From 5f7e3b85e0a6c33ec1aa8284c2ddaba0eb99593a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Mar 2022 16:40:39 +0200
Subject: Started using new notation for landmark energies.

---
 stokes.tex | 34 ++++++++++++++++++++++------------
 1 file changed, 22 insertions(+), 12 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index d09ae87..10eda20 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -1698,7 +1698,7 @@ rank-1 saddles.
   \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
+    angle $\varphi$ for various $\Delta$ at $\epsilon_1=\epsilon_{k=2}$, 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
@@ -1707,11 +1707,21 @@ rank-1 saddles.
   }
 \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
+Below $\epsilon_{k=1}$, 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 $180^\circ$.
+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 $\epsilon_{k=1}$, $\epsilon_2-\epsilon_1\propto
+-|\epsilon_1-\epsilon_{k=1}|^2$, $\varphi-90^\circ\propto-|\epsilon_1-\mathcal
+E_1|^{1/2}$, and $\Delta_\textrm{min}\propto|\epsilon_1-\epsilon_{k=1}|$. 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_closest_energy.pdf}
@@ -1720,7 +1730,7 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce
     The energy $\epsilon_2$ of the nearest neighbor stationary points in the
     complex plane to a given real stationary point of energy $\epsilon_1$. The
     dashed line shows $\epsilon_2=\epsilon_1$. The nearest neighbor energy
-    coincides with the dashed line until $\mathcal E_1$, the energy where
+    coincides with the dashed line until $\epsilon_{k=1}$, the energy where
     rank-one saddles vanish, where it peels off.
   }
 \end{figure}
@@ -1735,15 +1745,15 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce
     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
+    any other angle. Below $\epsilon_{k=2}$ but above $\epsilon_{k=1}$, stationary
     points are found at arbitrarily close distance but only at $90^\circ$.
-    Below $\mathcal E_1$, neighboring stationary points are separated by a
+    Below $\epsilon_{k=1}$, 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}
 
-\section{The {\it p}-spin spherical models: numerics}
+\subsection{Pure {\it p}-spin: numerics}
 
 To study Stokes lines numerically, we approximated them by parametric curves.
 If $z_0$ and $z_1$ are two stationary points of the action with
@@ -1937,17 +1947,17 @@ Each integral will be dominated by its value near the maximum of the real part o
 As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can
 only be reliably evaluated with a saddle point when either the period of
 oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore
-expect changes in behavior when $\epsilon=\mathcal E_0$, the ground state energy.
+expect changes in behavior when $\epsilon=\epsilon_{k=0}$, the ground state energy.
 The temperature at which this happens is
 \begin{eqnarray}
-  \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\mathcal E_0+\frac12\frac p{p-1}\sqrt{\mathcal E_0^2-\mathcal E_\mathrm{th}^2}\\
-  \operatorname{Re}\beta_B&=-\mathcal E_0
+  \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_{k=0}+\frac12\frac p{p-1}\sqrt{\epsilon_{k=0}^2-\epsilon_\mathrm{th}^2}\\
+  \operatorname{Re}\beta_B&=-\epsilon_{k=0}
 \end{eqnarray}
 which for all $p\geq2$ has $\operatorname{Re}\beta_A>\operatorname{Re}\beta_B$.
 Therefore, the emergence of zeros in $Z_A$ does not lead to the emergence of
 zeros in the partition function as a whole, because $Z_B$ still produces a
-coherent result (despite the unknown constant factor $\eta(\mathcal E_0)$). It is
-only at $\operatorname{Re}\beta_B=-\mathcal E_0$ where both terms contributing to
+coherent result (despite the unknown constant factor $\eta(\epsilon_{k=0})$). It is
+only at $\operatorname{Re}\beta_B=-\epsilon_{k=0}$ where both terms contributing to
 the partition function at large $N$ involve incoherent integrals near the
 maximum, and only here where the density of zeros is expected to become
 nonzero.
-- 
cgit v1.2.3-70-g09d2

