From cb6072491121d3482fe7dca44eeb2aa344fb5e43 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 30 Mar 2022 16:49:29 +0200
Subject: Updates to matrix ensemble section.

---
 stokes.tex | 178 +++++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 108 insertions(+), 70 deletions(-)

(limited to 'stokes.tex')

diff --git a/stokes.tex b/stokes.tex
index 10eda20..b98c46b 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -528,14 +528,14 @@ These two facts allow us to find the hessian at stationary points using a
 simpler technique, that of Lagrange multipliers.
 
 Suppose that our complex manifold $\tilde\Omega$ is defined by all points
-$z\in\mathbb C^N$ such that $g(z)=0$ for some holomorphic function $z$. In the
+$z\in\mathbb C^N$ such that $g(z)=0$ for some holomorphic function $g$. In the
 case of the spherical models, $g(z)=\frac12(z^Tz-N)$. Introducing the Lagrange
 multiplier $\mu$, we define the constrained action
 \begin{equation}
   \tilde\mathcal S(z)=\mathcal S(z)-\mu g(z)
 \end{equation}
 The condition for a stationary point is that $\partial\tilde\mathcal S=0$. This implies that, at any stationary point,
-$\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\neq0$, we find the value for $\mu$ as
+$\partial\mathcal S=\mu\partial g$. In particular, if $\partial g^T\partial g\neq0$, we find the value for the Lagrange multiplier $\mu$ as
 \begin{equation} \label{eq:multiplier}
   \mu=\frac{\partial g^T\partial\mathcal S}{\partial g^T\partial g}
 \end{equation}
@@ -554,7 +554,9 @@ $P\operatorname{Hess}\mathcal SP^T$. For notational simplicity we will not
 include this here.
 
 In order to describe the structure of thimbles, one must study the Hessian of
-$\operatorname{Re}\beta\mathcal S$.  We first pose the problem problem as one
+$\operatorname{Re}\beta\mathcal S$, since it is the upward directions in the
+flow on the real action in the vicinity of stationary points which define the thimbles in the first place.  We first
+pose the problem problem as one
 of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$, the hessian of the
 real part of the action with respect to these real variables is
 \begin{equation} \label{eq:real.hessian}
@@ -608,7 +610,7 @@ the more manifestly complex way
 where $\operatorname{Hess}\mathcal S$ is the hessian with respect to $z$ given
 in \eqref{eq:complex.hessian}.
 
-The eigenvalues and eigenvectors of the Hessian are important for evaluating
+The eigenvalues and eigenvectors of the hessian are important for evaluating
 thimble integrals, because those associated with upward directions provide a
 local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb
 R^N$ are such that
@@ -635,7 +637,7 @@ 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, whose solutions are called \emph{Takagi values and vectors} \cite{Takagi_1924_On}. If we did not know the eigenvalues were real, we could
+eigenvalue problem whose solutions are called the \emph{Takagi vectors} of $\operatorname{Hess}\mathcal S$ \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 $\lambda^*=\lambda$.
@@ -645,16 +647,18 @@ in its complex form is that each eigenvalue comes in a pair, since
 \begin{equation}
   \beta\operatorname{Hess}\mathcal S(iv)=i\lambda v^*=-\lambda(iv)
 \end{equation}
-Therefore, if $\lambda$ is an eigenvalue of the hessian with eigenvector $v$,
-than so is $-\lambda$, with associated eigenvector $iv$, rotated in the complex
+Therefore, if $\lambda$ satisfies \eqref{eq:generalized.eigenproblem} with Takagi vector $v$,
+than so does $-\lambda$, with associated Takagi $iv$, rotated in the complex
 plane. It follows that each stationary point has an equal number of descending
-and ascending directions, e.g. the index of each stationary point is $N$. For a
-stationary point in a real problem this might seem strange, because there are
+and ascending directions, e.g., the index of each stationary point is $N$. For
+a stationary point in a real problem this might seem strange, because there are
 clear differences between minima, maxima, and saddles of different index.
-However, we can quickly see here that for a such a stationary point, its $N$
-real eigenvectors which determine its index in the real problem are accompanied
-by $N$ purely imaginary eigenvectors, pointing into the complex plane and each
-with the negative eigenvalue of its partner.
+However, for a such a stationary point, its $N$ real Takagi vectors that
+determine its index in the real problem are accompanied by $N$ purely imaginary
+Takagi vectors, pointing into the complex plane and each with the negative
+eigenvalue of its partner. A real minimum on the real manifold therefore has
+$N$ downward directions alongside its $N$ upward ones, all pointing directly
+into complex configuration space.
 
