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| -rw-r--r-- | bezout.bib | 7 | ||||
| -rw-r--r-- | bezout.tex | 27 |
2 files changed, 20 insertions, 14 deletions
@@ -11,6 +11,13 @@ url = {https://doi.org/10.1007%2Fjhep12%282016%29071}, doi = {10.1007/jhep12(2016)071} } +@book{livan2018introduction, + title={Introduction to random matrices: theory and practice}, + author={Livan, Giacomo and Novaes, Marcel and Vivo, Pierpaolo}, + volume={26}, + year={2018}, + publisher={Springer} +} @article{Antenucci_2015_Complex, author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.}, @@ -161,7 +161,7 @@ partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an 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 +singular values of $\partial\partial H$, the distribution of square-rooted eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$. @@ -172,7 +172,7 @@ study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact. A useful property of the Gaussian distributions is that gradient and Hessian -may be seen to be independent \cite{Bray_2007_Statistics, +for given $\epsilon$ 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. The $\delta$-functions are converted to exponentials by the introduction of @@ -260,7 +260,7 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of \partial_i\partial_jH_0 =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, \end{equation} -which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances +{\bf \color{red} restricting to directions proportional to $z$, i.e. orthogonal to the constraint}, these makes its ensemble that of Gaussian complex symmetric matrices. Given its variances $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse @@ -274,17 +274,17 @@ spectrum of $\partial\partial H$ therefore is that of an ellipse in the complex plane whose center lies at $-p\epsilon$. Examples of these distributions are shown in the insets of Fig.~\ref{fig:spectra}. -The eigenvalue spectrum of the Hessian of the real part, or equivalently the -eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the +The eigenvalue spectrum of the Hessian of the real part is the one we need for our Kac-Rice formula. It is different from the spectrum $\partial\partial H$, but rather equivalent to the +square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$,in other words, the 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 +knowledge a closed form is not in the literature. We have worked out an implicit form for this spectrum using the saddle point of a replica symmetric calculation for the Green function. Introducing replicas to bring the partition function to -the numerator of the Green function gives +the numerator of the Green function \cite{livan2018introduction} gives \begin{widetext} \begin{equation} \label{eq:green.replicas} G(\sigma)=\frac1N\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} @@ -313,7 +313,7 @@ the numerator of the Green function gives \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 $\mathop{\mathrm{Re}}\alpha_0$ appears gives the correct +smallest value of $\mathop{\mathrm{Re}}\alpha_0$ 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 @@ -345,19 +345,18 @@ may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$, the resulting expression is always maximized for $a=\infty$. Taking this saddle gives \begin{equation} \label{eq:bezout} - \overline{\mathcal N}(\kappa,\epsilon) - =e^{N\log(p-1)} - =(p-1)^N. + \ln \overline{\mathcal N}(\kappa,\epsilon) + ={N\log(p-1)} \end{equation} This is precisely the Bézout bound, the maximum number of solutions that $N$ equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is gained by looking at the count as a function of $a$, defined by \begin{equation} \label{eq:count.def.marginal} - \overline{\mathcal N}(\kappa,\epsilon) - =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a) + {\mathcal N}(\kappa,\epsilon,a) + ={\mathcal N}(\kappa,\epsilon/ \sum_i y_i^2<Na) \end{equation} and likewise the $a$-dependant complexity -$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$. In +$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}_a(\kappa,\epsilon,a)$ the large-$N$ limit, the $a$-dependant expression may be considered the cumulative number of critical points up to the value $a$. |