From eccc17d8d17c2069a0ef3cd5417f5def9614e8d2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 15:48:40 +0100
Subject: Added derivation of the spectrum of the Hessian.

---
 bezout.tex | 24 +++++++++++++++++++++++-
 1 file changed, 23 insertions(+), 1 deletion(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 7d7d4ec..4fb5641 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -77,7 +77,29 @@ form
     = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
       |\det\partial\partial H|^2.
 \end{equation}
-$\langle|\partial_i\partial_j H_0|^2\rangle=p(p-1)|z|^2/2N$ and $\langle(\partial_i\partial_j H_0)^2\rangle=p(p-1)\kappa z^2/2N$
+
+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, whose
+spectrum is constant inside the support of a certain ellipse and zero
+everywhere else \cite{Nguyen_2014_The}. 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 $\theta=\frac12\arg\kappa$.
 
 \bibliographystyle{apsrev4-2}
 \bibliography{bezout}
-- 
cgit v1.2.3-70-g09d2

