From 25891cd9dfd49dc8bb1e23553fafed36419bfe25 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 8 Dec 2020 18:03:18 +0100
Subject: Spectra figure added.

---
 bezout.tex | 32 +++++++++++++++++++++++++++++++-
 1 file changed, 31 insertions(+), 1 deletion(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 880c490..eb04d2a 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -131,7 +131,7 @@ singular values of $\partial\partial H$, while both are the same as the
 distribution of square-rooted eigenvalues of $(\partial\partial
 H)^\dagger\partial\partial H$.
 
-{\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as 
+{\color{red} {\bf perhaps not here} This expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as 
 $N \Sigma= 
 \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
 \ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact.
@@ -167,6 +167,36 @@ where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
 spectrum of $\partial\partial H$ therefore is that of an ellipse whose center
 is shifted by $p\epsilon$.
 
+\begin{figure}
+  \centering
+
+  \raisebox{60pt}{$|\epsilon|=0$}
+  \hfill
+  \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_0.0.pdf}}
+  \hspace{2pt}
+  \includegraphics{fig/spectra_singular_0.0.pdf}\\ 
+  \raisebox{60pt}{$|\epsilon|=\frac12|\epsilon_{\mathrm{th}}|$}
+  \hfill
+  \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_0.5.pdf}}
+  \hspace{2pt}
+  \includegraphics{fig/spectra_singular_0.5.pdf}\\
+  \raisebox{60pt}{$|\epsilon|=|\epsilon_{\mathrm{th}}|$}
+  \hfill
+  \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_1.0.pdf}}
+  \hspace{2pt}
+  \includegraphics{fig/spectra_singular_1.0.pdf}\\ 
+  \raisebox{60pt}{$|\epsilon|=\frac32|\epsilon_{\mathrm{th}}|$}
+  \hfill
+  \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_1.5.pdf}}
+  \hspace{2pt}
+  \includegraphics{fig/spectra_singular_1.5.pdf}
+
+  \caption{
+    Eigenvalue and singular value spectra for the matrix $\partial\partial H$
+    for $a=\frac54$, $\kappa=\frac34e^{i3\pi/4}$, and $\epsilon=i|\epsilon|$ for values values of $|\epsilon|$.
+  } \label{fig:spectra}
+\end{figure}
+
 The eigenvalue spectrum of the Hessian of the real part, or equivalently the
 eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the
 singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
-- 
cgit v1.2.3-54-g00ecf

