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| -rw-r--r-- | bezout.tex | 8 |
1 files changed, 3 insertions, 5 deletions
@@ -289,14 +289,13 @@ Introducing replicas to bring the partition function to the numerator of the Green function \cite{Livan_2018_Introduction} gives \begin{widetext} \begin{equation} \label{eq:green.replicas} - G(\sigma)=\frac1N\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} + G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} \exp\left\{ \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma -\Re\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right) \right] \right\} \end{equation} - \textcolor{red}{\textbf{Not sure if the $N$ belongs here...}} with sums taken over repeated latin indices. The average can then be made over $J$ and Hubbard--Stratonovich used to change variables to replica matrices @@ -306,7 +305,7 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives and 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} - \overline G(\sigma)=\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+\frac{p(p-1)}{16}a^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2) +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right) @@ -320,12 +319,11 @@ solution. A detailed analysis of the saddle point integration is needed to understand why this is so. Given such $\alpha_0$, the density of singular values follows from the jump across the cut, or \begin{equation} \label{eq:spectral.density} - \rho(\sigma)=\frac1{i\pi}\left( + \rho(\sigma)=\frac1{i\pi N}\left( \lim_{\mathop{\mathrm{Im}}\sigma\to0^+}\overline G(\sigma) -\lim_{\mathop{\mathrm{Im}}\sigma\to0^-}\overline G(\sigma) \right) \end{equation} -\textcolor{red}{\textbf{Missing a factor of two? Please check...}} The transition from a one-cut to two-cut singular value spectrum naturally corresponds to the origin leaving the support of the eigenvalue spectrum. |