From aafde5c787252a47633ac8bcc4fbf09a28d22ca1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 8 Feb 2021 15:09:30 +0100
Subject: Added paragraph describing typical norm of critical points.

---
 bezout.tex | 9 +++++++++
 1 file changed, 9 insertions(+)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index d7f4dc7..9222449 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -402,6 +402,15 @@ $\epsilon$ is varied.
   } \label{fig:desert}
 \end{figure}
 
+In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
+critical points are concentrated at infinite $a$, i.e., at complex vectors with
+very large squared norm. For finite $N$ the expectation value $\langle
+a\rangle$ is likewise finite. By differentiating $\overline{\mathcal N}$ with
+respect to $a$ and normalizing, one has an approximation for the distribution
+of critical points as a function of $a$. The expectation value this yields is
+$\langle a\rangle\propto N^{1/2}+O(N^{-1/2})$. One therefore expects typical
+critical points to have a norm that grows modestly with system size.
+
 These qualitative features carry over to nonzero $\epsilon$. In
 Fig.~\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. When $\kappa=1$---the analytic
-- 
cgit v1.2.3-54-g00ecf

