From e3a50546d952cb192b8ff63d1a22879c97ad4cc7 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 13:50:08 +0100
Subject: Some abstract work.

---
 bezout.tex | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 6fc863c..552dd93 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -24,18 +24,18 @@
 
 \begin{abstract}
   We study the saddle-points of the $p$-spin model -- the best understood
-  example of `complex (rugged) landscape' -- in the space in which all its $N$
-  variables are allowed to be complex. The problem becomes a system of $N$
-  random equations of degree $p-1$.  We solve for quantities averaged over
-  randomness in the $N \rightarrow \infty$ limit.  We show that the number of
-  solutions saturates the Bézout bound $\ln {\cal{N}}\sim N \ln (p-1)$
-  \cite{Bezout_1779_Theorie}.  The Hessian of each saddle is given by a random
-  matrix of the form $M=a A^2+b B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$
-  are GOE matrices and $a-d$ real. Its spectrum  has a transition from one-cut
-  to two-cut that generalizes the notion  of `threshold level' that is
-  well-known in the real problem.  In the case that the disorder is itself
-  real, only the square-root of the total  number solutions are real.  In terms
-  of real and imaginary parts of the energy, the solutions are divided in
+  example of a `complex' (rugged) landscape -- when its $N$ variables are
+  complex. These points are the solutions to a system of $N$ random equations
+  of degree $p-1$.  We solve for $\overline{\mathcal{N}}$, the number of
+  solutions averaged over randomness in the $N\to\infty$ limit.  We find that
+  it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$.
+  The Hessian of each saddle is given by a random matrix of the form $M=a A^2+b
+  B^2 +ic [A,B]_-+ d [A,B]_+$, where $A$ and $B$ are GOE matrices and $a-d$
+  real.  Its spectrum has a transition from one-cut to two-cut that generalizes
+  the notion  of `threshold level' that is well-known in the real problem.  The
+  results from the real problem are recovered in the limit of real disorder. In
+  this case, only the square-root of the total number solutions are real.  In
+  terms of real and imaginary parts of the energy, the solutions are divided in
   sectors where the saddles have different topological properties.
 \end{abstract}
 
-- 
cgit v1.2.3-54-g00ecf

