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-rw-r--r--bezout.bib14
-rw-r--r--bezout.tex23
2 files changed, 36 insertions, 1 deletions
diff --git a/bezout.bib b/bezout.bib
index 0318f58..7f52032 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -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}
+}
+
diff --git a/bezout.tex b/bezout.tex
index e4aead7..d4e3225 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -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}