 The effect of changing the phase of $\beta$ is revealed by
 \eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and
@@ -664,27 +668,27 @@ dividing both sides by $|\beta|e^{i\phi/2}$, one finds
   =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^*
   =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^*
 \end{equation}
-Therefore, one only needs to consider solutions to the eigenproblem for the
+Therefore, one only needs to consider solutions to the Takagi problem for the
 action alone, $\operatorname{Hess}\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the
-resulting vectors by a constant phase corresponding to half the phase of
+resulting Takagi vectors by a constant phase corresponding to half the phase of
 $\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and
 \ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from
 left to right produces a clockwise rotation in the thimbles in the
 complex-$\theta$ plane.
 
-These eigenvalues and vectors can be further related to properties of the
+These eigenvalues associated with the Takagi vectors can be further related to properties of the
 complex symmetric matrix $\beta\operatorname{Hess}\mathcal S$. Suppose that
 $u\in\mathbb R^N$ satisfies the eigenvalue equation
 \begin{equation}
   (\beta\operatorname{Hess} S)^\dagger(\beta\operatorname{Hess} S)u
   =\sigma u
 \end{equation}
-for some positive real $\sigma$ (real because $(\beta\operatorname{Hess}
+for some positive real $\sigma$ (because $(\beta\operatorname{Hess}
 S)^\dagger(\beta\operatorname{Hess} S)$ is self-adjoint). The square root of these
 numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of
 $\beta\operatorname{Hess}\mathcal S$. A direct relationship between these singular
-values and the eigenvalues of the hessian immediately follows by taking an
-eigenvector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem},
+values and the eigenvalues of the real hessian immediately follows by taking a
+Takagi vector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem},
 and writing
 \begin{equation}
   \eqalign{
@@ -698,8 +702,8 @@ and writing
 \end{equation}
 Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues
 of the real hessian are the singular values of the complex matrix
-$\beta\operatorname{Hess}\mathcal S$, and their eigenvectors coincide up to a
-constant complex factor.
+$\beta\operatorname{Hess}\mathcal S$, and the Takagi vectors coincide with the
+eigenvectors of the singular value problem up to a constant complex factor.
 
 \subsection{The conditions for Stokes points}
 
@@ -729,7 +733,9 @@ This is because these stationary points are not adjacent: they are separated
 from each other by the thimbles of other stationary points. This is a
 consistent story in one complex dimension, since the codimension of the
 thimbles is the same as the codimension of the constant imaginary energy
-surface is one, and such a surface can divide space into regions. However, in higher dimensions thimbles do not have codimension high enough to divide space into regions.
+surface is one, and such a surface can divide space into regions. However, in
+higher dimensions thimbles do not have codimension high enough to divide space
+into regions.
 
 \begin{figure}
   \hspace{5pc}
@@ -742,14 +748,35 @@ surface is one, and such a surface can divide space into regions. However, in hi
     descent of the two minima, while the red and orange lines trace those of
     the two saddles. The location of the maxima are marked as points, but their
     thimbles are not shown.
-  }
+  } \label{fig:3d.thimbles}
 \end{figure}
 
+Despite the fact that in higher dimensions, the level sets of constant
+imaginary energy appear to usually be globally connected, thimble intersections
+are still governed by a requirement for adjacency. Fig.~\ref{fig:3d.thimbles}
+shows a projection of the thimbles of an $N=3$ 2-spin model, which is defined
+on the sphere. Because of an inversion symmetry of the model, stationary points
+on opposite sides of the sphere have identical energies, and therefore also
+share the same imaginary energy. However, their thimbles (blue and green in the
+figure) do not intersect. Here, they could not possibly intersect, since the
+real parts of their energy are also the same, and upward flow could therefore
+not connect them.
+
+Stokes lines, when they manifest, are persistent: if a Stokes line connects two
+stationary points and the action is smoothly modified under the constraint that
+the imaginary parts of the two stationary points is held equal, the Stokes line
+will continue to connect them so long as the flow of a third stationary point
+does not sever their connection. This implies that despite being relatively
+low-dimensional surfaces of codimension $N$, thimble connections are seen with
+only a codimension one tuning of parameters, modulo the topological adjacency
+requirement. This means that Stokes points can generically appear when a
+dimension-one curve is followed in parameter space.
+
 \subsection{Evaluating thimble integrals}
 
