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authorkurchan.jorge <kurchan.jorge@gmail.com>2020-12-09 12:28:00 +0000
committeroverleaf <overleaf@localhost>2020-12-09 12:28:56 +0000
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Update on Overleaf.
-rw-r--r--bezout.bib31
-rw-r--r--bezout.tex10
2 files changed, 36 insertions, 5 deletions
diff --git a/bezout.bib b/bezout.bib
index 659e92b..37e8f33 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -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}
+}
diff --git a/bezout.tex b/bezout.tex
index 6b817fd..753b65b 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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