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diff --git a/figs/bound.pdf b/figs/bound.pdf Binary files differnew file mode 100644 index 0000000..0e6814f --- /dev/null +++ b/figs/bound.pdf @@ -1455,7 +1455,33 @@ variable $\delta$. Though the value of $\delta$ is bounded by $\Delta$ by $|\delta|\leq\Delta$, in reality this bound is not the relevant one, because we are confined on the -manifold $N=z^2$. +manifold $N=z^2$. This bound is most easily established by returning to a +$2N$-dimensional real problem, with $x=x_1$ and $z=x_2+iy_2$. The constraint gives $x_2^Ty_2=0$, $x_1^Tx_1=1$, and $x_2^Tx_2=1+y_2^Ty_2$. Then +\begin{equation} + \Delta=1+x_2^Tx_2+y_2^Ty_2-2x_1^Tx_2=2(1+y_2^Ty_2-x_1^Tx_2) +\end{equation} +\begin{equation} + \Delta=2(1+|y_2|^2-\sqrt{1-|y_2|^2}\cos\theta_{xx}) +\end{equation} +\begin{eqnarray} + \delta&=2-2x_1^Tx_2-2ix_1^Ty_2=2(1-|x_2|\cos\theta_{xx}-i|y_2|\cos\theta_{xy}) \\ + &=2(1-\sqrt{1-|y_2|^2}\cos\theta_{xx}-i|y_2|\cos\theta_{xy}) +\end{eqnarray} +$\cos^2\theta_{xy}\leq1-\cos^2\theta_{xx}$ +These equations along with the inequality produce the required bound on $|\delta|$ as a function of $\Delta$ and $\arg\delta$. + +\begin{figure} + \includegraphics{figs/bound.pdf} + + \caption{ + The line bounding $\delta$ in the complex plane as a function of + $\Delta=1,2,\ldots,6$ (inner to outer). Notice that for $\Delta\leq4$, + $|\delta|=\Delta$ is saturated for positive real $\delta$, but is not for + $\Delta>4$, and $\Delta=4$ has a cusp in the boundary. This is due to + $\Delta=4$ corresponding to the maximum distance between any two points on + the real sphere. + } +\end{figure} \section{The $p$-spin spherical models: numerics} |