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@@ -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 @@ -325,26 +325,26 @@ 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 gap 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 $\sim(a-1)^N$. +{\color{teal} {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap 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 exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around `deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the $J$'s could produce a real, unusually deep solution for the real problem, in a region where we expect this not to happen. } -\begin{figure}[htpb]\label{desert} + +\begin{figure}[htpb] \centering \includegraphics{fig/desert.pdf} \caption{ 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} |