0 100 200 300 400 500 600 700 800 900 1000
Figure 4-6. Fitting Monte-Carlo data to exp. The data from the Monte-Carlo simulations
is fitted to an exponential in order to read off the mass-gap. Each point on the
graph gives the correlation between a fixed up-spin at (i, j) = (1,500) and a
spin at various location on the space-slice i = 1. The simulated system here
had M=40 and N 1000 and we used about 106 sweeps. The The top plot is
from a low coupling Monte-Carlo run (g/p2 = 0.9), whereas the lower plot
is from strong coupling (g/p 2 = 0.4). Notice how much weaker the overall
correlation is for low coupling, i.e., the signal strength in the exp falloff is
weak.
various starting states such as the "almost empty" state (all spins down except for a long
line of up spins through the entire lattice), or some random spin state. We saw how these
observables, although starting at some values converged to the same "equilibrium" value
irrespective of the starting point. When this value had been reached within statistical
accuracy, we claimed the system to be relaxed. A systematic and detailed analysis was
done by observing magnetization and correlations and we found that about 105 or 106
sweeps was usually more than enough, even if starting from the "almost empty" state, a
state which we believe had little overlap with the "ground" state, or "equilibrium" state.
The equilibrium state of course depends on the coupling so we took up the standard
of running at least 105 sweeps before data was sampled, even when we started from