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| -rw-r--r-- | bezout.bib | 31 | ||||
| -rw-r--r-- | bezout.tex | 10 |
2 files changed, 36 insertions, 5 deletions
@@ -174,3 +174,34 @@ year={2004}, publisher={APS} } +@article{crisanti1995thouless, + title={Thouless-Anderson-Palmer approach to the spherical p-spin spin glass model}, + author={Crisanti, Andrea and Sommers, H-J}, + journal={Journal de Physique I}, + volume={5}, + number={7}, + pages={805--813}, + year={1995}, + publisher={EDP Sciences} +} +@article{crisanti1992sphericalp, + title={The sphericalp-spin interaction spin glass model: the statics}, + author={Crisanti, Andrea and Sommers, H-J}, + journal={Zeitschrift f{\"u}r Physik B Condensed Matter}, + volume={87}, + number={3}, + pages={341--354}, + year={1992}, + publisher={Springer} +} + +@article{cugliandolo1993analytical, + title={Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model}, + author={Cugliandolo, Leticia F and Kurchan, Jorge}, + journal={Physical Review Letters}, + volume={71}, + number={1}, + pages={173}, + year={1993}, + publisher={APS} +} @@ -39,8 +39,7 @@ 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 - +The most tractable family of these are the mean-field spherical p-spin models \cite{crisanti1992sphericalp} 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}, @@ -50,11 +49,12 @@ to a sphere $\sum_i z_i^2=N$. If there is a single term of a given $p$, this is 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, +compute the Boltzmann--Gibbs distribution\cite{crisanti1992sphericalp}, a Kac--Rice \cite{Kac_1943_On, Rice_1939_The, Fyodorov_2004_Complexity} procedure (similar to the Fadeev--Popov -integral) to compute the number of saddle-points of the energy function, and +integral) to compute the number of saddle-points of the energy function +\cite{crisanti1995thouless}, and the gradient-descent -- or more generally Langevin -- dynamics staring from a -high-energy configuration. Thanks to the relative simplicity of the energy, +high-energy configuration \cite{cugliandolo1993analytical}. Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit. In this paper we shall extend the study to the case where $z\in\mathbb C^N$ are and $J$ is a symmetric tensor |