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| -rw-r--r-- | bezout.tex | 69 |
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@@ -124,7 +124,7 @@ or the norm squared of that of an $N\times N$ complex symmetric matrix. These equivalences belie a deeper connection between the spectra of the corresponding matrices: each eigenvalue of the real matrix has a negative partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an -eigenvalue of the Hermitian matrix. Finally, $|\lambda|$ is a \emph{singular +eigenvalue of the Hermitian matrix and $|\lambda|$ is a \emph{singular value} of the complex symmetric matrix. The distribution of positive eigenvalues of the Hessian is therefore the same as the distribution of singular values of $\partial\partial H$, while both are the same as the @@ -140,7 +140,56 @@ study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{ A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{Bray_2007_Statistics, Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the -Hessians as independent. We compute each by taking the saddle point. +Hessians as independent. We compute each by taking the saddle point. The +$\delta$-functions are converted to exponentials by the introduction of +auxiliary fields $\hat z=\hat x+i\hat y$. The average over $J$ can then be +performed. A generalized Hubbard--Stratonovich then allows a change of +variables from the $4N$ original and auxiliary fields to eight bilinears +defined by +\begin{equation} + \begin{aligned} + Na=z^*\cdot z + && + N\hat c=\hat z\cdot\hat z + && + Nb=\hat z^*\cdot z \\ + N\hat a=\hat z^*\cdot\hat z + && + Nd=\hat z\cdot z + \end{aligned} +\end{equation} +and their conjugates. The result is, to leading order in $N$, +\begin{equation} \label{eq:saddle} + \overline{\mathcal N_J}(\kappa,\epsilon) + = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)}, +\end{equation} +where +\begin{widetext} + \textcolor{red}{\textbf{[appendix?? I'm putting too much right now so as to trim later...]}} + \begin{equation} + f=2+\frac12\log\det\frac12\begin{bmatrix} + 1 & a & d & b \\ + a & 1 & b^* & d^* \\ + 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\} + +\int d\lambda\,\rho(\lambda)\log|\lambda|^2 + \end{equation} + where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of + $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in + $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with + $\kappa\in\mathbb R$) + \begin{equation} + \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, + \end{aligned} + \end{equation} +\end{widetext} +This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes as $a\to1$, where we should recover known results for the real $p$-spin. + The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial H=\partial\partial H_0-p\epsilon I$, or the Hessian of @@ -168,7 +217,7 @@ where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue spectrum of $\partial\partial H$ therefore is that of an ellipse whose center is shifted by $p\epsilon$. -\begin{figure} +\begin{figure}[htpb] \centering \raisebox{60pt}{$|\epsilon|=0$} @@ -210,8 +259,18 @@ singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this corresponds to a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and to our knowledge a closed form is not known. We have worked out an implicit form for -this spectrum using the saddle point of a replica calculation for the Green -function. blah blah blah\dots +this spectrum using the saddle point of a replica symmetric calculation for the +Green function. The result is +\begin{widetext} + \begin{equation} + G(\sigma)=\lim_{n\to0}\int d\alpha\,d\chi\,d\chi^*\frac\alpha2 + \exp nN\left\{ + 1+\frac{p(p-1)}{16}a^{p-2}\alpha^2-\frac{\alpha\sigma}2+\frac12\log(\alpha^2-|\chi|^2) + +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi^2-\epsilon^*\chi\right) + \right\} + \end{equation} +\end{widetext} +The argument of the exponential has several saddles, but the one with the smallest value of $\mathop{\mathrm{Re}}\alpha$ gives the correct solution \textcolor{red}{\textbf{why????? we never figured this out...}}. The transition from a one-cut to two-cut singular value spectrum naturally corresponds to the origin leaving the support of the eigenvalue spectrum. |