From ddf2f6ed656635c40ec35027f6050ef401fc8d03 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 29 Mar 2022 14:30:17 +0200
Subject: Lots of writing in the last two sections.

---
 stokes.tex | 156 ++++++++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 113 insertions(+), 43 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 1c60a0d..8565ac2 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -842,7 +842,7 @@ imaginary eigenvectors that contribute to its thimble. The matrix of
 eigenvectors can therefore be written $U=i^kO$ for an orthogonal matrix $O$,
 and with all eigenvectors canonically oriented $\det O=1$. We therefore have
 $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action before any Stokes points,
-\begin{equation}
+\begin{equation} \label{eq:real.thimble.partition.function}
   Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)}
 \end{equation}
 
@@ -1713,6 +1713,18 @@ Below $\mathcal E_1$, where the rank-1 saddle complexity vanishes, the complexit
 \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 $180^\circ$.
 
+\begin{figure}
+  \includegraphics{figs/neighbor_closest_energy.pdf}
+
+  \caption{
+    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
+    rank-one saddles vanish, where it peels off.
+  }
+\end{figure}
+
 \begin{figure}
   \includegraphics{figs/neighbor_plot.pdf}
 
@@ -1731,13 +1743,87 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce
   } \label{fig:nearest.properties}
 \end{figure}
 
+\section{The {\it p}-spin spherical models: 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
+$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we
+take the curve
+\begin{equation}
+  z(t)
+  =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_iP_i^{(1,1)}(2t-1)
+\end{equation}
+where the $g$s are undetermined complex vectors and the $P_i^{(1,1)}(x)$ are the
+Jacobi polynomials, orthogonal on the interval $[-1,1]$ under the weight
+$(1-x)(1+x)$. The Jacobi polynomials are used because they are orthogonal with
+respect to integration over precisely the term they appear inside above. These
+are fixed by minimizing a cost function, which has a global minimum only for
+Stokes lines. Defining
+\begin{equation}
+  \mathcal L(t)
+  = 1-\frac{\operatorname{Re}[\dot z(z(t))^\dagger z'(t)]}{|\dot z(z(t))||z'(t)|}
+\end{equation}
+where $\dot z$ is the flow at $z$ given by \eqref{eq:flow}, this cost is given by
+\begin{equation} \label{eq:cost}
+  \mathcal C=\int_0^1 dt\,\mathcal L(t)
+\end{equation}
+$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$
+whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics.
+Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes
+lines.
+
+We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to
+the parameter vectors $g$. Stokes lines are found or not between points by
+using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$,
+and approximating the cost integral by a finite sum. To sample nearby
+stationary points and assess their propensity for Stokes points, we do the
+following. First, a saddle-finding routine based on Newton's method is run on
+the \emph{real} configuration space of the $p$-spin model. Then, a
+saddle-finding routine is run on the complex configuration space in the close
+vicinity of the real saddle, using random initial conditions in a slowly
+increasing radius of the real stationary point. When this process finds a new
+distinct stationary point, it is done. This method of sampling pairs heavily
+biases the statistics we report here in favor of seeing Stokes points.
+
+Once a pair of nearby stationary points has been found, one real and one in the
+complex plane, their energies are used to compute the phase $\theta$ necessary
+to give $\beta$ in order to set their imaginary energies to the same value, the
+necessary conditions for a Stokes line. A straight line (ignoring even the
+constraint) is thrown between them and then minimized using the cost function
+\eqref{eq:cost} for some initial $m=5$. Once a minimum is found, $m$ is
+iteratively increased several times, each time minimizing the cost in between,
+until $m=20$. If at some point in this process the cost blows up, indicating
+that the solution is running away, the pair is thrown out; this happens
+infrequently. At the end, there are several ways to asses whether a given
+minimized line is a Stokes line: the value of the cost, the integrated
+deviation from the constraint, the integrated deviation from the phase of the
+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 $m$.
+
 \subsection{Pure {\it p}-spin: is analytic continuation possible?}
 
+After this work, one is motivated to ask: can analytic continuation be done in
+even a simple complex model like the pure $p$-spin? Numeric and analytic
+evidence indicates that the project is hopeless if ungapped stationary points
+take a significant weight in the partition function, since for these Stokes
+lines proliferate at even small continuation and there is no hope of tracking
+them. However, for gapped stationary points we have seen compelling evidence
+that suggests they will not participate in Stokes points, at least until a
+large phase rotation of the parameter being continued. This gives some hope for
+continuation of the low-temperature thermodynamic phase of the $p$-spin, where
+weight is concentrated in precisely these points.
+
+Recalling our expression of the single-thimble contribution to the partition
+function for a real stationary point of a real action expanded to lowest order
+in large $|\beta|$ \eqref{eq:real.thimble.partition.function}, we can write for
+the $p$-spin after an infintesimal rotation of $\beta$ into the complex plane
+(before any Stokes points have been encountered)
 \begin{equation}
   \eqalign{
     Z(\beta)
     &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta) \\
-    &=\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}
+    &\simeq\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}
       |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}
       e^{-\beta\mathcal S(s_\sigma)} \\
     &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k)
@@ -1745,17 +1831,20 @@ The result in $\Delta_\textrm{min}$ and the corresponding $\varphi$ that produce
       |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} e^{-\beta N\epsilon}
   }
 \end{equation}
-Following Derrida \cite{Derrida_1991_The},
+where $\mathcal N_\mathrm{typ}(\epsilon,k)$ is the typical number of stationary
+points in a sample of the real $p$-spin model in the energy range $\epsilon$ to
+$\epsilon+d\epsilon$ and with index $k$.
+Following Derrida \cite{Derrida_1991_The}, this is related to the
+\emph{average} number of stationary points in this range at large $N$ by
 \begin{equation}
   \mathcal N_\mathrm{typ}(\epsilon,k)=\overline\mathcal N(\epsilon,k)+\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2}
 \end{equation}
 where $\eta$ is a random, sample-dependant number of order one. This gives two
-contributions to the partition function, but as we shall see it is the
-fluctuations in the number of stationary points at a given energy level that
-governs things at large $|\beta|$, not its total. This gives two terms to the typical partition function
+terms to the typical partition function
 \begin{equation}
   Z_\mathrm{typ}=Z_A+Z_B
 \end{equation}
+where
 \begin{eqnarray} \fl
   Z_A
   \simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k)
@@ -1769,7 +1858,7 @@ governs things at large $|\beta|$, not its total. This gives two terms to the ty
     |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} \\
     =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)}
 \end{eqnarray}
-for
+for functions $f_A$ and $f_B$ defined by
 \begin{eqnarray}
   f_A
   &=-\beta\epsilon+\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta
@@ -1778,23 +1867,32 @@ for
   &=-\beta\epsilon+\frac12\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta
     +i\frac\pi2P(\lambda<0\mid\epsilon)
 \end{eqnarray}
+and $P(\lambda<0\mid\epsilon)$ is the cumulative probability distribution of the eigenvalues of the spectrum given $\epsilon$,
+\begin{equation}
+  P(\lambda<0\mid\epsilon)=\int_{-\infty}^0 d\lambda'\,\rho(\lambda'\mid\epsilon)
+\end{equation}
+and produces the macroscopic index $k/N$.
 Each integral will be dominated by its value near the maximum of the real part of the exponential argument. Assuming that $\epsilon<\epsilon_\mathrm{th}$, this maximum occurs at
 \begin{equation}
-  0=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2}
+  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_A=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2}
 \end{equation}
 \begin{equation}
-  0=-\operatorname{Re}\beta-\epsilon^2
+  0=\frac{\partial}{\partial\epsilon}\operatorname{Re}f_B=-\operatorname{Re}\beta-\epsilon
 \end{equation}
-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 $\epsilon=\epsilon_0$, the ground state energy. The temperature at which this happens is
+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.
+The temperature at which this happens is
 \begin{eqnarray}
-  \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_0+\frac12\frac p{p-1}\sqrt{\epsilon_0^2-\epsilon_\mathrm{th}^2}\\
-  \operatorname{Re}\beta_B&=-\epsilon_0
+  \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
 \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 $\eta(\epsilon_0)$). It is
-only at $\operatorname{Re}\beta_B=-\epsilon_0$ where both terms contributing to
+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
 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.
@@ -1831,35 +1929,7 @@ large-$|\beta|$ saddle-point used to evaluate the thimble integrals. Taking the
 thimbles to the next order in $\beta$ may reveal more explicitly where Stokes
 points become important.
 
-\section{The {\it p}-spin spherical models: 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
-$\operatorname{Re}\mathcal S(z_0)>\operatorname{Re}\mathcal S(z_1)$, then we
-take the curve
-\begin{equation}
-  z(t)
-  =(1-t)z_0+tz_1+(1-t)t\sum_{i=0}^mg_it^i
-\end{equation}
-where the $g$s are undetermined complex vectors. These are fixed by minimizing
-a cost function, which has a global minimum only for Stokes lines. Defining
-\begin{equation}
-  \mathcal L(t)
-  = 1-\frac{\operatorname{Re}[\dot z^*(z(t))\cdot z'(t)]}{|\dot z(z(t))||z'(t)|}
-\end{equation}
-this cost is given by
-\begin{equation}
-  \mathcal C=\int_0^1 dt\,\mathcal L(t)
-\end{equation}
-$\mathcal C$ has minimum of zero, which is reached only by functions $z(t)$
-whose tangent is everywhere parallel to the direction $\dot z$ of the dynamics.
-Therefore, functions that minimize $\mathcal C$ are time-reparameterized Stokes
-lines.
-
-We explicitly computed the gradient and Hessian of $\mathcal C$ with respect to
-the parameter vectors $g$. Stokes lines are found or not between points by
-using the Levenberg--Marquardt algorithm starting from $g^{(i)}=0$ for all $i$,
-and approximating the cost integral by a finite sum.
+\section{Conclusion}
 
 \section*{References}
 \bibliographystyle{unsrt}
-- 
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