The Exit Distribution for Smart Kinetic Walk with Symmetric and Asymmetric Transition Probability

摘要

It has been proved that the distribution of the point where the Smart KineticWalk (SKW) exits a domain converges in distribution to harmonic measure on thehexagonal lattice. For other lattices, it is believed that this result stillholds, and there is good numerical evidence to support this conjecture. Here weexamine the effect of the symmetry and asymmetry of the transition probabilityon each step of the SKW on the square lattice and test if the exit distributionconverges in distribution to harmonic measure as well. From our simulations,the limiting exit distribution of the SKW with a non-uniform but symmetrictransition probability as the lattice spacing goes to zero is the harmonicmeasure. This result does not hold for asymmetric transition probability. Weare also interested in the difference between the SKW with symmetric transitionprobability exit distribution and harmonic measure. Our simulations providestrong support for a explicit conjecture about this first order difference. Theexplicit formula for the conjecture will be given below.