Interlacing sequences resulting from an interval split-merge dynamics and the induced probability measures

TL;DR

This paper investigates the complex limiting behavior of interlacing sequences generated by a split-merge process on partitions of the unit interval, revealing rich structures even under deterministic rules.

Contribution

It introduces a novel split-merge dynamics framework for interval partitions and analyzes their empirical distributions, both deterministic and randomized, from multiple perspectives.

Findings

Empirical distributions exhibit rich structures despite deterministic rules.

The limiting behavior of break points' distribution is characterized.

Randomized splitting rules lead to diverse asymptotic distributions.

Abstract

We study sequences of partitions of the unit interval into subintervals, starting from the trivial partition, in which each partition is obtained from the one before by splitting its subintervals in two, according to a given rule, and then merging pairs of subintervals at the break points of the old partition. The $n$th partition then comprises $n+1$ subintervals with $n$ break points, which inherently possess an interlacing property. The empirical distribution of these points reveals a surprisingly rich structure, even when the splitting rule is completely deterministic. We consider both deterministic and randomized splitting rules and we study from multiple angles the limiting behavior of the empirical distribution of the break points.