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author | kurchan.jorge <kurchan.jorge@gmail.com> | 2020-12-08 10:32:47 +0000 |
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committer | overleaf <overleaf@localhost> | 2020-12-08 10:32:49 +0000 |
commit | b128bec287a75df35db8bef28a4df4c480ab9ea5 (patch) | |
tree | 03ad5511a8529cc59aad7283b801f19d43ba8792 | |
parent | 9da8b2e769fe3b4e8957eae19d8a203a2b2a794f (diff) | |
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Update on Overleaf.
-rw-r--r-- | bezout.tex | 16 |
1 files changed, 16 insertions, 0 deletions
@@ -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} |