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| -rw-r--r-- | bezout.tex | 34 |
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@@ -99,18 +99,28 @@ of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). \end{equation} One might balk at the constraint $z^2=N$---which could appropriately be called -a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning -is twofold. First, at every point $z$ the energy \eqref{eq:bare.hamiltonian} -has a `radial' gradient of magnitude proportional to the energy, since -$z\cdot\partial H_0=pH_0$. This anomalous direction must be neglected if critical -points are to exist at any nonzero energy, and the constraint surface $z^2=N$ is -the unique surface whose normal is parallel to $z$ and which contains the -configuration space of the real $p$-spin model as a subspace. Second, taking -the constraint to be the level set of a holomorphic function means the -resulting configuration space is a \emph{bone fide} complex manifold, and -therefore permits easy generalization of the integration techniques referenced -above. The same cannot be said for the space defined by $|z|^2=N$, which is -topologically the $(2N-1)$-sphere and cannot admit a complex structure. +a \emph{hyperbolic} constraint---by comparison with $|z|^2=N$. The reasoning +behind the choice is twofold. + +First, we seek draw conclusions from our model that would be applicable to +generic holomorphic functions without any symmetry. Samples of $H_0$ nearly +provide this, save for a single anomaly: the value of the energy and its +gradient at any point $z$ correlate along the $z$ direction, with +$\overline{H_0\partial_iH_0}\propto\kappa(z^2)^{p-1}z_i$ and +$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. Besides being a +spurious correlation, in each sample there is also a `radial' gradient of +magnitude proportional to the energy, since $z\cdot\partial H_0=pH_0$. This +anomalous direction must be neglected if we are to draw conclusions about +generic functions, and the constraint surface $z^2=N$ is the unique surface +whose normal is parallel to $z$ and which contains the configuration space of +the real $p$-spin model as a subspace. + +Second, taking the constraint to be the level set of a holomorphic function +means the resulting configuration space is a \emph{bone fide} complex manifold, +and therefore permits easy generalization of the integration techniques +referenced above. The same cannot be said for the space defined by $|z|^2=N$, +which is topologically the $(2N-1)$-sphere and cannot admit a complex +structure. The critical points are of $H$ given by the solutions to the set of equations \begin{equation} \label{eq:polynomial} |