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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 |