From 6df768b8aa400f6e0ab2017797c928dcc831cd64 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 7 Jun 2024 17:18:14 +0200
Subject: Modified large deviation figure.

---
 marginal.tex | 22 ++++++++++++++++++----
 1 file changed, 18 insertions(+), 4 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index c461f81..e2bd24f 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -288,13 +288,27 @@ corresponds to the intersection of the average spectrum with zero, i.e., a
 pseudogap.
 
 \begin{figure}
-  \includegraphics[width=\columnwidth]{figs/large_deviation.pdf}
+  \hspace{1.3em}
+  \includegraphics{figs/spectrum_less.pdf}
+  \hspace{-2em}
+  \includegraphics{figs/spectrum_eq.pdf}
+  \hspace{-2em}
+  \includegraphics{figs/spectrum_more.pdf}
+  \\
+  \includegraphics{figs/large_deviation.pdf}
   \caption{
-    The large deviation function $G_\sigma(\mu)$ defined in
+    The large deviation function $G_0(\mu)$ defined in
     \eqref{eq:large.dev} as a function of the shift $\mu$ to the
-    GOE diagonal. As expected, $G_\sigma(2\sigma)=0$, while for
+    GOE diagonal. $G_0(2\sigma)=0$, while for
     $\mu>2\sigma$ it is negative and for $\mu<2\sigma$ it gains an
-    imaginary part.
+    imaginary part. The top panels show schematically what happens to the
+    spectral density in each of these regimes. For $\mu<2\sigma$, an $N^2$
+    large deviation would be required to fix the smallest eigenvalue to zero
+    and the calculation breaks down, leading to the imaginary part. For
+    $\mu>2\sigma$ the spectrum can satisfy the constraint on the smallest
+    eigenvalue by isolating a single eigenvalue at zero at the cost of an
+    order-$N$ large deviation. At $\mu=2\sigma$, the transition point, the
+    spectrum is pseudogapped or marginal.
   } \label{fig:large.dev}
 \end{figure}
 
-- 
cgit v1.2.3-70-g09d2

