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@@ -841,28 +841,19 @@ function is defined by \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 -$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The -Hessian of the unconstrained Hamiltonian is -\begin{equation} \label{eq:bare.hessian} - \partial_i\partial_jH_0 - =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, -\end{equation} -which makes its ensemble that of Gaussian complex symmetric matrices, when the -anomalous direction normal to the constraint surface is neglected. Given its variances -$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and -$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is -constant inside the ellipse +We will consider an ensemble of random matrices $A=A_0+\lambda_0I$, where the +entries of $A_0$ are complex-normal distributed with variance $\Gamma=1/N$ and +$\lambda_0$ is some 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 \begin{equation} \label{eq:ellipse} - \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{r^{p-2}+|\kappa|}\right)^2+ - \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{r^{p-2}-|\kappa|}\right)^2 - <\frac{p(p-1)}{2r^{p-2}} + \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{\Gamma+|C|}\right)^2+ + \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{\Gamma-|C|}\right)^2 + <1 \end{equation} -where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue -spectrum of $\partial\partial H$ is therefore constant inside the same ellipse -translated so that its center lies at $-p\epsilon$. Examples of these +where $\theta=\frac12\arg C$ \cite{Nguyen_2014_The}. The eigenvalue +spectrum of $A$ is therefore constant inside the same ellipse +translated so that its center lies at $\lambda_0$. Examples of these distributions are shown in the insets of Fig.~\ref{fig:spectra}. The eigenvalue spectrum of the Hessian of the real part is not the @@ -1117,6 +1108,20 @@ which can in turn be used to compute the determinant. Then we will treat the $\delta$-functions and the resulting saddle point equations. The results of these calculations begin around \eqref{eq:bezout}. +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 +$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The +Hessian of the unconstrained Hamiltonian is +\begin{equation} \label{eq:bare.hessian} + \partial_i\partial_jH_0 + =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, +\end{equation} +which makes its ensemble that of Gaussian complex symmetric matrices, when the +anomalous direction normal to the constraint surface is neglected. Given its variances +$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and +$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is +constant inside the ellipse We will now address the $\delta$-functions of \eqref{eq:complex.kac-rice}. These are converted to exponentials by the introduction of auxiliary fields |