About Thinning Invariant Partition Structures

Bernoulli-$p$ thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences $(X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.})$; (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.