About Thinning Invariant Partition Structures

Abstract

Bernoulli-p thinning has been well-studied for point processes. Here we consider three other cases: (1) sequences (X1, X2, . . . ); (2) gaps of such sequences (Xn+1 − X1) 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.