \hspace{4pc} \includegraphics{figs/re_complexity.pdf} @@ -1408,6 +1407,14 @@ opposite sign of the imaginary part of $u$. } \label{fig:p-spin.complexity} \end{figure} +Contours of this complexity for the pure 3-spin are plotted in +Fig.~\ref{fig:p-spin.complexity} for pure and imaginary energy. The thick black +line shows the contour of zero complexity, where stationary points are no +longer found at large $N$. As the magnitude of the imaginary part of the spin +taken greater, more stationary points are found, and at a wider array of +energies. This is also true in other directions into the complex energy plane, +where the story is qualitatively the same. + \begin{figure} \hspace{2pc} \includegraphics{figs/ground_complexity.pdf} @@ -1425,6 +1432,30 @@ opposite sign of the imaginary part of $u$. } \label{fig:ground.complexity} \end{figure} +Something more interesting is revealed if we zoom in on the complexity around +the ground state, shown in Fig.~\ref{fig:ground.complexity}. Here, the region +where most stationary points have a gapped hessian is shaded. The line +separating gapped from ungapped distribution corresponds to the threshold +energy $\epsilon_\mathrm{th}$ in the limit of $(\operatorname{Im}s)^2\to0$. +Above the threshold, the limit of the complexity to zero imaginary component +(or equivalently $r\to1$) also approaches the real complexity, plotted under +the horizontal axis. However, below the threshold this is no longer the case: +here the limit of $(\operatorname{Im}s)^2\to0$ of the complexity of complex +stationary points corresponds to the complexity of \emph{rank one saddles} in +the real problem, and their complexity becomes zero at $\epsilon_1$, where the +complexity of rank one saddles becomes zero \cite{Auffinger_2012_Random}. + +There are several interesting features of the complexity. First is this +inequivalence between the real complexity and the limit of the complex +complexity to zero complex part. It implies, among other things, a desert of +stationary points in the complex plan surrounding the lowest minima, something +we shall see more explicitly in the next section. Second, there is only a small +collection of stationary points that appear with positive complexity and a +gapped spectrum: the small region in Fig.~\ref{fig:ground.complexity} that is +both to the right of the thick line and brightly shaded. We suspect that these +are the only stationary points that are somewhat protected from participation +in Stokes points. + \subsection{Pure \textit{p}-spin: where are my neighbors?} The problem of counting the density of Stokes points in an analytic |