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authorJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 15:32:16 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 15:32:16 +0100
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Messy mathop/mathrm replaced by operatorname.
-rw-r--r--bezout.tex44
1 files changed, 22 insertions, 22 deletions
diff --git a/bezout.tex b/bezout.tex
index bfabdec..0c395c3 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -118,38 +118,38 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ an
Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical point of $\Im H_0$. The number of
critical points of $H$ is therefore the number of critical points of
-$\mathop{\mathrm{Re}}H$. From each critical point emerges a gradient line of $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
+$\operatorname{Re}H$. From each critical point emerges a gradient line of $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
-Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
+Writing $z=x+iy$, $\operatorname{Re}H$ 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}
\begin{aligned}
\mathcal N_J(\kappa,\epsilon)
- &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\
+ &= \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{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
+ \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
+ \partial_y\partial_x\operatorname{Re}H & \partial_y\partial_y\operatorname{Re}H
\end{bmatrix}\right|.
\end{aligned}
\end{equation}
The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
a manifestly complex way. 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
-$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. Carrying
+$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and
+$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying
these transformations through, we have
\begin{equation} \label{eq:complex.kac-rice}
\begin{aligned}
\mathcal N_J&(\kappa,\epsilon)
- = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) \\
+ = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
&\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
- \mathop{\mathrm{Re}}\partial\partial H & -\mathop{\mathrm{Im}}\partial\partial H \\
- -\mathop{\mathrm{Im}}\partial\partial H & -\mathop{\mathrm{Re}}\partial\partial H
+ \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
+ -\operatorname{Im}\partial\partial H & -\operatorname{Re}\partial\partial H
\end{bmatrix}\right| \\
- &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+ &= \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H)
\left|\det[(\partial\partial H)^\dagger\partial\partial H]\right| \\
- &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+ &= \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H)
|\det\partial\partial H|^2.
\end{aligned}
\end{equation}
@@ -208,7 +208,7 @@ where
d & b^* & \hat c & \hat a \\
b & d^* & \hat a & \hat c^*
\end{bmatrix}
- +\mathop{\mathrm{Re}}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
+ +\operatorname{Re}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
+\int d\lambda\,\rho(\lambda)\log|\lambda|^2
\end{equation}
where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of
@@ -218,8 +218,8 @@ where
\begin{equation} \label{eq:free.energy.a}
\begin{aligned}
f(a)&=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 \\
- &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Re}}\epsilon)^2
- -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Im}}\epsilon)^2,
+ &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Re}\epsilon)^2
+ -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Im}\epsilon)^2,
\end{aligned}
\end{equation}
\end{widetext}
@@ -244,8 +244,8 @@ $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ 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
+ \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
+ \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
<\frac{p(p-1)}{2a^{p-2}}
\end{equation}
where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
@@ -291,7 +291,7 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
\exp\left\{
\frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
- -\Re\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
+ -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
\right]
\right\}
\end{equation}
@@ -307,20 +307,20 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
\overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
\exp\left\{nN\left[
1+\frac{p(p-1)}{16}a^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
- +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
+ +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
\right]\right\}.
\end{equation}
\end{widetext}
The argument of the exponential has several saddles. The solutions $\alpha_0$
are the roots of a sixth-order polynomial, but the root with the
-smallest value of $\mathop{\mathrm{Re}}\alpha_0$ in all the cases we studied gives the correct
+smallest value of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct
solution. A detailed analysis of the saddle point integration is needed to
understand why this is so. Given such $\alpha_0$, the density of singular
values follows from the jump across the cut, or
\begin{equation} \label{eq:spectral.density}
\rho(\sigma)=\frac1{i\pi N}\left(
- \lim_{\mathop{\mathrm{Im}}\sigma\to0^+}\overline G(\sigma)
- -\lim_{\mathop{\mathrm{Im}}\sigma\to0^-}\overline G(\sigma)
+ \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma)
+ -\lim_{\operatorname{Im}\sigma\to0^-}\overline G(\sigma)
\right)
\end{equation}