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 ∈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.