From edf64fefc8e0b20f54570dd9095a1d7d01d388b0 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Mon, 7 Dec 2020 14:11:37 +0000
Subject: Update on Overleaf.
---
bezout.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/bezout.tex b/bezout.tex
index 5b39134..b9189a9 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -23,7 +23,7 @@
\date\today
\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
+ We study the saddle-points of the $p$-spin model, the best understood example of `complex (rugged) landscape' in the space in which all
saturates the Bézout bound \cite{Bezout_1779_Theorie}.
\end{abstract}
--
cgit v1.2.3-70-g09d2
From 3d7fdf5bcaac1deb024f1318c9b711b340fa30f8 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Mon, 7 Dec 2020 14:48:43 +0000
Subject: Update on Overleaf.
---
bezout.tex | 13 +++++++++++--
1 file changed, 11 insertions(+), 2 deletions(-)
diff --git a/bezout.tex b/bezout.tex
index b9189a9..6c56d09 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -23,12 +23,21 @@
\date\today
\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
- saturates the Bézout bound \cite{Bezout_1779_Theorie}.
+ 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 sectors where the saddles have
+different topological properties.
\end{abstract}
\maketitle
+
\begin{equation} \label{eq:bare.hamiltonian}
H_0 = \frac1{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
\end{equation}
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cgit v1.2.3-70-g09d2