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| -rw-r--r-- | bezout.tex | 5 |
1 files changed, 3 insertions, 2 deletions
@@ -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 |