From b93590a71ed9a511b1d2d311bd359865ad47d5a6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 12:21:16 +0100
Subject: Added some equation labels and shorted some lines.

---
 bezout.tex | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 67544c4..f4ebf7e 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -278,7 +278,7 @@ Green function.
 Introducing replicas to bring the partition function to
 the numerator of the Green function gives
 \begin{widetext}
-  \begin{equation}
+  \begin{equation} \label{eq:green.replicas}
     G(\sigma)=\frac1N\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
@@ -295,7 +295,7 @@ the numerator of the Green function gives
   vectors. Taking the replica-symmetric ansatz leaves all off-diagonal elements
   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}
+  \begin{equation}\label{eq:green.saddle}
     \overline G(\sigma)=\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)
@@ -309,8 +309,11 @@ smallest value of $\mathop{\mathrm{Re}}\alpha_0$ appears gives the correct
 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}
-  \rho(\sigma)=\frac1{i\pi}\left(\lim_{\mathop{\mathrm{Im}}\sigma\to0^+}\overline G(\sigma)-\lim_{\mathop{\mathrm{Im}}\sigma\to0^-}\overline G(\sigma)\right)
+\begin{equation} \label{eq:spectral.density}
+  \rho(\sigma)=\frac1{i\pi}\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...}}
 
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