From 61b5ac271085fe9ecf5d1fb57d95180b90a48973 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 11 Dec 2020 13:36:59 +0100
Subject: Switched all fonts to Times.

---
 bezout.tex | 53 +++++++++++++++++++++++++++--------------------------
 1 file changed, 27 insertions(+), 26 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 6c24058..896e705 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -2,6 +2,7 @@
 
 \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode?
 \usepackage[T1]{fontenc} % vector fonts plz
+\usepackage{newtxtext,newtxmath}
 \usepackage[
   colorlinks=true,
   urlcolor=purple,
@@ -9,7 +10,7 @@
   filecolor=purple,
   linkcolor=purple
 ]{hyperref} % ref and cite links with pretty colors
-\usepackage{amsmath, amssymb, graphicx, xcolor} % standard packages
+\usepackage{amsmath, graphicx, xcolor} % standard packages
 
 \begin{document}
 
@@ -199,7 +200,6 @@ where the argument of the exponential is
     +\operatorname{Re}\left\{
       \frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b
     \right\}.
-    \nonumber % He's too big!
   \end{equation}
   The integral of the distribution $\rho$ of eigenvalues of $\partial\partial
   H$ comes from the Hessian and is dependant on $a$ alone. This function has a
@@ -251,28 +251,6 @@ is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and
 to our knowledge a closed form is not in the literature.  We have worked out an
 implicit form for this spectrum using the replica method.
 
-\begin{figure}[htpb]
-  \centering
-
-  \includegraphics{fig/spectra_0.0.pdf}
-  \includegraphics{fig/spectra_0.5.pdf}\\
-  \includegraphics{fig/spectra_1.0.pdf}
-  \includegraphics{fig/spectra_1.5.pdf}
-
-  \caption{
-    Eigenvalue and singular value spectra of the matrix $\partial\partial H$
-    for $p=3$, $a=\frac54$, and $\kappa=\frac34e^{-i3\pi/4}$ with (a)
-    $\epsilon=0$, (b) $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
-    $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
-    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each inset
-    shows the support of the eigenvalue distribution. The solid line on each
-    plot shows the distribution of singular values, while the overlaid
-    histogram shows the empirical distribution from $2^{10}\times2^{10}$ complex
-    normal matrices with the same covariance and diagonal shift as
-    $\partial\partial H$.
-  } \label{fig:spectra}
-\end{figure}
-
 Introducing replicas to bring the partition function into the numerator of the
 Green function \cite{Livan_2018_Introduction} gives
 \begin{widetext}
@@ -301,6 +279,29 @@ Green function \cite{Livan_2018_Introduction} gives
     \nonumber % He's too long, and we don't cite him (now)!
   \end{equation}
 \end{widetext}
+
+\begin{figure}[b]
+  \centering
+
+  \includegraphics{fig/spectra_0.0.pdf}
+  \includegraphics{fig/spectra_0.5.pdf}\\
+  \includegraphics{fig/spectra_1.0.pdf}
+  \includegraphics{fig/spectra_1.5.pdf}
+
+  \caption{
+    Eigenvalue and singular value spectra of the matrix $\partial\partial H$
+    for $p=3$, $a=\frac54$, and $\kappa=\frac34e^{-i3\pi/4}$ with (a)
+    $\epsilon=0$, (b) $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
+    $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
+    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each inset
+    shows the support of the eigenvalue distribution. The solid line on each
+    plot shows the distribution of singular values, while the overlaid
+    histogram shows the empirical distribution from $2^{10}\times2^{10}$ complex
+    normal matrices with the same covariance and diagonal shift as
+    $\partial\partial H$.
+  } \label{fig:spectra}
+\end{figure}
+
 The argument of the exponential has several saddles. The solutions $\alpha_0$
 are the roots of a sixth-order polynomial, and the root with the smallest value
 of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct
@@ -380,7 +381,7 @@ already underlined in \cite{Bogomolny_1992_Distribution}). The full
 $\epsilon$-dependence of the real $p$-spin is recovered by this limit as
 $\epsilon$ is varied.
 
-\begin{figure}[htpb]
+\begin{figure}[b]
   \centering
   \includegraphics{fig/desert.pdf}
   \caption{
@@ -402,7 +403,7 @@ tells us that this is a necessity: otherwise a small perturbation of the $J$s
 could produce an unusually deep solution to the real problem, in a region where
 this should not happen.
 
-\begin{figure}[htpb]
+\begin{figure}[t]
   \centering
 
   \includegraphics{fig/threshold_2.000.pdf}
-- 
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