Source localisation in simple random walks

TL;DR

This paper investigates the problem of locating the starting point of a simple random walk on lattices and vertex-transitive graphs, showing that in high dimensions the source can be identified with high probability, unlike in low dimensions.

Contribution

It provides new results on source localization probabilities for simple random walks in various dimensions and extends the analysis to general vertex-transitive graphs.

Findings

In dimensions d ≥ 5, the source can be identified with probability bounded away from 0 using one guess.

In dimensions d ≥ 5, the source can be identified with probability arbitrarily close to 1 using a constant number of guesses.

In dimensions d ≤ 2, the source cannot be located with positive constant probability.

Abstract

We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one guess, and (b) with probability arbitrarily close to $1$ using a constant number of guesses. On the other hand, for dimensions $d \leq 2$, we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs.