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| -rw-r--r-- | bezout.tex | 11 |
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@@ -333,17 +333,6 @@ limit as $\epsilon$ is varied. {\color{teal} {\bf somewhere else} -Another instrument we have to study this problem is to compute the following partition function: -\begin{equation} - \begin{aligned} - Z(a,\beta)&=\int dx\, dy \, e^{-\mathop{\mathrm{Re}}(\beta H_0)}\\ - &\qquad\delta(\sum_i z_i^2-N) \delta\left(\sum_i y_i^2 -N \frac{a-1}{2}\right). - \end{aligned} -\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 the number of configurations of a given $(a,H_0)$. -This problem may be solved exactly with replicas, {\em but it may also be simulated} \cite{Bray_1980_Metastable}. -Consider for example the ground-state energy for given $a$, that is, the energy in the limit $\beta_R \rightarrow \infty$ taken adjusting $\beta_I$ so that $\Im H_0=0$ . For $a=1$ this coincides with the ground-state of the real problem. {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$. For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation |