diff options
| -rw-r--r-- | bezout.tex | 58 |
1 files changed, 34 insertions, 24 deletions
@@ -97,6 +97,40 @@ points it has is given by the usual Kac--Rice formula: \end{bmatrix}\right|. \end{aligned} \end{equation} +The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in +a manifestly complex way. Using the Wirtinger derivative +$\partial=\partial_x-i\partial_y$, one can write +$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and +$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. Carrying +these transformations through, we have +\begin{equation} \label{eq:complex.kac-rice} + \begin{aligned} + \mathcal N_J&(\kappa,\epsilon) + = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) \\ + &\qquad\qquad\qquad\times\left|\det\begin{bmatrix} + \mathop{\mathrm{Re}}\partial\partial H & -\mathop{\mathrm{Im}}\partial\partial H \\ + -\mathop{\mathrm{Im}}\partial\partial H & -\mathop{\mathrm{Re}}\partial\partial H + \end{bmatrix}\right| \\ + &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) + \left|\det[(\partial\partial H)^\dagger\partial\partial H]\right| \\ + &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) + |\det\partial\partial H|^2. + \end{aligned} +\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, +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 +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 +distribution of square-rooted eigenvalues of $(\partial\partial +H)^\dagger\partial\partial H$. + {\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as $N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all, of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim @@ -107,30 +141,6 @@ so that we may treat the delta-functions and the Hessians as independent. } -The Cauchy--Riemann relations imply that the matrix is of the form: -\begin{equation} \label{eq:real.kac-rice1} - \begin{bmatrix} - \bar A & \bar B \\ - \bar B & -\bar A - \end{bmatrix} -\end{equation} -with $\bar A=-\mathop{\mathrm{Re}}\partial\partial H$ and $\bar B=-\mathop{\mathrm{Im}}\partial\partial H$ Gaussian real symmetric matrices, \emph{correlated}, as we shall see.. -Using the Wirtinger -derivative $\partial=\partial_x-i\partial_y$, one can write -$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and -$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. With similar -transformations, the eigenvalue spectrum of the Hessian of -$\mathop{\mathrm{Re}}H$ can be shown to be equivalent to the singular value -spectrum of the Hessian $\partial\partial H$ of $H$, and as a result the -determinant that appears above is equivalent to $|\det\partial\partial H|^2$. -This allows us to write \eqref{eq:real.kac-rice} in the manifestly complex -form -\begin{equation} \label{eq:complex.kac-rice} - \mathcal N_J(\kappa,\epsilon) - = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) - |\det\partial\partial H|^2. -\end{equation} - The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial H=\partial\partial H_0-p\epsilon I$, or the Hessian of \eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The |