Wrote more.

1 files changed, 64 insertions, 33 deletions

diff --git a/stokes.tex b/stokes.tex
index 447fb3b..5a58265 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -137,52 +137,60 @@ resulting weights.
 \section{Gradient descent dynamics}
 
 For a holomorphic Hamiltonian $H$, dynamics are defined by gradient descent on
-$\operatorname{Re}H$. In hermitian geometry, the gradient is given by raising
+$\operatorname{Re}\beta H$. In hermitian geometry, the gradient is given by raising
 an index of the conjugate differential, or
 $\operatorname{grad}f=(\partial^*f)^\sharp$. This implies that, in terms of
 coordinates $u:M\to\mathbb C^N$, gradient descent follows the dynamics
 \begin{equation} \label{eq:flow.raw}
   \dot z^i
-  =-(\partial^*_{\beta}\operatorname{Re}H)h^{\beta\alpha}\partial_\alpha z^i
-  =-\tfrac12(\partial_\beta H)^*h^{\beta\alpha}\partial_\alpha z^i
+  =-(\partial^*_{\gamma}\operatorname{Re}\beta H)h^{\gamma\alpha}\partial_\alpha z^i
+  =-\tfrac12(\beta\partial_\gamma H)^*h^{\gamma\alpha}\partial_\alpha z^i
 \end{equation}
 where $h$ is the Hermitian metric and we have used the fact that, for holomorphic $H$, $\partial^*H=0$.
 
-This can be simplied  by noting that $h^{\beta\alpha}=h^{-1}_{\beta\alpha}$ for
-$h_{\beta\alpha}=(\partial_\beta z^i)^*\partial_\alpha z^i=(J^\dagger
-J)_{\beta\alpha}$ where $J$ is the Jacobian of the coordinate map. Writing
+This can be simplied  by noting that $h^{\gamma\alpha}=h^{-1}_{\gamma\alpha}$ for
+$h_{\gamma\alpha}=(\partial_\gamma z^i)^*\partial_\alpha z^i=(J^\dagger
+J)_{\gamma\alpha}$ where $J$ is the Jacobian of the coordinate map. Writing
 $\partial H=\partial H/\partial z$ and inserting Jacobians everywhere they
 appear, \eqref{eq:flow.raw} becomes
 \begin{equation} \label{eq:flow}
-  \dot z=-\tfrac12(\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T
-  =-\tfrac12(\partial H)^\dagger P
+  \dot z=-\tfrac12(\beta\partial H)^\dagger J^*[J^\dagger J]^{-1}J^T
+  =-\tfrac12\beta^*(\partial H)^\dagger P
 \end{equation}
 which is nothing but the projection of $(\partial H)^*$ into the tangent space of the manifold, with $P=J^*[J^\dagger J]^{-1}J^T$. Note that $P$ is hermitian: $P^\dagger=(J^*[J^\dagger J]^{-1}J^T)^\dagger=J^*[J^\dagger J]^{-1}J^T=P$.
 
-Gradient descent on $\operatorname{Re}H$ is equivalent to Hamiltonian dynamics
-with the Hamiltonian $\operatorname{Im}H$. This is because $(M, h)$ is Kähler
+Gradient descent on $\operatorname{Re}\beta H$ is equivalent to Hamiltonian dynamics
+with the Hamiltonian $\operatorname{Im}\beta H$. This is because $(M, h)$ is Kähler
 and therefore admits a symplectic structure, but that the flow conserves
-$\operatorname{Im}H$ can be shown using \eqref{eq:flow} and the holomorphic property of $H$:
+$\operatorname{Im}\beta H$ can be shown using \eqref{eq:flow} and the holomorphic property of $H$:
 \begin{equation}
   \begin{aligned}
-    \frac d{dt}&\operatorname{Im}H
-    =\dot z\partial\operatorname{Im}H+\dot z^*\partial^*\operatorname{Im}H \\
+    \frac d{dt}&\operatorname{Im}\beta H
+    =\dot z\partial\operatorname{Im}\beta H+\dot z^*\partial^*\operatorname{Im}\beta H \\
     &=\frac i4\left(
-      (\partial H)^\dagger P\partial H-(\partial H)^TP^\dagger(\partial H)^*
+      \beta^*\beta(\partial H)^\dagger P\partial H-\beta\beta^*(\partial H)^TP^\dagger(\partial H)^*
     \right) \\
-    &=\frac i4\left(
+    &=\frac i4|\beta|^2\left(
       (\partial H)^\dagger P\partial H-[(\partial H)^\dagger P\partial H]^*
     \right) \\
-    &=\frac i4\left(
-      \|\partial H\|-\|\partial H\|^*
+    &=\frac i4|\beta|^2\left(
+      \|\partial H\|^2-(\|\partial H\|^*)^2
     \right)=0.
   \end{aligned}
 \end{equation}
-As a result of this conservation law, surfaces of constant $\operatorname{Im}H$
-will be important when evaluting the possible endpoints of trajectories.
+As a result of this conservation law, surfaces of constant $\operatorname{Im}\beta H$
+will be important when evaluting the possible endpoints of trajectories. A consequence of this conservation is that the flow in the energy takes a simple form:
+\begin{equation}
+  \dot H
+  =\dot z\partial H
+  =-\frac12\beta^*(\partial H)^\dagger P H
+  =-\frac12\beta^*\|\partial H\|^2.
+\end{equation}
+In the complex-$H$ plane, dynamics is occurs along straight lines whose
+direction is the same as $\arg \beta^*$.
 
 Let us consider the generic case, where the critical points of $H$ have
-distinct energies. What is the topology of the $C=\operatorname{Im}H$ level
+distinct energies. What is the topology of the $C=\operatorname{Im}\beta H$ level
 set? We shall argue its form by construction. Consider initially the situation
 in the absence of any critical point. In this case the level set consists of a
 single simply connected surface, locally diffeomorphic to $\mathbb R^{2N}$. Now, `place' a generic
@@ -191,27 +199,36 @@ point, the flow is locally
 \begin{equation}
   \begin{aligned}
     \dot z
-    &\simeq-\frac12(\partial\partial H)^\dagger P(z-z_0)
+    &\simeq-\tfrac12\beta^*(\partial\partial H)^\dagger P(z-z_0)^*
   \end{aligned}
 \end{equation}
 The matrix $(\partial\partial H)^\dagger P$ has a spectrum identical to that of
 $(\partial\partial H)^\dagger$ save marginal directions corresponding to the normals to
-manifold. Assuming we are working in a diagonal basis, we find
+manifold. Assuming we are working in a diagonal basis, this becomes
 \begin{equation}
-  \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2)
+  \dot z_i=-\tfrac12\beta^*\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2)
 \end{equation}
 Breaking into real and imaginary parts gives
 \begin{equation}
   \begin{aligned}
     \frac{d\Delta x_i}{dt}&=-\frac12\left(
-      \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\lambda_i\Delta y_i
+      \operatorname{Re}\beta\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i
     \right) \\
     \frac{d\Delta y_i}{dt}&=-\frac12\left(
-      \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\lambda_i\Delta y_i
+      \operatorname{Im}\beta\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i
     \right)
   \end{aligned}
 \end{equation}
-Therefore, in the complex plane defined by each eigenvector of $(\partial\partial H)^\dagger P$ there is a separatrix flow of the form in Figure \ref{fig:local_flow}. The effect of these separatrices in each complex direction of the tangent space $T_{z_0}M$ is to separate that space into four quadrants: two disconnected pieces with greater imaginary part than the critical point, and two with lesser imaginary part. This partitioning implies that the level set of $\operatorname{Im} H=C$ for $C\neq\operatorname{Im}H(z_0)$ is splint into two disconnected pieces, one lying in each of two quadrants corresponding with its value relative to that at the critical point.
+Therefore, in the complex plane defined by each eigenvector of
+$(\partial\partial H)^\dagger P$ there is a separatrix flow of the form in
+Figure \ref{fig:local_flow}. The effect of these separatrices in each complex
+direction of the tangent space $T_{z_0}M$ is to separate that space into four
+quadrants: two disconnected pieces with greater imaginary part than the
+critical point, and two with lesser imaginary part. This partitioning implies
+that the level set of $\operatorname{Im}\beta H=C$ for
+$C\neq\operatorname{Im}\beta H(z_0)$ is split into two disconnected pieces, one
+lying in each of two quadrants corresponding with its value relative to that at
+the critical point.
 
