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| -rw-r--r-- | bezout.tex | 40 |
1 files changed, 21 insertions, 19 deletions
@@ -108,9 +108,9 @@ z$ would only be satisfied for $\epsilon=0$. The critical points are given by the solutions to the set of equations \begin{equation} - \frac{p}{p!}\sum_{j_2\cdots j_p}^NJ_{ij_2\cdots j_p}z_{j_2}\cdots z_{j_p} = \epsilon z_i + \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}} = \epsilon z_i \end{equation} -for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ are a set of $N$ +for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ is a set of $N$ equations of degree $p-1$, to which one must add the constraint. In this sense this study also provides a complement to the work on the distribution of zeroes of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ @@ -143,7 +143,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these transformations through, we have \begin{equation} \label{eq:complex.kac-rice} \begin{aligned} - \mathcal N_J&(\kappa,\epsilon) + &\mathcal N_J(\kappa,\epsilon) = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\ &\qquad\qquad\qquad\times\left|\det\begin{bmatrix} \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\ @@ -169,10 +169,10 @@ singular values of $\partial\partial H$, the distribution of square-rooted eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$. The expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as -$N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation +$N \Sigma= \overline{\log\mathcal N} = \int dJ \, \log \mathcal N_J$, a calculation that involves the replica trick. In most the parameter-space that we shall -study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{ -\mathcal N_J} = \ln \int dJ \; N_J$ is exact. +study here, the {\em annealed approximation} $N \Sigma \sim \log \overline{ +\mathcal N} = \log\int dJ \, \mathcal N_J$ is exact. A useful property of the Gaussian distributions is that gradient and Hessian for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics, @@ -256,12 +256,14 @@ that of the same ellipse 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 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 in the literature. We have worked out an implicit form for +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 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. @@ -348,7 +350,7 @@ 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} - \ln \overline{\mathcal N}(\kappa,\epsilon) + \log\overline{\mathcal N}(\kappa,\epsilon) ={N\log(p-1)} \end{equation} This is precisely the Bézout bound, the maximum number of solutions that $N$ @@ -383,14 +385,14 @@ $a$-dependence gives a cumulative count, this implies a $\delta$-function density of critical points along the line $y=0$. The number of critical points contained within is \begin{equation} - \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a) - \sim (p-1)^{N/2}, + \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a) + = \frac12N\log(p-1), \end{equation} -the square root of \eqref{eq:bezout} and precisely the number of critical +half of \eqref{eq:bezout} and corresponding precisely to the number of critical points of the real pure spherical $p$-spin model. (note the role of conjugation -symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In -fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is -recovered by this limit as $\epsilon$ is varied. +symmetry, already underlined in \cite{Bogomolny_1992_Distribution}). The full +$\epsilon$-dependence of the real $p$-spin is recovered by this limit as +$\epsilon$ is varied. \begin{figure}[htpb] \centering |