summaryrefslogtreecommitdiff
diff options
context:
space:
mode:
authorkurchan.jorge <kurchan.jorge@gmail.com>2020-12-08 10:32:47 +0000
committeroverleaf <overleaf@localhost>2020-12-08 10:32:49 +0000
commitb128bec287a75df35db8bef28a4df4c480ab9ea5 (patch)
tree03ad5511a8529cc59aad7283b801f19d43ba8792
parent9da8b2e769fe3b4e8957eae19d8a203a2b2a794f (diff)
downloadPRR_3_023064-b128bec287a75df35db8bef28a4df4c480ab9ea5.tar.gz
PRR_3_023064-b128bec287a75df35db8bef28a4df4c480ab9ea5.tar.bz2
PRR_3_023064-b128bec287a75df35db8bef28a4df4c480ab9ea5.zip
Update on Overleaf.
-rw-r--r--bezout.tex16
1 files changed, 16 insertions, 0 deletions
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}