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authorJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 15:25:18 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 15:25:18 +0100
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Nudging Figure 1 around to minimize blank space...
-rw-r--r--bezout.tex45
1 files changed, 22 insertions, 23 deletions
diff --git a/bezout.tex b/bezout.tex
index 949a55b..bfabdec 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -228,29 +228,6 @@ z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter
vanishes as $a\to1$, where we should recover known results for the real
$p$-spin.
-\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}
-
-
The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
H=\partial\partial H_0-p\epsilon I$, or the Hessian of
\eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The
@@ -285,6 +262,28 @@ knowledge a closed form is not in the literature. We have worked out an implici
this spectrum using the saddle point of a replica symmetric calculation for the
Green function.
+\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 to
the numerator of the Green function \cite{Livan_2018_Introduction} gives
\begin{widetext}