Random walk in a finite directed graph subject to a road coloring

TL;DR

This paper establishes that a finite directed graph's random walk driven by road colors is measurable with respect to the road colors if and only if the coloring is synchronizing, linking graph structure to probabilistic properties.

Contribution

It proves a necessary and sufficient condition for measurability of the walk based on the synchronizing property of the road coloring.

Findings

Measurability holds iff the road coloring is synchronizing.

Non-synchronizing colorings lead to uniform distribution on a partition.

The paper characterizes the probabilistic behavior of the walk under different colorings.

Abstract

A necessary and sufficient condition for a random walk in a finite directed graph subject to a road coloring to be measurable with respect to the driving random road colors is proved to be that the road coloring is synchronizing. For this, the random walk subject to a non-synchronizing road coloring is proved to have uniform law on a certain partition of the state space.