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| -rw-r--r-- | bezout.tex | 75 |
1 files changed, 50 insertions, 25 deletions
@@ -183,7 +183,7 @@ where $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with $\kappa\in\mathbb R$) - \begin{equation} + \begin{equation} \label{eq:free.energy.a} \begin{aligned} f(a)&=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 \\ &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Re}}\epsilon)^2 @@ -275,22 +275,62 @@ geometry problem, and yields \end{equation} for $\delta=\kappa a^{-(p-2)}$. -% This is kind of a boring definition... +With knowledge of this distribution, the integral in \eqref{eq:free.energy.a} +may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$, +the resulting expression is always maximized for $a=\infty$, that is, for +unbounded complex spins. Taking this saddle gives +\begin{equation} \label{eq:bezout} + \overline{\mathcal N}(\kappa,\epsilon) + =e^{N\log(p-1)} + =(p-1)^N. +\end{equation} +This is precisely the Bézout bound, the maximum number of solutions that $N$ +equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is gained by looking at the count as a function of $a$, defined by \begin{equation} \label{eq:count.def.marginal} \overline{\mathcal N}(\kappa,\epsilon) =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a) \end{equation} - -\begin{equation} \label{eq:count.zero.energy} - \overline{\mathcal N}(\kappa,0,a) - =\left[(p-1)a^{p-1}\sqrt{\frac{1-a^{-2}}{a^{2(p-1)}-|\kappa|^2}}\right]^N +and likewise the $a$-dependant complexity +$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$. In +the large-$N$ limit, the $a$-dependant expression may be considered the +cumulative number of critical points up to the value $a$. + +The integral in \eqref{eq:free.energy.a} can only be performed explicitly for +certain ellipse geometries. One of these is at $\epsilon=0$ any values of +$\kappa$ and $a$, which yields the $a$-dependent complexity +\begin{equation} \label{eq:complexity.zero.energy} + \Sigma(\kappa,0,a) + =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2a^{-2(p-1)}}{1-a^{-2}}\right). \end{equation} - +Notice that the limit of this expression as $a\to\infty$ corresponds with +\eqref{eq:bezout}, as expected. The complexity at zero energy can be seen +plotted as a function of $a$ for several values of $\kappa$ in +Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to +negative infinity as $a\to1$, i.e., as the spins are restricted to the reals. +However, when the result is analytically continued to $\kappa=1$ (which +corresponds to real $J$) something novel occurs: the complexity has a finite +value at $a=1$. Interpreting this $a$-dependence as a cumulative count, this +implies a $\delta$-function density of critical points along the line $y=0$. +The number of critical points contained within is \begin{equation} - \overline{\mathcal N}(\kappa,\epsilon) - =\lim_{a\to\infty}\overline{\mathcal N}(\kappa,\epsilon,a) - =(p-1)^N + \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a) + =(p-1)^{N/2}, \end{equation} +the square root of \eqref{eq:bezout} and precisely the number of critical +points of the real pure spherical $p$-spin model. In fact, the full +$\epsilon$-dependence of the real pure spherical $p$-spin is recovered by this +limit as $\epsilon$ is varied. + + +\begin{figure}[htpb] + \centering + \includegraphics{fig/complexity.pdf} + \caption{ + The complexity of the pure 3-spin model at $\epsilon=0$ as a function of + $a$ at several values of $\kappa$. The dashed line shows + $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. + } \label{fig:complexity} +\end{figure} For $|\kappa|<1$, \begin{equation} @@ -298,11 +338,6 @@ For $|\kappa|<1$, =0 \end{equation} -\begin{equation} - \lim_{a\to1}\overline{\mathcal N}(1,0,a) - =(p-1)^{N/2} -\end{equation} - {\color{teal} {\bf somewhere else} @@ -318,16 +353,6 @@ is the logarithm of the number of configurations of a given $(a,H_0)$. This problem may be solved exactly with replicas, {\em but it may also be simulated} \cite{Bray_1980_Metastable}. Consider for example the ground-state energy for given $a$, that is, the energy in the limit $\beta_R \rightarrow \infty$ taken adjusting $\beta_I$ so that $\Im H_0=0$ . For $a=1$ this coincides with the ground-state of the real problem. -\begin{figure}[htpb] - \centering - \includegraphics{fig/complexity.pdf} - \caption{ - The complexity of the pure 3-spin model at $\epsilon=0$ as a function of - $a$ at several values of $\kappa$. The dashed line shows - $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. - } \label{fig:complexity} -\end{figure} - {\color{teal} {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$. For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation is more interesting. In the range of values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. |