Scenery Reconstruction on Finite Abelian Groups

TL;DR

This paper investigates conditions under which a random walk on finite abelian groups can be used to reconstruct binary labelings, extending previous results from cycles to more general groups.

Contribution

It establishes the necessity of a known reconstructibility condition for prime-sized cycles with rational probabilities and extends this to broader abelian groups.

Findings

Reconstructibility condition is necessary for prime cycles larger than 5 with rational probabilities.

The condition is not necessary for all cycles, only under specific prime and size constraints.

Extension of results from cycles to general finite abelian groups.

Abstract

We consider the question of when a random walk on a finite abelian group with a given step distribution can be used to reconstruct a binary labeling of the elements of the group, up to a shift. Matzinger and Lember (2006) give a sufficient condition for reconstructibility on cycles. While, as we show, this condition is not in general necessary, our main result is that it is necessary when the length of the cycle is prime and larger than 5, and the step distribution has only rational probabilities. We extend this result to other abelian groups.