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| -rw-r--r-- | bezout.tex | 29 |
1 files changed, 19 insertions, 10 deletions
@@ -41,28 +41,28 @@ Spin-glasses have long been considered the paradigm of `complex landscapes' of m 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 defined by the energy: \begin{equation} \label{eq:bare.hamiltonian} - E = \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}, + H_o = \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 $\sum_i z_i^2=N$. +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. This problem has been attacked from several angles: the replica trick to compute the Boltzmann-Gibbs distribution, a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral) to compute the number of saddle-points of the energy function, and the gradient-descent -- or more generally Langevin -- dynamics staring from a high-energy configuration. Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit. -In th -where $z\in\mathbb C^N$ is constrained by $z^2=N$ and $J$ is a symmetric tensor +In this paper we shall extend the study to the case where $z\in\mathbb C^N$ are and $J$ is a symmetric tensor whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and -$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter -$|\kappa|<1$. The constraint is enforced using the method of Lagrange +$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter $|\kappa|<1$. + +It is enforced using the method of Lagrange multipliers: introducing the $\epsilon\in\mathbb C$, this gives \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). \end{equation} -At any critical point $\epsilon=H/N$, the average energy. +It is easy to see that {\em for a pure $p$ spin}, at any critical point $\epsilon=H/N$, the average energy. -When compared with $z^*z=N$, the constraint $z^2=N$ may seem an unnatural -extension of the real $p$-spin spherical model. However, a model with this +For the constraint we shall choose here $z^2=N$, rather than $|z^2|=N$, in order to preserve the holomorphic nature +of the functions.. In addition, the nonholomorphic spherical constraint has a disturbing lack of critical points nearly everywhere, since $0=\partial H/\partial z^*=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. @@ -81,7 +81,16 @@ points it has is given by the usual Kac--Rice formula: \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H \end{bmatrix}\right|. \end{equation} -The Cauchy--Riemann relations can be used to simplify this. Using the Wirtinger +The Cauchy--Riemann relations imply that the matrix is of the form: +\begin{equation} \label{eq:real.kac-rice1} + \left|\det\begin{bmatrix} + A & B \\ + B & -A + \end{bmatrix}\right|. +\end{equation} + + +Using the Wirtinger derivative $\partial=\partial_x-i\partial_y$, one can write $\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and $\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. With similar |