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| -rw-r--r-- | bezout.tex | 8 |
1 files changed, 4 insertions, 4 deletions
@@ -103,11 +103,11 @@ 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 +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 +$\overline{H_0(\partial_iH_0)^*}\propto|z|^{2(p-1)}z_i$. 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 @@ -132,7 +132,7 @@ equations of degree $p-1$, to which one must add the constraint. In this sense this study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ and $p\to\infty$. We see from \eqref{eq:polynomial} that at any critical -point, $\epsilon=H/N$, the average energy. +point $\epsilon=H/N$, the average energy. Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a critical point of $\operatorname{Im}H$. The number of critical points of $H$ is @@ -187,7 +187,7 @@ $(\partial\partial H)^\dagger\partial\partial H$. The expression \eqref{eq:complex.kac-rice} is to be averaged over $J$ to give the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \, -\log \mathcal N_J$, a calculation that involves the replica trick. In most the +\log \mathcal N_J$, a calculation that involves the replica trick. In most of the parameter-space that we shall study here, the \emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is exact. |