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| -rw-r--r-- | bezout.tex | 34 |
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@@ -52,8 +52,23 @@ Thanks to the relative simplicity of the energy, all these approaches are possib 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 $|\kappa|<1$. +$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ 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. + +There is however a more fundamental reason for this study: we know from experience that extending a problem to the complex plane often uncovers an underlying simplicity that is hidden in the purely real case. Consider, for example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a polynomial of degree $p$ chosen to have simple, known roots. Because we are working in +complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. With real +variables minima of functions appear and disappear, and this procedure is not possible. The same idea may be implemented by +performing diffusion in the $J$'s, and following the roots, in complete analogy with Dyson's stochastic dynamics. + + +Let us go back to our model. +For the constraint we 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 $0=\partial^* H=-p\epsilon z$ is only +satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. It is enforced using the method of Lagrange multipliers: introducing the $\epsilon\in\mathbb C$, this gives \begin{equation} \label{eq:constrained.hamiltonian} @@ -61,11 +76,6 @@ multipliers: introducing the $\epsilon\in\mathbb C$, this gives \end{equation} It is easy to see that {\em for a pure $p$ spin}, at any critical point $\epsilon=H/N$, the average energy. -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 $0=\partial^* H=-p\epsilon z$ is only -satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. Since $H$ is holomorphic, a point is a critical point of its real part if and only if it is also a critical point of its imaginary part. The number of @@ -83,13 +93,11 @@ points it has is given by the usual Kac--Rice formula: \end{equation} 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} - - + \begin{bmatrix} + \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.. 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 |