About Thinning Invariant Partition Structures

TL;DR

This paper characterizes distributions invariant under Bernoulli-$p$ thinning for sequences and explores conjectures for related structures, connecting to spin glass models and extending prior work on point processes.

Contribution

It provides a complete characterization for sequence invariance under Bernoulli-$p$ thinning and proposes conjectures for gaps and partition structures, with a new perspective on spin glass relations.

Findings

Characterization of sequence distributions invariant under Bernoulli-$p$ thinning.

Conjectures for invariance in gaps and partition structures.

Connection to spin glass models and previous research.

Abstract

Bernoulli-$p$ thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences $(X_1,X_2,...)$; (2) gaps of such sequences $(X_{n+1}-X_1)_{n\in\mathbb{N}}$; (3) partition structures. For the first case we characterize the distributions which are simultaneously invariant under Bernoulli-$p$ thinning for all $p \in (0,1]$. Based on this, we make conjectures for the latter two cases, and provide a potential approach for proof. We explain the relation to spin glasses, which is complementary to important previous work of Aizenman and Ruzmaikina, Arguin, and Shkolnikov.