From 6ea986fc46fd860a8ac36999ab7e6ff2dd11466e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Dec 2020 15:14:26 +0100
Subject: Added discussion of solution.

---
 bezout.tex | 75 +++++++++++++++++++++++++++++++++++++++++---------------------
 1 file changed, 50 insertions(+), 25 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index b0ead4e..8286208 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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 $a=1$. 
-- 
cgit v1.2.3-54-g00ecf

