Simulating a Random Walk with Constant Error

TL;DR

This paper examines Jim Propp's P-machine, a deterministic process that effectively simulates a random walk on integer lattices with a bounded error, using probabilistic estimates and conjectures about related functions.

Contribution

It provides a proof that the P-machine approximates a random walk within a constant error and discusses open conjectures on function properties in the context of random walks.

Findings

Error in simulation is bounded by a constant

Uses estimates from simple random walk theory

Presents conjectures on sign-changes and unimodality

Abstract

We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,x) : x \in Z^d\}$, where $p(n,x)$ is the probability that a walk beginning from the origin arrives at $x$ at time $n$.