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| -rw-r--r-- | stokes.tex | 128 |
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@@ -830,8 +830,10 @@ distribution in one complex variable $Z$ whose variances are $\overline{Z^*Z}=\overline{|Z|^2}=\Gamma$ and $\overline{Z^2}=C$. $\Gamma$ is positive, and $|C|\leq\Gamma$. The special case of $C=\Gamma$, where the variance of the complex variable and its covariance with its conjugate are the -same, reduces to the ordinary normal distribution. Its probability density -function is defined by +same, reduces to the ordinary normal distribution. The case where $C=0$ results +in the real and imaginary parts of $Z$ being uncorrelated, in what is known as +the standard complex normal distribution. Its probability density function is +defined by \begin{equation} p(z\mid\Gamma,C)= \frac1{\pi\sqrt{\Gamma^2-|C|^2}}\exp\left\{ @@ -841,41 +843,83 @@ function is defined by \right\} \end{equation} -We will consider an ensemble of random matrices $A=A_0+\lambda_0I$, where the -entries of $A_0$ are complex-normal distributed with variance $\Gamma=1/N$ and -$\lambda_0$ is some constant shift to its diagonal. The eigenvalue distribution -of these matrices is already known to take the form of an elliptical ensemble, -with constant support inside the ellipse defined by +We will consider an ensemble of random $N\times N$ matrices $B=A+\lambda_0I$, where the +entries of $A$ are complex-normal distributed with variances +$\overline{|A|^2}=A_0/N$ and $\overline{A^2}=C_0/N$, and $\lambda_0$ is a +constant shift to its diagonal. The eigenvalue distribution of these matrices +is already known to take the form of an elliptical ensemble, with constant +support inside the ellipse defined by \begin{equation} \label{eq:ellipse} - \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{\Gamma+|C|}\right)^2+ - \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{\Gamma-|C|}\right)^2 - <1 + \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{1+|C_0|/A_0}\right)^2+ + \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{1-|C_0|/A_0}\right)^2 + <A_0 \end{equation} -where $\theta=\frac12\arg C$ \cite{Nguyen_2014_The}. The eigenvalue -spectrum of $A$ is therefore constant inside the same ellipse +where $\theta=\frac12\arg C_0$ \cite{Nguyen_2014_The}. The eigenvalue +spectrum of $B$ is therefore constant inside the same ellipse translated so that its center lies at $\lambda_0$. Examples of these distributions are shown in the insets of Fig.~\ref{fig:spectra}. -The eigenvalue spectrum of the Hessian of the real part is not the -spectrum $\rho(\lambda)$ of $\partial\partial H$, but instead the -square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$; -in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial -H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this -is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and -to our knowledge a closed form is not in the literature. We have worked out an -implicit form for the singular value spectrum using the replica method. +When $C=0$ and the elements of $A$ are standard complex normal, the singular +value distribution of $B$ is a complex Wishart distribution. For $C\neq0$ the +problem changes, and to our knowledge a closed form is not in the literature. +We have worked out an implicit form for the singular value spectrum using the +replica method. + +The singular values of $B$ correspond with the square-root of the eigenvalues +of $B^\dagger B$, but also they correspond to the absolute value of the +eigenvalues of the real $2N\times2N$ block matrix +\begin{equation} + \left[\matrix{\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right] +\end{equation} +as we saw in \S\ref{sec:stationary.hessian}. The eigenvalue spectrum of this +block matrix can be studied by ordinary means. Defining the `partition function' +\begin{equation} + Z(\sigma)=\int dx\,dy\,\exp\left\{ + -\frac12\left[\matrix{x&y}\right] + \left(\sigma I- + \left[\matrix{\hphantom{-}\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right] + \right) + \left[\matrix{x\cr y}\right] + \right\} +\end{equation} +implies a Green function +\begin{equation} + G(\sigma)=\frac\partial{\partial\sigma}\log Z(\sigma) +\end{equation} +This can be put into a manifestly complex form in the same way it was done in \S\ref{sec:stationary.hessian}, using the same linear transformation of $x$ and $y$ into $z$ and $z^*$. This gives +\begin{eqnarray} + Z(\sigma) + &=\int dz\,dz^*\,\exp\left\{ + -\frac12\left[\matrix{z^*&-iz}\right] + \left(\sigma I- + \left[\matrix{0&(iB)^*\cr iB&0}\right] + \right) + \left[\matrix{z\cr iz^*}\right] + \right\} \\ + &=\int dz\,dz^*\,\exp\left\{ + -\frac12\left( + 2z^\dagger z\sigma-z^\dagger B^*z^*-z^TBz + \right) + \right\} \\ + &=\int dz\,dz^*\,\exp\left\{ + -z^\dagger z\sigma+\operatorname{Re}(z^TBz) + \right\} +\end{eqnarray} +which is a general expression for the singular values $\sigma$ of a symmetric +complex matrix $B$. + Introducing replicas to bring the partition function into the numerator of the Green function \cite{Livan_2018_Introduction} gives \begin{equation} \label{eq:green.replicas} - G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} + G(\sigma)=\lim_{n\to0}\int dz\,dz^*\,(z^{(0)})^\dagger z^{(0)} \exp\left\{ - \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma - -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right) + -\sum_\alpha^n\left[(z^{(\alpha)})^\dagger z^{(\alpha)}\sigma + +\operatorname{Re}\left((z^{(\alpha)})^TBz^{(\alpha)}\right) \right] \right\}, \end{equation} -with sums taken over repeated Latin indices. The average is then made over +The average is then made over $J$ and Hubbard--Stratonovich is used to change variables to the replica matrices $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^\dagger\zeta^{(\beta)}$ and $N\chi_{\alpha\beta}=(\zeta^{(\alpha)})^T\zeta^{(\beta)}$, and a series of @@ -885,8 +929,8 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is \begin{equation}\label{eq:green.saddle} \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}r^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2) - +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right) + 1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2) + +\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right) \right]\right\}. \end{equation} @@ -899,18 +943,15 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is \includegraphics{figs/spectra_1.5.pdf} \caption{ - Eigenvalue and singular value spectra of the Hessian $\partial\partial H$ - of the $3$-spin model with $\kappa=\frac34e^{-i3\pi/4}$. Pictured - distributions are for critical points at `radius' $r=\sqrt{5/4}$ and with - energy per spin (a) $\epsilon=0$, (b) - $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c) - $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d) - $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each + Eigenvalue and singular value spectra of a random matrix $A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $\overline{|A|^2}=A_0=\sqrt{5/4}$ and $\overline{A^2}=C_0=\frac34e^{-i3\pi/4}$. + The diaginal shifts differ in each plot, with (a) $\lambda_0=0$, (b) + $\lambda_0=\frac12|\lambda_{\mathrm{th}}|$, (c) + $\lambda_0=|\lambda_{\mathrm{th}}|$, and (d) + $\lambda_0=\frac32|\lambda_{\mathrm{th}}|$. The shaded region of each inset shows the support of the eigenvalue distribution \eqref{eq:ellipse}. The solid line on each plot shows the distribution of singular values \eqref{eq:spectral.density}, while the overlaid histogram shows the - empirical distribution from $2^{10}\times2^{10}$ complex normal matrices - with the same covariance and diagonal shift as $\partial\partial H$. + empirical distribution from $2^{10}\times2^{10}$ complex normal matrices. } \label{fig:spectra} \end{figure} @@ -939,12 +980,11 @@ of singular values becomes gapped---is reduced to the geometry problem of determining when the boundary of the ellipse defined in \eqref{eq:ellipse} intersects the origin, and yields \begin{equation} \label{eq:threshold.energy} - |\epsilon_{\mathrm{th}}|^2 - =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2r^{p-2}} - {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)} + |\lambda_{\mathrm{th}}|^2 + =\frac{(1-|\delta|^2)^2} + {1+|\delta|^2-2|\delta|\cos(\arg C_0+2\arg\lambda_0)} \end{equation} -for $\delta=\kappa r^{-(p-2)}$. Notice that the threshold depends on both the -energy per spin $\epsilon$ on the `radius' $r$ of the saddle. +for $\delta=C_0/A_0$. \section{The \textit{p}-spin spherical models} @@ -1314,13 +1354,7 @@ physical dynamics, are a problem we hope to address in future work. } \end{figure} -\subsection{(2 + 4)-spin} - -\begin{equation} - H_2+H_4 -\end{equation} - -\section{Numerics} +\section{The $p$-spin spherical models: numerics} To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically. |