Cokernels of random matrix products and flag Cohen--Lenstra heuristic

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective flag of Zₚ-modules modeled by cok(M₁⋯ Mₖ) cok(M₁) for independent random matrices M₁,,Mₖ∈ Matₙ( Zₚ) satisfies the Cohen--Lenstra heuristic. We answer the question affirmatively when M₁,,Mₖ follow the Haar measure, and our proof demonstrates how classical ideas in Cohen--Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices.