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author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2020-12-09 13:23:05 +0100 |
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committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2020-12-09 13:23:05 +0100 |
commit | c87f43d8dc8b161c4c8bbbd5b9df3756c18d69c0 (patch) | |
tree | 20fb88e3739efa28fcf567726f53b3df088b1e05 | |
parent | 57ad09701c6b6b8b4362b0dd4fadef48d26e13b4 (diff) | |
parent | c3c4d44881d1fc80b1d7cb18406a78d172bf19c7 (diff) | |
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Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484
-rw-r--r-- | bezout.tex | 5 |
1 files changed, 4 insertions, 1 deletions
@@ -39,12 +39,15 @@ 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 family of these are the mean-field spherical p-spin models defined by the energy: +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 = \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 real Gaussian variables and the $z_i$ are real and constrained to a sphere $\sum_i z_i^2=N$. If there is a single term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall study here. +Also in the algebra \cite{cartwright2013number} and probability literature \cite{auffinger2013complexity,auffinger2013random} This problem has been attacked from several angles: the replica trick to compute the Boltzmann--Gibbs distribution, a Kac--Rice \cite{Kac_1943_On, |