 After all the work of decomposing an integral into a sub over thimbles, one
 eventually wants to actually evaluate it. For large $|\beta|$ and in the
-absence of any Stokes points, one can come to a nice asymptotic expression. For
+absence of any Stokes points, one can come to a nice asymptotic expression. For a
 thorough account of evaluating these integrals (including \emph{at} Stokes
 points), see Howls \cite{Howls_1997_Hyperasymptotics}.
 
@@ -855,16 +882,17 @@ $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvector
 
 \section{The ensemble of symmetric complex-normal matrices}
 
-We will now begin dealing with the implications of actions defined in very many
-dimensions. We saw in \S\ref{sec:stationary.hessian} that the singular values
-of the complex hessian of the action at any stationary point are important in
-the study of thimbles. Hessians are symmetric matrices by construction. For
-real actions of real variables, the study random symmetric matrices with
-Gaussian entries provides insight into a wide variety of problems. In our case,
-we will find the relevant ensemble is that of random symmetric matrices with
-\emph{complex-normal} entries. In this section, we will introduce this
-distribution, review its known properties, and derive its singular value
-distribution in the large-matrix limit.
+Having introduced the generic method for analytic continuation, we will now
+begin dealing with the implications of actions defined in very many dimensions
+with disorder.  We saw in \S\ref{sec:stationary.hessian} that the singular
+values of the complex hessian of the action at each stationary point are
+important in the study of thimbles. Hessians are symmetric matrices by
+construction. For real actions of real variables, the study of random symmetric
+matrices with Gaussian entries provides insight into a wide variety of
+problems. In our case, we will find the relevant ensemble is that of random
+symmetric matrices with \emph{complex-normal} entries. In this section, we will
+introduce this distribution, review its known properties, and derive its
+singular value distribution in the large-matrix limit.
 
 The complex normal distribution with zero mean is the unique Gaussian
 distribution in one complex variable $Z$ whose variances are
@@ -883,17 +911,21 @@ defined by
     }\right]^{-1}\left[\matrix{z\cr z^*}\right]
   \right\}
 \end{equation}
+This is the same as writing $Z=X+iY$ and requiring that the mutual distribution
+in $X$ and $Y$ be normal with $\overline{X^2}=\Gamma+\operatorname{Re}C$,
+$\overline{Y^2}=\Gamma-\operatorname{Re}C$, and
+$\overline{XY}=\operatorname{Im}C$.
 
 We will consider an ensemble of random $N\times N$ matrices $B=A+\lambda_0I$, where the
 entries of $A$ are complex-normal distributed with variances
-$\overline{|A|^2}=A_0/N$ and $\overline{A^2}=C_0/N$, and $\lambda_0$ is a
-constant shift to its diagonal. The eigenvalue distribution of these matrices
-is already known to take the form of an elliptical ensemble, with constant
-support inside the ellipse defined by
+$\overline{|A_{ij}|^2}=\Gamma_0/N$ and $\overline{A_{ij}^2}=C_0/N$, and $\lambda_0$ is a
+constant shift to its diagonal. The eigenvalue distribution the matrices $A$
+is already known to take the form of an elliptical ensemble in the large-$N$
+limit, with constant support inside the ellipse defined by
 \begin{equation} \label{eq:ellipse}
-  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{1+|C_0|/A_0}\right)^2+
-  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{1-|C_0|/A_0}\right)^2
-  <A_0
+  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{1+|C_0|/\Gamma_0}\right)^2+
+  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{1-|C_0|/\Gamma_0}\right)^2
+  <\Gamma_0
 \end{equation}
 where $\theta=\frac12\arg C_0$ \cite{Nguyen_2014_The}. The eigenvalue
 spectrum of $B$ is therefore constant inside the same ellipse
@@ -904,7 +936,7 @@ When $C=0$ and the elements of $A$ are standard complex normal, the singular
 value distribution of $B$ is a complex Wishart distribution. For $C\neq0$ the
 problem changes, and to our knowledge a closed form is not in the literature.
 We have worked out an implicit form for the singular value spectrum using the
-replica method.
+replica method, first published in \cite{Kent-Dobias_2021_Complex}.
 
 The singular values of $B$ correspond with the square-root of the eigenvalues
 of $B^\dagger B$, but also they correspond to the absolute value of the
@@ -912,8 +944,10 @@ eigenvalues of the real $2N\times2N$ block matrix
 \begin{equation}
   \left[\matrix{\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right]
 \end{equation}
-as we saw in \S\ref{sec:stationary.hessian}. The eigenvalue spectrum of this
-block matrix can be studied by ordinary means.  Defining the `partition function'
+as we saw in \S\ref{sec:stationary.hessian}. The $2N\times2N$ problem is easier
+to treat analytically than the $N\times N$ one because the matrix under study
+is linear in the entries of $B$. The eigenvalue spectrum of this block matrix
+can be studied by ordinary techniques from random matrix theory.  Defining the `partition function'
 \begin{equation} \fl \qquad
   Z(\sigma)=\int dx\,dy\,\exp\left\{
     -\frac12\left[\matrix{x&y}\right]
@@ -927,23 +961,25 @@ implies a Green function
 \begin{equation}
   G(\sigma)=\frac\partial{\partial\sigma}\log Z(\sigma)
 \end{equation}
-This can be put into a manifestly complex form in the same way it was done in \S\ref{sec:stationary.hessian}, using the same linear transformation of $x$ and $y$ into $z$ and $z^*$. This gives
+This can be put into a manifestly complex form in the same way it was done in
+\S\ref{sec:stationary.hessian}, using the same linear transformation of
+$x,y\in\mathbb R^N$ into $z\in\mathbb C^N$. This gives
 \begin{equation}
   \eqalign{
     Z(\sigma)
-    &=\int dz\,dz^*\,\exp\left\{
+    &=\int dz^*dz\,\exp\left\{
       -\frac12\left[\matrix{z^*&-iz}\right]
       \left(\sigma I-
       \left[\matrix{0&(iB)^*\cr iB&0}\right]
     \right)
       \left[\matrix{z\cr iz^*}\right]
     \right\} \\
-    &=\int dz\,dz^*\,\exp\left\{
+    &=\int dz^*dz\,\exp\left\{
       -\frac12\left(
         2z^\dagger z\sigma-z^\dagger B^*z^*-z^TBz
       \right)
     \right\} \\
-    &=\int dz\,dz^*\,\exp\left\{
+    &=\int dz^*dz\,\exp\left\{
       -z^\dagger z\sigma+\operatorname{Re}(z^TBz)
     \right\}
   }
@@ -951,30 +987,30 @@ This can be put into a manifestly complex form in the same way it was done in \S
 which is a general expression for the singular values $\sigma$ of a symmetric
 complex matrix $B$.
 
-
 Introducing replicas to bring the partition function into the numerator of the
 Green function \cite{Livan_2018_Introduction} gives
 \begin{equation} \label{eq:green.replicas} \fl\quad
-  G(\sigma)=\lim_{n\to0}\int dz\,dz^*\,(z^{(0)})^\dagger z^{(0)}
+  G(\sigma)=\lim_{n\to0}\int dz^*dz\,z_0^\dagger z_0
     \exp\left\{
-    -\sum_\alpha^n\left[(z^{(\alpha)})^\dagger z^{(\alpha)}\sigma
-      +\operatorname{Re}\left((z^{(\alpha)})^TBz^{(\alpha)}\right)
+    -\sum_\alpha^n\left[z_\alpha^\dagger z_\alpha\sigma
+      +\operatorname{Re}\left(z_\alpha^TBz_\alpha\right)
     \right]
   \right\},
 \end{equation}
 The average is then made over
-$J$ and Hubbard--Stratonovich is used to change variables to the replica matrices
-$N\alpha_{\alpha\beta}=(z^{(\alpha)})^\dagger z^{(\beta)}$ and
-$N\chi_{\alpha\beta}=(z^{(\alpha)})^Tz^{(\beta)}$, and a series of
+the entries of $B$ and Hubbard--Stratonovich is used to change variables to the
+replica matrices
+$N\alpha_{\alpha\beta}=z_\alpha^\dagger z_\beta$ and
+$N\chi_{\alpha\beta}=z_\alpha^Tz_\beta$, and a series of
 replica vectors. The replica-symmetric ansatz leaves all replica vectors
 zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
 $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
 \begin{equation}\label{eq:green.saddle}
   \eqalign{
-    \overline G(\sigma)=N\lim_{n\to0}\int &d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
+    \overline G(\sigma)=N\lim_{n\to0}\int &d\alpha_0\,d\chi_0^*\,d\chi_0\,\alpha_0
     \exp\left\{nN\left[
-      1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2\right.\right.\cr
-         &\left.\left.+\frac12\log(\alpha_0^2-|\chi_0|^2)+\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right)
+      1+\frac18\Gamma_0\alpha_0^2-\frac{\alpha_0\sigma}2\right.\right.\cr
+         &\left.\left.+\frac12\log(\alpha_0^2-|\chi_0|^2)+\frac12\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right)
     \right]\right\}.
   }
 \end{equation}
@@ -982,13 +1018,13 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
 \begin{figure}
   \centering
 
