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| -rw-r--r-- | bezout.tex | 6 |
1 files changed, 3 insertions, 3 deletions
@@ -249,7 +249,7 @@ elements of $J$ are standard complex normal, this corresponds to a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and to our knowledge a closed form is not known. We have worked out an implicit form for this spectrum using the saddle point of a replica symmetric calculation for the -Green function. The result is +Green function. {\color{red} the calculation is standard, we outline it in appendix xx} The result is \begin{widetext} \begin{equation} G(\sigma)=\lim_{n\to0}\int d\alpha\,d\chi\,d\chi^*\frac\alpha2 @@ -259,7 +259,7 @@ Green function. The result is \right\} \end{equation} \end{widetext} -The argument of the exponential has several saddles, but the one with the smallest value of $\mathop{\mathrm{Re}}\alpha$ gives the correct solution \textcolor{red}{\textbf{why????? we never figured this out...}}. +The argument of the exponential has several saddles, but the one with the smallest value of $\mathop{\mathrm{Re}}\alpha$ gives the correct solution \textcolor{red}{\textbf{we have checked this, but a detailed analysis of the saddle-point integration is still needed to justify it.}}. The transition from a one-cut to two-cut singular value spectrum naturally corresponds to the origin leaving the support of the eigenvalue spectrum. @@ -328,7 +328,7 @@ Consider for example the ground-state energy for given $a$, that is, the energy } \end{figure} -\begin{figure}[htpb] +\begin{figure}[htpb]\label{ \centering \includegraphics{fig/desert.pdf} \caption{ |