Simulating a Random Walk with Constant Error
Joshua Cooper, Joel Spencer · Combinatorics Probability Computing · 2006
We analyse Jim Propp's $P$ -machine, a simple deterministic process that simulates a random walk on ${\mathbb Z}^d$ 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 $\{p(\cdot,{\bf x}) : {\bf x} \in {\mathbb Z}^d\}$ , where $p(n,{\bf x})$ is the probability that a walk beginning from the origin arrives at ${\bf x}$ at time $n$ .