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authorkurchan.jorge <kurchan.jorge@gmail.com>2020-12-07 14:53:04 +0000
committeroverleaf <overleaf@localhost>2020-12-07 14:53:28 +0000
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Update on Overleaf.
-rw-r--r--bezout.tex5
1 files changed, 3 insertions, 2 deletions
diff --git a/bezout.tex b/bezout.tex
index dac5927..ce3e850 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -39,10 +39,11 @@ different topological properties.
Spin-glasses have long been considered the paradigm of `complex landscapes' of many variables, a subject that
includes Neural Networks and optimization problems, most notably Constraint Satisfaction ones.
-The most tractable, yet very rich model
+The most tractable family of these are the mean-field spherical p-spin models defined by the energy:
\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},
+ E = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
\end{equation}
+where the $J_{i_1\cdots i_p}$ are
where $z\in\mathbb C^N$ is constrained by $z^2=N$ and $J$ is a symmetric tensor
whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and
$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter