Influence of the random walk finite step on the first-passage probability

TL;DR

This paper investigates how the finite step size in a 2D random walk affects the accuracy of first-passage probability estimates, revealing deviations from the ideal case due to step size limitations.

Contribution

It introduces a numerical simulation approach to quantify the impact of finite step size on first-passage probabilities in 2D random walks.

Findings

Finite step size causes measurable deviations from theoretical first-passage probabilities.

The deviation pattern is regular and quantifiable.

Results highlight the importance of step size in accurate random walk modeling.

Abstract

A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius $R$ in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.