From d40ca90a33d98d2bc3ec76463497dea1b5cccdf4 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 15:23:05 +0100
Subject: Figured out that N doesn't belong in the Green function.

---
 bezout.tex | 8 +++-----
 1 file changed, 3 insertions(+), 5 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index e6ea9c2..949a55b 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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.
-- 
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