From e89513ed62967929dcf09b2944c8301451366e99 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Dec 2020 14:22:49 +0100
Subject: Updated bibliography with DOIs and fixed label.

---
 bezout.tex | 18 +++++++++---------
 1 file changed, 9 insertions(+), 9 deletions(-)

(limited to 'bezout.tex')

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}
 
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