  It has been proved that the distribution of the point where the Smart Kinetic
Walk (SKW) exits a domain converges in distribution to harmonic measure on the
hexagonal lattice. For other lattices, it is believed that this result still
holds, and there is good numerical evidence to support this conjecture. Here we
examine the effect of the symmetry and asymmetry of the transition probability
on each step of the SKW on the square lattice and test if the exit distribution
converges in distribution to harmonic measure as well. From our simulations,
the limiting exit distribution of the SKW with a non-uniform but symmetric
transition probability as the lattice spacing goes to zero is the harmonic
measure. This result does not hold for asymmetric transition probability. We
are also interested in the difference between the SKW with symmetric transition
probability exit distribution and harmonic measure. Our simulations provide
strong support for a explicit conjecture about this first order difference. The
explicit formula for the conjecture will be given below.
