Started generic introduction to matrix ensemble.

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@@ -462,6 +462,7 @@ In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in
 a direction set by the argument of $\beta$.
 
 \subsection{The structure of stationary points}
+\label{sec:stationary.hessian}
 
 The shape of each thimble in the vicinity of a stationary point can be
 described using an analysis of the hessian of the real part of the action at
@@ -813,6 +814,33 @@ $\det U=i^k$. 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.
+
+The complex normal distribution with zero mean is the unique Gaussian
+distribution in one complex variable $Z$ whose variances are
+$\overline{Z^*Z}=\overline{|Z|^2}=\Gamma$ and $\overline{Z^2}=C$. $\Gamma$ is
+positive, and $|C|\leq\Gamma$. The special case of $C=\Gamma$, where the
+variance of the complex variable and its covariance with its conjugate are the
+same, reduces to the ordinary normal distribution. Its probability density
+function is defined by
+\begin{equation}
+  p(z\mid\Gamma,C)=
+  \frac1{\pi\sqrt{\Gamma^2-|C|^2}}\exp\left\{
+    \frac12\left[\matrix{z^*&z}\right]\left[\matrix{
+        \Gamma & C \cr C^* & \Gamma
+    }\right]^{-1}\left[\matrix{z\cr z^*}\right]
+  \right\}
+\end{equation}
+
 The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
 the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
 distribution $\rho$ is therefore related to the unconstrained distribution