Updated bibliography with DOIs and fixed label.

2 files changed, 108 insertions, 92 deletions

diff --git a/bezout.bib b/bezout.bib
index a23955b..5b61d1e 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -26,6 +26,34 @@
  doi = {10.1103/physreva.91.053816}
 }
 
+@article{Auffinger_2012_Random,
+ author = {Auffinger, Antonio and Arous, Gérard Ben and Černý, Jiří},
+ title = {Random Matrices and Complexity of Spin Glasses},
+ journal = {Communications on Pure and Applied Mathematics},
+ publisher = {Wiley},
+ year = {2012},
+ month = {9},
+ number = {2},
+ volume = {66},
+ pages = {165--201},
+ url = {https://doi.org/10.1002%2Fcpa.21422},
+ doi = {10.1002/cpa.21422}
+}
+
+@article{Auffinger_2013_Complexity,
+ author = {Auffinger, Antonio and Arous, Gerard Ben},
+ title = {Complexity of random smooth functions on the high-dimensional sphere},
+ journal = {The Annals of Probability},
+ publisher = {Institute of Mathematical Statistics},
+ year = {2013},
+ month = {11},
+ number = {6},
+ volume = {41},
+ pages = {4214--4247},
+ url = {https://doi.org/10.1214%2F13-aop862},
+ doi = {10.1214/13-aop862}
+}
+
 @book{Bezout_1779_Theorie,
  author = {Bézout, Etienne},
  title = {Théorie générale des équations algébriques},
@@ -63,6 +91,76 @@
  doi = {10.1103/physrevlett.98.150201}
 }
 
+@article{Cartwright_2013_The,
+ author = {Cartwright, Dustin and Sturmfels, Bernd},
+ title = {The number of eigenvalues of a tensor},
+ journal = {Linear Algebra and its Applications},
+ publisher = {Elsevier BV},
+ year = {2013},
+ month = {1},
+ number = {2},
+ volume = {438},
+ pages = {942--952},
+ url = {https://doi.org/10.1016%2Fj.laa.2011.05.040},
+ doi = {10.1016/j.laa.2011.05.040}
+}
+
+@article{Castellani_2005_Spin-glass,
+ author = {Castellani, Tommaso and Cavagna, Andrea},
+ title = {Spin-glass theory for pedestrians},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ publisher = {IOP Publishing},
+ year = {2005},
+ month = {5},
+ number = {05},
+ volume = {2005},
+ pages = {P05012},
+ url = {https://doi.org/10.1088%2F1742-5468%2F2005%2F05%2Fp05012},
+ doi = {10.1088/1742-5468/2005/05/p05012}
+}
+
+@article{Crisanti_1992_The,
+ author = {Crisanti, A. and Sommers, H. -J.},
+ title = {The spherical $p$-spin interaction spin glass model: the statics},
+ journal = {Zeitschrift für Physik B Condensed Matter},
+ publisher = {Springer Science and Business Media LLC},
+ year = {1992},
+ month = {10},
+ number = {3},
+ volume = {87},
+ pages = {341--354},
+ url = {https://doi.org/10.1007%2Fbf01309287},
+ doi = {10.1007/bf01309287}
+}
+
+@article{Crisanti_1995_Thouless-Anderson-Palmer,
+ author = {Crisanti, A. and Sommers, H. -J.},
+ title = {Thouless-Anderson-Palmer Approach to the Spherical p-Spin Spin Glass Model},
+ journal = {Journal de Physique I},
+ publisher = {EDP Sciences},
+ year = {1995},
+ month = {7},
+ number = {7},
+ volume = {5},
+ pages = {805--813},
+ url = {https://doi.org/10.1051%2Fjp1%3A1995164},
+ doi = {10.1051/jp1:1995164}
+}
+
+@article{Cugliandolo_1993_Analytical,
+ author = {Cugliandolo, L. F. and Kurchan, J.},
+ title = {Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {1993},
+ month = {7},
+ number = {1},
+ volume = {71},
+ pages = {173--176},
+ url = {https://doi.org/10.1103%2Fphysrevlett.71.173},
+ doi = {10.1103/physrevlett.71.173}
+}
+
 @article{Fyodorov_2004_Complexity,
  author = {Fyodorov, Yan V.},
  title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices},
@@ -132,86 +230,4 @@
  doi = {10.1007/bf01456804}
 }
 
