Fragmentations with erasure

TL;DR

This work studies the asymptotics of break-point distributions in fragmentation with erasure, a process that builds partitions of $(0,1]$ by splitting subintervals and then erasing prior break points. The authors encode break points via a time-inhomogeneous random walk (or a random walk in random environment) and establish weak convergence of the empirical break-point distributions to a measure supported at the endpoints $0$ and $1$, with the endpoint mass determined by the average splitting proportions. They then refine the picture by showing a CLT-based, away-from-endpoints scaling leading to a continuous density given by the derivative $Q'$, where $Q=\

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, and 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 study the limiting behavior of the empirical distribution of the break points, from multiple angles.

Figures (1)

• Figure 1: Illustration of the iterative definition of $(\mathcal{P}_n)_{n\ge0}$.

Theorems & Definitions (29)

• Proposition 2.1
• proof
• Proposition 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1
• proof
• Remark 4.2