Cutting and Shuffling of a Line Segment: Effect of Variation in Cut Location

We present a computational study of the impact of variation in cut location on finite-Time mixing of a line segment by cutting and shuffling, which is a one-dimensional piecewise isometry (PWI), also known as an interval exchange transformation (IET). A line segment of unit length is repeatedly cut into subsegments and shuffled according to any one of a variety of permutations. To mimic practical process error, variations drawn from a normal distribution are used to perturb cut locations. Illustrative examples of the mixing behaviors and finite-Time measures of mixing are used to analyze the effect of variation in cut location for different permutations of subsegment mixing order. Mixing is significantly improved under irreducible nonrotational permutations when the dynamics show a resonance-like structure without variation. Specifically, the requirement of an irrational subsegment length ratio for good mixing can be relaxed as the underlying periodic dynamics is perturbed by the stochastic variation in cut location. Thus, good mixing can occur even with only four subsegments of roughly the same length for most irreducible nonrotational permutations.