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@@ -1273,6 +1273,45 @@ related directly to the magnitude of the imaginary part of $z$, since $z^\dagger z=x^Tx+y^Ty=N+2y^Ty$. We therefore define $Y=\frac1N(z^\dagger z-N)$, the specific measure of the distance into the complex plane from the real sphere. The complexity can then be written +\begin{equation} + \Sigma + = + \log(p-1)-\frac12\log\left( + \frac{1-r^{-2(p-1)}}{1-r^{-2}} + \right) + -\frac{(\operatorname{Re}\epsilon)^2}{R_+^2}-\frac{(\operatorname{Im}\epsilon)^2}{R_-^2} + +I_p(\epsilon/\epsilon_\mathrm{th}) +\end{equation} +\begin{equation} + R_\pm^2=\frac{p-1}2\frac{(r^{p-2}\pm1)\left[ + r^{2(p-1)}\pm(p-1)r^{p-2}(r^2-1)-1 + \right]}{ + 1+r^{2(p-2)}\left[p(p-2)(r^2-1)-1\right] + } +\end{equation} +$I_p(u)=0$ if $|\epsilon|^2<|\epsilon_\mathrm{th}|^2$. +\begin{eqnarray} + I_p(u) + &= + \left(\frac12+\frac1{r^{p-2}-1}\right)^{-1}(\operatorname{Re}u)^2 + - + \left(\frac12-\frac1{r^{p-2}+1}\right)^{-1}(\operatorname{Im}u)^2\\ + &\qquad-\log\left( + r^{p-2}\left| + u+\sqrt{u^2-1} + \right|^2 + \right)+2\operatorname{Re} + \left( + u\sqrt{u^2-1} + \right) +\end{eqnarray} +where the branch of the square roots are chosen such that the real part of the +root has the opposite sign as the real part of $u$, e.g., if +$\operatorname{Re}u<0$ then $\operatorname{Re}\sqrt{u^2-1}>0$. If the real part +is zero, then the sign is taken so that the imaginary part of the root has the +opposite sign of the imaginary part of $u$. + + \subsection{Pure \textit{p}-spin: where are my neighbors?} |