 \begin{figure}
   \includegraphics[width=\columnwidth]{figs/local_flow.pdf}
@@ -225,17 +242,17 @@ Therefore, in the complex plane defined by each eigenvector of $(\partial\partia
 Continuing to `insert' critical points whose imaginary energy differs from $C$,
 one repeatedly partitions the space this way with each insertion. Therefore,
 for the generic case with $\mathcal N$ critical points, with $C$ differing in
-value from all critical points, the level set $\operatorname{Im}H=C$ has
+value from all critical points, the level set $\operatorname{Im}\beta H=C$ has
 $\mathcal N+1$ connected components, each of which is simply connected,
-diffeomorphic to $\mathbb R^{2(N-1)-1}$ and connects two sectors of infinity to
+diffeomorphic to $\mathbb R^{2N-1}$ and connects two sectors of infinity to
 each other.
 
 When $C$ is brought to the same value as the imaginary part of some critical
 point, two of these disconnected surfaces pinch grow nearer and pinch together
-at the critical point when $C=\operatorname{Im}H$, as in the black lines of
+at the critical point when $C=\operatorname{Im}\beta H$, as in the black lines of
 Figure \ref{fig:local_flow}. The unstable manifold of the critical point, which
-corresponds with the portion of this surface that flows away, is known as a
-\emph{Lefshetz thimble}.
+corresponds with the portion of this surface that flows away, produce the
+Lefschetz thimble associated with that critical point.
 
 Stokes lines are trajectories that approach distinct critical points as time
 goes to $\pm\infty$. From the perspective of dynamics, these correspond to
@@ -245,12 +262,26 @@ must have the same imaginary energy if they are to be connected by a Stokes
 line. This is not a generic phenomena, but will happen often as one model
 parameter is continuously varied. When two critical points do have the same
 imaginary energy and $C$ is brought to that value, the level set
-$C=\operatorname{Im}H$ sees formally disconnected surfaces pinch together in
+$C=\operatorname{Im}\beta H$ sees formally disconnected surfaces pinch together in
 two places. We shall argue that generically, a Stokes line will exist whenever
 the two critical points in question lie on the same connected piece of this
 surface.
 
-What are the ramifications of this for disordered Hamiltonians? When some process brings two critical points to the same imaginary energy, whether a Stokes line connects them depends on whether the points are separated from each other by the separatrices of one or more intervening critical points. Therefore, we expect that in regions where critical points with the same value of $\operatorname{Im}H$ tend to be nearby, Stokes lines will proliferate, while in regions where critical points with the same value of $\operatorname{Im}H$ tend to be distant compared to those with different $\operatorname{Im}H$, Stokes lines will be rare.
+What are the ramifications of this for disordered Hamiltonians? When some
+process brings two critical points to the same imaginary energy, whether a
+Stokes line connects them depends on whether the points are separated from each
+other by the separatrices of one or more intervening critical points.
+Therefore, we expect that in regions where critical points with the same
+energies tend to be nearby, Stokes lines will proliferate, while in regions
+where critical points with the same energies tend to be distant compared to
+those with different energies, Stokes lines will be rare.
+
+\textcolor{teal}{
+  Here we make a generic argument that, for high-dimensional landscapes with
+  exponentially many critical points, the existence of exponentially many
+  Stokes points depends on the spectrum of the Hessian $\partial\partial H$ of
+  critical points. 
+}
 
 \section{p-spin spherical models}