Checkerboarding

Is there any particularly good order in which to sweep through the lattice? A natural approach runs through the sites in typewriter order, i.e. all the $x$ values in ascending order for $y = 0$, then all the $x$ values for $y = 1$, etc. Notice, however, that this procedure updates a site and then immediately uses the new spin value when updating the neighbor. This process risks introducing a directional bias. The effect of updating is always propagated in the positive $x$ and positive $y$ directions. Any directional bias is easily avoided by a ``checkerboard'' updating order. We color the sites of the lattice in the same manner as a checkerboard (same as a chess board) alternating red and black. The red sites then have only black neighbors and vice versa. The checkerboard schemefirst updates all the red sites and then all the black sites to complete one sweep.