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???; (3) partition structures. For the first case we characterize the distributions which are simultaneously invariant under Bernoulli-p thinning for all p??(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.