-  \includegraphics{figs/spectra_0.0.pdf}
-  \includegraphics{figs/spectra_0.5.pdf}\\
-  \includegraphics{figs/spectra_1.0.pdf}
-  \includegraphics{figs/spectra_1.5.pdf}
+  \includegraphics{figs/spectra_00.pdf}
+  \includegraphics{figs/spectra_05.pdf}\\
+  \includegraphics{figs/spectra_10.pdf}
+  \includegraphics{figs/spectra_15.pdf}
 
   \caption{
-    Eigenvalue and singular value spectra of a random matrix $A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $\overline{|A|^2}=A_0=5/4$ and $\overline{A^2}=C_0=\frac34e^{-i3\pi/4}$.
+    Eigenvalue and singular value spectra of a random matrix $B=A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $\overline{|A_{ij}|^2}=\Gamma_0=1$ and $\overline{A_{ij}^2}=C_0=\frac7{10}e^{i\pi/8}$.
     The diaginal shifts differ in each plot, with (a) $\lambda_0=0$, (b)
     $\lambda_0=\frac12|\lambda_{\mathrm{th}}|$, (c)
     $\lambda_0=|\lambda_{\mathrm{th}}|$, and (d)
@@ -1020,16 +1056,18 @@ the origin leaving the support of the eigenvalue spectrum.  Weyl's theorem
 requires that the product over the norm of all eigenvalues must not be greater
 than the product over all singular values \cite{Weyl_1912_Das}.  Therefore, the
 absence of zero eigenvalues implies the absence of zero singular values. The
-determination of the threshold energy---the energy at which the distribution
-of singular values becomes gapped---is reduced to the geometry problem of
+determination of the constant shift $\lambda_0$ at which the distribution
+of singular values becomes gapped is reduced to the geometry problem of
 determining when the boundary of the ellipse defined in \eqref{eq:ellipse}
 intersects the origin, and yields
 \begin{equation} \label{eq:threshold.energy}
-  |\lambda_{\mathrm{th}}|^2
-  =\frac{(1-|\delta|^2)^2}
-  {1+|\delta|^2-2|\delta|\cos(\arg C_0+2\arg\lambda_0)}
+  |\lambda_{\mathrm{gap}}|^2
+  =\Gamma_0\frac{(1-|\delta|^2)^2}
+  {1+|\delta|^2-2|\delta|\cos(\arg\delta+2\arg\lambda_0)}
 \end{equation}
-for $\delta=C_0/A_0$.
+for $\delta=C_0/\Gamma_0$. Because the support is an ellipse, this naturally
+depends on the argument of $\lambda_0$, or the direction in the complex plane
+in which the distribution is shifted.
 
 \section{The \textit{p}-spin spherical models}
 
@@ -1239,7 +1277,7 @@ the results of \S\ref{sec:stationary.hessian} appropriately scaled. The variance
 \end{equation}
 As expected for a real problem, the two variances coincide when $Y=0$.
 The diagonal shift is $-p\epsilon$. In the language of
-\S\ref{sec:stationary.hessian}, this means that $A_0=p(p-1)(1+Y)^{p-2}/2N$,
+\S\ref{sec:stationary.hessian}, this means that $\Gamma_0=p(p-1)(1+Y)^{p-2}/2N$,
 $C_0=p(p-1)2N$, and $\lambda_0=-p\epsilon$.
 
 \begin{equation}
-- 
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