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| -rw-r--r-- | figs/numerics_angle_gap_32.pdf | bin | 0 -> 12979 bytes | |||
| -rw-r--r-- | figs/numerics_prob_eigenvalue.pdf | bin | 0 -> 12534 bytes | |||
| -rw-r--r-- | stokes.bib | 28 | ||||
| -rw-r--r-- | stokes.tex | 46 |
4 files changed, 73 insertions, 1 deletions
diff --git a/figs/numerics_angle_gap_32.pdf b/figs/numerics_angle_gap_32.pdf Binary files differnew file mode 100644 index 0000000..bdbc71f --- /dev/null +++ b/figs/numerics_angle_gap_32.pdf diff --git a/figs/numerics_prob_eigenvalue.pdf b/figs/numerics_prob_eigenvalue.pdf Binary files differnew file mode 100644 index 0000000..0215b44 --- /dev/null +++ b/figs/numerics_prob_eigenvalue.pdf @@ -271,6 +271,21 @@ doi = {10.1063/1.1703862} } +@article{Folena_2020_Rethinking, + author = {Folena, Giampaolo and Franz, Silvio and Ricci-Tersenghi, Federico}, + title = {Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed $p$-Spin Model}, + journal = {Physical Review X}, + publisher = {American Physical Society}, + year = {2020}, + month = {8}, + volume = {10}, + pages = {031045}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.10.031045}, + doi = {10.1103/PhysRevX.10.031045}, + issue = {3}, + numpages = {26} +} + @book{Forstneric_2017_Stein, author = {Forstnerič, Franc}, title = {{Stein} Manifolds and Holomorphic Mappings}, @@ -454,6 +469,19 @@ series = {Proceedings of Science} } +@article{Takagi_1924_On, + author = {Takagi, Teiji}, + title = {On an Algebraic Problem Related to an Analytic Theorem of {Carathéodory} and {Fejér} and on an Allied Theorem of {Landau}}, + journal = {Japanese journal of mathematics: transactions and abstracts}, + publisher = {Mathematical Society of Japan (JST)}, + year = {1924}, + number = {0}, + volume = {1}, + pages = {83--93}, + url = {https://doi.org/10.4099%2Fjjm1924.1.0_83}, + doi = {10.4099/jjm1924.1.0_83} +} + @article{Takahashi_2013_Zeros, author = {Takahashi, K and Obuchi, T}, title = {Zeros of the partition function and dynamical singularities in spin-glass systems}, @@ -635,7 +635,7 @@ It therefore follows that the eigenvalues and vectors of the real hessian satisf \beta\operatorname{Hess}\mathcal S v=\lambda v^* \end{equation} a sort of generalized -eigenvalue problem. If we did not know the eigenvalues were real, we could +eigenvalue problem, whose solutions are called \emph{Takagi values and vectors} \cite{Takagi_1924_On}. If we did not know the eigenvalues were real, we could still see it from the second implied equation, $(\beta\operatorname{Hess}\mathcal S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. @@ -1801,6 +1801,50 @@ two stationary points. Among minimized lines these values fall into doubly-peaked histograms which well-separated prospective Stokes lines into `good' and `bad' values for the given level of approximation $m$. +One cannot explicitly study the effect of crossing various landmark energies on +the $p$-spin in the system sizes that were accessible to our study, up to +around $N=64$, as the presence of, e.g., the threshold energy separating +saddles from minima is not noticeable until much larger size +\cite{Folena_2020_Rethinking}. However, we are +able to examine the effect of its symptoms: namely, the influence of the +spectrum of the stationary point in question on the likelihood that a randomly +chosen neighbor will share a Stokes line. + +\begin{figure} + \includegraphics{figs/numerics_prob_eigenvalue.pdf} + + \caption{ + The probability $P_\mathrm{Stokes}$ that a real stationary point will share + a Stokes line with its randomly chosen neighbor as a function of + $|\lambda_\textrm{min}|$, the magnitude of the minimum eigenvalue of the + hessian at the real stationary point. The horizontal axis has been rescaled + to collapse the data at different system sizes $N$. + } \label{fig:numeric.prob.eigenvalue} +\end{figure} + +Data for the likelihood of a Stokes line as a function of the empirical gap +$|\lambda_\textrm{min}|$ of the real stationary point is shown in +Fig.~\ref{fig:numeric.prob.eigenvalue}. There, one sees that the probability of +finding a Stokes line with a near neighbor falls off as an exponential in the +magnitude of the smallest eigenvalue. As a function of system size, the tail +contracts like $N^{-1/2}$, which means that in the thermodynamic limit one +expects the probability of finding such a Stokes line will approach zero +everywhere expect where $\lambda_\textrm{min}\ll1$. This supports the idea that +gapped minima are unlikely to see Stokes lines. + +\begin{figure} + \includegraphics{figs/numerics_angle_gap_32.pdf} + + \caption{ + The probability density function for identified Stokes points as a function + of $|\theta|$, the magnitude of the phase necessary to add to $\beta$ to + reach the Stokes point, at $N=32$ and for several binned + $|\lambda_\textrm{min}|$. As the empirical gap is increased, the population + of discovered Stokes points becomes more concentrated around + $|\theta|=\pi$. + } \label{fig:numeric.angle.gap} +\end{figure} + \subsection{Pure {\it p}-spin: is analytic continuation possible?} After this work, one is motivated to ask: can analytic continuation be done in |