From b82d78af197676c90fa17e669bea7b7805b480c2 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 10 Dec 2020 15:32:16 +0100 Subject: Messy mathop/mathrm replaced by operatorname. --- bezout.tex | 44 ++++++++++++++++++++++---------------------- 1 file changed, 22 insertions(+), 22 deletions(-) (limited to 'bezout.tex') 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} -- cgit v1.2.3-54-g00ecf