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| -rw-r--r-- | bezout.bib | 14 | ||||
| -rw-r--r-- | bezout.tex | 23 |
2 files changed, 36 insertions, 1 deletions
@@ -7,4 +7,18 @@ address = {rue S. Jacques, Paris} } +@article{Nguyen_2014_The, + author = {Nguyen, Hoi H. and O'Rourke, Sean}, + title = {The Elliptic Law}, + journal = {International Mathematics Research Notices}, + publisher = {Oxford University Press (OUP)}, + year = {2014}, + month = {10}, + number = {17}, + volume = {2015}, + pages = {7620--7689}, + url = {https://doi.org/10.1093%2Fimrn%2Frnu174}, + doi = {10.1093/imrn/rnu174} +} + @@ -60,7 +60,7 @@ At any critical point $\epsilon=H/N$, the average energy. When compared with $z^*z=N$, the constraint $z^2=N$ may seem an unnatural extension of the real $p$-spin spherical model. However, a model with this nonholomorphic spherical constraint has a disturbing lack of critical points -nearly everywhere, since $0=\partial H/\partial z^*=-p\epsilon z$ is only +nearly everywhere, since $0=\partial^* H=-p\epsilon z$ is only satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint. Since $H$ is holomorphic, a point is a critical point of its real part if and @@ -93,6 +93,27 @@ form |\det\partial\partial H|^2. \end{equation} +The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial_i\partial_j +H=\partial_i\partial_j H_0-p\epsilon\delta_{ij}$, or the Hessian of +\eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The +eigenvalue distribution $\rho$ of the constrained Hessian is therefore related +to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar +shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of +\eqref{eq:bare.hamiltonian} is +\begin{equation} \label{eq:bare.hessian} + \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 +$\langle|\partial_i\partial_j H_0|^2\rangle=p(p-1)a^{p-2}/2N$ and +$\langle(\partial_i\partial_j H_0)^2\rangle=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse +\begin{equation} \label{eq:ellipse} + \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{1+|\kappa|/a^{p-2}}\right)^2+ + \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{1-|\kappa|/a^{p-2}}\right)^2 + <\frac12p(p-1)a^{p-2} +\end{equation} +where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. + \bibliographystyle{apsrev4-2} \bibliography{bezout} |