diff options
| -rw-r--r-- | bezout.tex | 2 |
1 files changed, 1 insertions, 1 deletions
@@ -345,7 +345,7 @@ 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. -{\color{teal} {\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 $\sim(a-1)^N$. +{\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 is more interesting. In the range of values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around |