accurate then the Monte-Carlo methods computation time only increases polynomially as
a function of the size of the problem K.
Comparing the results of D&K to the Monte-Carlo simulation is especially easy, since
their setup is so much like the Lightcone World Sheet formulation of the same theory.
The action is the same, so the mass and coupling can be directly related and furthermore,
the momentum P+ is even discretized in the same way, so that the D&K K is precisely
our M. However, D&K work in continuous "time" so that we must take N very large
compared to M to be in the same region of parameter space for the comparison (recall
that the world sheet in the Monte-Carlo simulation has N-by-M sites). Another issue of
great importance to the comparison is D&K's concentration on the stringy states, i.e.,
those which are singlets under the global SU(Nc) of the matrix indices and under the
cyclic permutations mentioned above. For the comparison to be meaningful we must
also be working with the very same states, which by construction we are. Firstly, the
single string state is automatic because of our choice of external legs, i.e., the simulation
starts with, and conserves, a single sheet coming in and a single sheet going out. And
secondly, because we take periodic boundary conditions for the spins in the pt direction,
i.e., wrapping the sheet up into a roll. This ensures that we are working with the same
very states as in the D&K work. Notice however, that we did not do this in the exact
treatment of the M = 2 and M = 3 cases above. In those cases we worked with all states,
cyclicly symmetric or not, which accounts for the special treatment there.
In their work, D&K plot the energy levels found from diagonalizing Eq. (4-41) as
well as the splitting of the first two energy levels as a function of the parameter x in that
equation. They do this for K = 8, 10 and 12. Their plot is reproduced in Figure (4-11)
together with our Monte-Carlo results plotted on the same graph. The results are in
relatively good agreement, especially when taking account of the statistical accuracy of the
stochastic simulation.