From b128bec287a75df35db8bef28a4df4c480ab9ea5 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Tue, 8 Dec 2020 10:32:47 +0000
Subject: Update on Overleaf.

---
 bezout.tex | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/bezout.tex b/bezout.tex
index ebcc4a7..621a52f 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -142,6 +142,9 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
   \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}
+
+{\color{red} \bf here I would explain the question of the det and also of the appearance of the gap, would draw a picture of ellipse etc, and would send the reader to an appendix for most of this part of the calculation}
+
 which makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
 $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
 $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
@@ -205,6 +208,19 @@ For $|\kappa|<1$,
 \end{equation}
 for $\delta=\kappa a^{-(p-2)}$.
 
+
+{\color{teal} {\bf somewhere else}
+
+Another instrument we have to study this problem is to compute the following partition function:
+
+\begin{equation}
+    Z= \int \Pi_i dx_i dy_i \; e^{-\beta_{R} \Re H_0 -\beta_I \Im H_0}
+    \delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right) 
+\end{equation}
+The energy $\Re H_0, \Im H_0$ are in a one-to one relation with the temperatures $\beta_R,\beta_I$. The entropy $S(a,H_0) = \ln Z+ +\beta_{R} \langle \Re H_0 \rangle +\beta_I \langle \Im H_0\rangle$
+is the logarithm of thnumber of configurations of a given 
+
+}
 \bibliographystyle{apsrev4-2}
 \bibliography{bezout}
 
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