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| -rw-r--r-- | bezout.tex | 32 |
1 files changed, 13 insertions, 19 deletions
@@ -128,8 +128,7 @@ whole space.) Where it might be a problem, we introduce the additional constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex manifold with boundary. We shall see that the `radius' $r$ proves an insightful knob in our present problem, revealing structure as it is varied. Note -that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem -to that of the ordinary $p$-spin. +that taking $r=1$ reduces the problem to that of the ordinary $p$-spin. The critical points are of $H$ given by the solutions to the set of equations \begin{equation} \label{eq:polynomial} @@ -143,18 +142,14 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ and $p\to\infty$. We see from \eqref{eq:polynomial} that at any critical point $\epsilon=H/N$, the average energy. -Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a -critical point of $\operatorname{Im}H$. The number of critical points of $H$ is -therefore the same as that of $\operatorname{Re}H$. From each saddle -emerge gradient lines of $\operatorname{Re}H$, which are also ones of constant -$\operatorname{Im}H$ and therefore constant phase. - -Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function -of $2N$ real variables. Its number of saddle-points is given by the usual -Kac--Rice formula: +Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also +one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for +$x,y\in\mathbb R^N$, $\operatorname{Re}H$ can be considered a real-valued +function of $2N$ real variables. The number of critical points of $H$ is thus given by the +usual Kac--Rice formula applied to $\operatorname{Re}H$: \begin{equation} \label{eq:real.kac-rice} \begin{aligned} - \mathcal N_J&(\kappa,\epsilon,r) + \mathcal N&(\kappa,\epsilon,r) = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\ &\hspace{6pc}\times\left|\det\begin{bmatrix} \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\ @@ -169,7 +164,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these transformations through, we have \begin{equation} \label{eq:complex.kac-rice} \begin{aligned} - \mathcal N_J&(\kappa,\epsilon,r) + \mathcal N&(\kappa,\epsilon,r) = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\ &\hspace{6pc}\times\left|\det\begin{bmatrix} \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\ @@ -183,7 +178,7 @@ transformations through, we have \end{equation} This gives three equivalent expressions for the determinant of the Hessian: as that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix, -i.e. the norm squared of that of an $N\times N$ complex symmetric matrix. +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 positive eigenvalue of the real matrix has a @@ -195,11 +190,10 @@ $\partial\partial H$, or 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 $J$ to give -the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \, -\log \mathcal N_J$, a calculation that involves the replica trick. In most of the -parameter-space that we shall study here, the \emph{annealed approximation} $N -\Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is -exact. +the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N}$, a calculation +that involves the replica trick. In most of the parameter-space that we shall +study here, the \emph{annealed approximation} $N \Sigma \sim \log \overline{ +\mathcal N}$ is exact. A useful property of the Gaussian $J$ is that gradient and Hessian at fixed $\epsilon$ are statistically independent \cite{Bray_2007_Statistics, |