About Thinning Invariant Partition Structures

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