  We study three different random walk models on several two-dimensional
lattices by Monte Carlo simulations. One is the usual nearest neighbor random
walk. Another is the nearest neighbor random walk which is not allowed to
backtrack. The final model is the smart kinetic walk. For all three of these
models the distribution of the point where the walk exits a simply connected
domain $D$ in the plane converges weakly to harmonic measure on $\partial D$ as
the lattice spacing $\delta \rightarrow 0$. Let $\omega(0,|dz|;D)$ be harmonic
measure for $D$, and let $\omega_\delta(0,|dz|;D)$ be the discrete harmonic
measure for one of the random walk models. Our definition of the random walk
models is unusual in that we average over the orientation of the lattice with
respect to the domain. We are interested in the limit of
$(\omega_\delta(0,|dz|;D)- \omega(0,|dz|;D))/\delta$. Our Monte Carlo
simulations of the three models lead to the conjecture that this limit equals
$c_{M,L} \, \rho_D(z) |dz|$, where the function $\rho_D(z)$ depends on the
domain, but not on the model or lattice, and the constant $c_{M,L}$ depends on
the model and on the lattice, but not on the domain. So there is a form of
universality for this first order correction. We also give an explicit formula
for the conjectured density $\rho_D$.
