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author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2020-12-07 14:06:10 +0100 |
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committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2020-12-07 14:06:10 +0100 |
commit | 938299611b8543b5fd50ce235f0643e45107863e (patch) | |
tree | a77245d83f8d6aa2e32f1f5232a9c3449a0e719a | |
parent | 608b4004a8ea90b7adc2fc080cd2c028c2d307a8 (diff) | |
download | PRR_3_023064-938299611b8543b5fd50ce235f0643e45107863e.tar.gz PRR_3_023064-938299611b8543b5fd50ce235f0643e45107863e.tar.bz2 PRR_3_023064-938299611b8543b5fd50ce235f0643e45107863e.zip |
Added explicit κ-dependence to the count.
-rw-r--r-- | bezout.tex | 4 |
1 files changed, 2 insertions, 2 deletions
@@ -48,7 +48,7 @@ $\mathop{\mathrm{Re}}H$. 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(\epsilon) + \mathcal N(\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 \\ @@ -66,7 +66,7 @@ determinant that appears above is equivalent to $|\det\partial\partial H|^2$. This allows us to write the \eqref{eq:real.kac-rice} in the manifestly complex form \begin{equation} \label{eq:complex.kac-rice} - \mathcal N(\epsilon) + \mathcal N(\kappa,\epsilon) = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) |\det\partial\partial H|^2. \end{equation} |