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| -rw-r--r-- | bezout.tex | 33 |
1 files changed, 18 insertions, 15 deletions
@@ -41,7 +41,7 @@ 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} - 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}, + 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. @@ -51,8 +51,8 @@ a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral) 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 -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 becomes $z^2=N$. +whose elements are complex normal with $\overline{|J|^2}=p!/2N^{p-1}$ and +$\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$. The motivations for this paper are of two types. On the practical side, there are situations in which complex variables have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Leuzzi-Crisanti, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos-Anous-Denef} modeling Black Hole horizons in the zero-temperature limit. @@ -84,12 +84,14 @@ $\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be interpreted as a real function of $2N$ real variables. The number of critical points it has is given by the usual Kac--Rice formula: \begin{equation} \label{eq:real.kac-rice} - \mathcal N_J(\kappa,\epsilon) - = \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) - \left|\det\begin{bmatrix} - \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\ - \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H - \end{bmatrix}\right|. + \begin{aligned} + \mathcal N_J(\kappa,\epsilon) + &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\ + &\qquad\times\left|\det\begin{bmatrix} + \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\ + \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H + \end{bmatrix}\right|. + \end{aligned} \end{equation} {\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as $N \Sigma= @@ -107,7 +109,8 @@ The Cauchy--Riemann relations imply that the matrix is of the form: \bar A & \bar B \\ \bar B & -\bar A \end{bmatrix} -\end{equation}with $\bar A$ and $\bar B$ Gaussian real symmetric matrices, {\em correlated}, as we shall see.. +\end{equation} +with $\bar A=-\mathop{\mathrm{Re}}\partial\partial H$ and $\bar B=-\mathop{\mathrm{Im}}\partial\partial H$ Gaussian real symmetric matrices, \emph{correlated}, as we shall see.. 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 @@ -119,13 +122,13 @@ determinant that appears above is equivalent to $|\det\partial\partial H|^2$. This allows us to write \eqref{eq:real.kac-rice} in the manifestly complex form \begin{equation} \label{eq:complex.kac-rice} - \mathcal N(\kappa,\epsilon) + \mathcal N_J(\kappa,\epsilon) = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) |\det\partial\partial H|^2. \end{equation} -The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial_i\partial_j -H=\partial_i\partial_j H_0-p\epsilon\delta_{ij}$, or the Hessian of +The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial +H=\partial\partial H_0-p\epsilon I$, or the Hessian of \eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The eigenvalue distribution $\rho$ of the constrained Hessian is therefore related to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar @@ -136,8 +139,8 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, \end{equation} which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances -$\langle|\partial_i\partial_j H_0|^2\rangle=p(p-1)a^{p-2}/2N$ and -$\langle(\partial_i\partial_j H_0)^2\rangle=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse +$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and +$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse \begin{equation} \label{eq:ellipse} \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+ \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2 |