The first order correction to the exit distribution for some random walks

TL;DR

This paper investigates the first order correction to the exit distribution of various 2D lattice-based random walks, revealing a universal form of correction that depends on the domain but not on the specific walk model.

Contribution

It introduces a conjecture that the first order correction to discrete harmonic measure has a universal form, independent of the domain, depending only on the model and lattice.

Findings

Monte Carlo simulations support the conjecture of a universal correction form.

The correction involves a domain-dependent function and a model-dependent constant.

An explicit formula for the correction density is provided.

Abstract

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 $\rho_D$.