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| -rw-r--r-- | bezout.tex | 14 |
1 files changed, 3 insertions, 11 deletions
@@ -277,8 +277,8 @@ for $\delta=\kappa a^{-(p-2)}$. 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 +the resulting expression is always maximized for $a=\infty$. Taking this saddle +gives \begin{equation} \label{eq:bezout} \overline{\mathcal N}(\kappa,\epsilon) =e^{N\log(p-1)} @@ -309,7 +309,7 @@ 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 +value at $a=1$. Since the $a$-dependence gives 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} @@ -321,7 +321,6 @@ 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} @@ -332,13 +331,6 @@ limit as $\epsilon$ is varied. } \label{fig:complexity} \end{figure} -For $|\kappa|<1$, -\begin{equation} - \lim_{a\to1}\overline{\mathcal N}(\kappa,\epsilon,a) - =0 -\end{equation} - - {\color{teal} {\bf somewhere else} Another instrument we have to study this problem is to compute the following partition function: |