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-rw-r--r--bezout.tex53
-rw-r--r--fig/complexity.pdfbin20600 -> 12732 bytes
-rw-r--r--fig/desert.pdfbin21953 -> 14593 bytes
-rw-r--r--fig/spectra_0.0.pdfbin13286 -> 13106 bytes
-rw-r--r--fig/spectra_0.5.pdfbin13722 -> 13611 bytes
-rw-r--r--fig/spectra_1.0.pdfbin14268 -> 14073 bytes
-rw-r--r--fig/spectra_1.5.pdfbin14124 -> 14000 bytes
-rw-r--r--fig/threshold_1.000.pdfbin11523 -> 11450 bytes
-rw-r--r--fig/threshold_1.125.pdfbin14147 -> 14169 bytes
-rw-r--r--fig/threshold_1.325.pdfbin14594 -> 14562 bytes
-rw-r--r--fig/threshold_2.000.pdfbin15513 -> 15334 bytes
11 files changed, 27 insertions, 26 deletions
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}
diff --git a/fig/complexity.pdf b/fig/complexity.pdf
index 82291b2..b68f2cf 100644
--- a/fig/complexity.pdf
+++ b/fig/complexity.pdf
Binary files differ
diff --git a/fig/desert.pdf b/fig/desert.pdf
index 5b65874..e19484a 100644
--- a/fig/desert.pdf
+++ b/fig/desert.pdf
Binary files differ
diff --git a/fig/spectra_0.0.pdf b/fig/spectra_0.0.pdf
index 8844605..ca4ea49 100644
--- a/fig/spectra_0.0.pdf
+++ b/fig/spectra_0.0.pdf
Binary files differ
diff --git a/fig/spectra_0.5.pdf b/fig/spectra_0.5.pdf
index 235ae97..205d964 100644
--- a/fig/spectra_0.5.pdf
+++ b/fig/spectra_0.5.pdf
Binary files differ
diff --git a/fig/spectra_1.0.pdf b/fig/spectra_1.0.pdf
index af2cea0..8eb9cb2 100644
--- a/fig/spectra_1.0.pdf
+++ b/fig/spectra_1.0.pdf
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diff --git a/fig/spectra_1.5.pdf b/fig/spectra_1.5.pdf
index 7859bd4..256c2f0 100644
--- a/fig/spectra_1.5.pdf
+++ b/fig/spectra_1.5.pdf
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diff --git a/fig/threshold_1.000.pdf b/fig/threshold_1.000.pdf
index 107ea10..9192481 100644
--- a/fig/threshold_1.000.pdf
+++ b/fig/threshold_1.000.pdf
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diff --git a/fig/threshold_1.125.pdf b/fig/threshold_1.125.pdf
index 0879251..1c7a075 100644
--- a/fig/threshold_1.125.pdf
+++ b/fig/threshold_1.125.pdf
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diff --git a/fig/threshold_1.325.pdf b/fig/threshold_1.325.pdf
index 83dfb0b..ad68068 100644
--- a/fig/threshold_1.325.pdf
+++ b/fig/threshold_1.325.pdf
Binary files differ
diff --git a/fig/threshold_2.000.pdf b/fig/threshold_2.000.pdf
index fb49ec3..50e6e5a 100644
--- a/fig/threshold_2.000.pdf
+++ b/fig/threshold_2.000.pdf
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