summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
-rw-r--r--bezout.tex14
1 files changed, 3 insertions, 11 deletions
diff --git a/bezout.tex b/bezout.tex
index 8286208..2368a1d 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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: