From b008d32d061c1fe729b04040d099623eb275753f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Mar 2022 15:37:48 +0200
Subject: Added some numeric evidence.

---
 figs/numerics_angle_gap_32.pdf    | Bin 0 -> 12979 bytes
 figs/numerics_prob_eigenvalue.pdf | Bin 0 -> 12534 bytes
 stokes.bib                        |  28 +++++++++++++++++++++++
 stokes.tex                        |  46 +++++++++++++++++++++++++++++++++++++-
 4 files changed, 73 insertions(+), 1 deletion(-)
 create mode 100644 figs/numerics_angle_gap_32.pdf
 create mode 100644 figs/numerics_prob_eigenvalue.pdf

diff --git a/figs/numerics_angle_gap_32.pdf b/figs/numerics_angle_gap_32.pdf
new file mode 100644
index 0000000..bdbc71f
Binary files /dev/null and b/figs/numerics_angle_gap_32.pdf differ
diff --git a/figs/numerics_prob_eigenvalue.pdf b/figs/numerics_prob_eigenvalue.pdf
new file mode 100644
index 0000000..0215b44
Binary files /dev/null and b/figs/numerics_prob_eigenvalue.pdf differ
diff --git a/stokes.bib b/stokes.bib
index eb5af66..437be14 100644
--- a/stokes.bib
+++ b/stokes.bib
@@ -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},
diff --git a/stokes.tex b/stokes.tex
index 8565ac2..7564b6b 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -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
-- 
cgit v1.2.3-70-g09d2