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| -rw-r--r-- | bezout.tex | 53 | ||||
| -rw-r--r-- | fig/complexity.pdf | bin | 20600 -> 12732 bytes | |||
| -rw-r--r-- | fig/desert.pdf | bin | 21953 -> 14593 bytes | |||
| -rw-r--r-- | fig/spectra_0.0.pdf | bin | 13286 -> 13106 bytes | |||
| -rw-r--r-- | fig/spectra_0.5.pdf | bin | 13722 -> 13611 bytes | |||
| -rw-r--r-- | fig/spectra_1.0.pdf | bin | 14268 -> 14073 bytes | |||
| -rw-r--r-- | fig/spectra_1.5.pdf | bin | 14124 -> 14000 bytes | |||
| -rw-r--r-- | fig/threshold_1.000.pdf | bin | 11523 -> 11450 bytes | |||
| -rw-r--r-- | fig/threshold_1.125.pdf | bin | 14147 -> 14169 bytes | |||
| -rw-r--r-- | fig/threshold_1.325.pdf | bin | 14594 -> 14562 bytes | |||
| -rw-r--r-- | fig/threshold_2.000.pdf | bin | 15513 -> 15334 bytes |
11 files changed, 27 insertions, 26 deletions
@@ -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 Binary files differindex 82291b2..b68f2cf 100644 --- a/fig/complexity.pdf +++ b/fig/complexity.pdf diff --git a/fig/desert.pdf b/fig/desert.pdf Binary files differindex 5b65874..e19484a 100644 --- a/fig/desert.pdf +++ b/fig/desert.pdf diff --git a/fig/spectra_0.0.pdf b/fig/spectra_0.0.pdf Binary files differindex 8844605..ca4ea49 100644 --- a/fig/spectra_0.0.pdf +++ b/fig/spectra_0.0.pdf diff --git a/fig/spectra_0.5.pdf b/fig/spectra_0.5.pdf Binary files differindex 235ae97..205d964 100644 --- a/fig/spectra_0.5.pdf +++ b/fig/spectra_0.5.pdf diff --git a/fig/spectra_1.0.pdf b/fig/spectra_1.0.pdf Binary files differindex af2cea0..8eb9cb2 100644 --- a/fig/spectra_1.0.pdf +++ b/fig/spectra_1.0.pdf diff --git a/fig/spectra_1.5.pdf b/fig/spectra_1.5.pdf Binary files differindex 7859bd4..256c2f0 100644 --- a/fig/spectra_1.5.pdf +++ b/fig/spectra_1.5.pdf diff --git a/fig/threshold_1.000.pdf b/fig/threshold_1.000.pdf Binary files differindex 107ea10..9192481 100644 --- a/fig/threshold_1.000.pdf +++ b/fig/threshold_1.000.pdf diff --git a/fig/threshold_1.125.pdf b/fig/threshold_1.125.pdf Binary files differindex 0879251..1c7a075 100644 --- a/fig/threshold_1.125.pdf +++ b/fig/threshold_1.125.pdf diff --git a/fig/threshold_1.325.pdf b/fig/threshold_1.325.pdf Binary files differindex 83dfb0b..ad68068 100644 --- a/fig/threshold_1.325.pdf +++ b/fig/threshold_1.325.pdf diff --git a/fig/threshold_2.000.pdf b/fig/threshold_2.000.pdf Binary files differindex fb49ec3..50e6e5a 100644 --- a/fig/threshold_2.000.pdf +++ b/fig/threshold_2.000.pdf |