The Monte-Carlo results are obtained in a similar way as before. The spin correlations
are used as a proxy to get to the energy gap of the system because it can be compared
to D&K. As has been discussed earlier however, it unfortunately does not give extremely
accurate answers, and relies heavily upon data fitting and has a rather weak signal to
noise ratio. The typical fitting was exemplified in Figure 4-6 and the procedure was
employed here to obtain the energy gap for a number of runs.
18 i i i
16- D&K
14 Monte-Carlo
12 -
10 -
8-
6-
4-
2-
011
0 02 04 06 08 1 12 14 16 18
Figure 4-11. M=12 D&K and MC comparison. Comparison between the energy splits of
the D&K paper and that of Lightcone World Sheet Monte-Carlo simulation.
Both cases shown have M 12 (equivalently K 12 in the notation of D&K),
the Monte-Carlo simulation is done with N 1000 whereas the D&K results
are done with continuous "time". In this context the results for M 10 and
M=8 are statistically indistinguishable in the Monte-Carlo simulation, which
is why we do not bother with the comparison for those values of M.
As a final note on the comparison of results presented here, we should point out
that even though the Monte Carlo method allows for estimations of the energy gap for
much broader range of M-s then there is no way of obtaining the actual energy levels of
the theory. Moreover, only the gap between the ground state and the first excited state
has been obtained with the Monte Carlo method, and although higher gaps certainly
leave a signature in the Monte Carlo data, their quantitative size are very difficult to
obtain, at least using the spin correlations as is done here. The Monte Carlo methods
as implemented here, are not appropriate for low coupling, because in this regime the