Split merge dynamics for expanding intervals and point processes on the real line

TL;DR

This paper investigates split-merge dynamics for partitions of intervals and point processes on the real line, analyzing convergence properties and invariant distributions under various conditions.

Contribution

It introduces new results on the limiting behavior of split-merge processes, including conditions for absolute continuity and characterization of invariant measures.

Findings

Empirical distributions converge to a singular measure supported at endpoints for fixed interval length.

When interval lengths vary regularly, the limit can be absolutely continuous.

Invariant distributions are characterized for split-merge dynamics on the real line.

Abstract

We study sequences of partitions of a non decreasing sequence I n of intervals 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 nth partition then comprises n+1 subintervals with n break points. When I n = [0, 1] is constant, the empirical distribution of these points was shown to converge weakly to a singular probability supported in {0, 1} in a previous article. When the length of the intervals is regularly varying with a positive index, we show in this article that the limit can be absolutely continuous. In the last part we extend the split merge dynamics to partitions of R. In this case we characterize invariant distributions and show that special instances of split merge dynamics for expanding intervals converge to these invariant measures vaguely in distribution.