-@article{auffinger2013random,
-  title={Random matrices and complexity of spin glasses},
-  author={Auffinger, Antonio and Arous, G{\'e}rard Ben and {\v{C}}ern{\`y}, Ji{\v{r}}{\'\i}},
-  journal={Communications on Pure and Applied Mathematics},
-  volume={66},
-  number={2},
-  pages={165--201},
-  year={2013},
-  publisher={Wiley Online Library}
-}
-
-@article{cartwright2013number,
-  title={The number of eigenvalues of a tensor},
-  author={Cartwright, Dustin and Sturmfels, Bernd},
-  journal={Linear algebra and its applications},
-  volume={438},
-  number={2},
-  pages={942--952},
-  year={2013},
-  publisher={Elsevier}
-}
-
-@article{auffinger2013complexity,
-  title={Complexity of random smooth functions on the high-dimensional sphere},
-  author={Auffinger, Antonio and Arous, Gerard Ben and others},
-  journal={The Annals of Probability},
-  volume={41},
-  number={6},
-  pages={4214--4247},
-  year={2013},
-  publisher={Institute of Mathematical Statistics}
-}
-@article{fyodorov2004complexity,
-  title={Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices},
-  author={Fyodorov, Yan V},
-  journal={Physical review letters},
-  volume={92},
-  number={24},
-  pages={240601},
-  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{castellani2005spin,
-  title={Spin-glass theory for pedestrians},
-  author={Castellani, Tommaso and Cavagna, Andrea},
-  journal={Journal of Statistical Mechanics: Theory and Experiment},
-  volume={2005},
-  number={05},
-  pages={P05012},
-  year={2005},
-  publisher={IOP Publishing}
-}
-
-@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 5fc340a..1a12025 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -39,22 +39,22 @@ 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 \cite{crisanti1992sphericalp} (for a review see \cite{castellani2005spin})
+The most tractable family of these  are the mean-field spherical p-spin models \cite{Crisanti_1992_The} (for a review see \cite{Castellani_2005_Spin-glass})
 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}.
+Also in the Algebra \cite{Cartwright_2013_The} and Probability  literature \cite{Auffinger_2012_Random, Auffinger_2013_Complexity}.
 
 This problem has been attacked from several angles: the replica trick to
-compute the Boltzmann--Gibbs distribution\cite{crisanti1992sphericalp}, a Kac--Rice \cite{Kac_1943_On,
+compute the Boltzmann--Gibbs distribution\cite{Crisanti_1992_The}, 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
-\cite{crisanti1995thouless}, and
+\cite{Crisanti_1995_Thouless-Anderson-Palmer}, and
 the gradient-descent -- or more generally Langevin -- dynamics staring from a
-high-energy configuration \cite{cugliandolo1993analytical}.  Thanks to the relative simplicity of the energy,
+high-energy configuration \cite{Cugliandolo_1993_Analytical}.  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
@@ -325,13 +325,14 @@ Consider for example the ground-state energy for given $a$, that is, the energy
     The complexity of the pure 3-spin model at $\epsilon=0$ as a function of
     $a$ at several values of $\kappa$. The dashed line shows
     $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
-  }
+  } \label{fig:complexity}
 \end{figure}
 
-{\color{teal} {\bf somewhere} In Figure \ref{desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
+\textcolor{teal}{ {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
 For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation
 is more interesting. In the range of  values of $\Re H_0$ where there are real solutions there are solutions
 all the way down to $a=1$: this is only possible if the density of solutions diverges at this value: this is natural, since. 
+}
 
 
 \begin{figure}[htpb]\label{desert}
@@ -341,10 +342,9 @@ all the way down to $a=1$: this is only possible if the density of solutions div
     The minimum value of $a$ for which the complexity is positive as a function
     of (real) energy $\epsilon$  for the pure 3-spin model at several values of
     $\kappa$.
-  }
+  } \label{fig:desert}
 \end{figure}
 
-}
 \bibliographystyle{apsrev4-2}
 \bibliography{bezout}