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 ${\mathbb{Z}}_p$-modules modeled by $\mathrm{c}\mathrm{o}\mathrm{k}(M_1\cdots M_k)\twoheadrightarrow \cdots \twoheadrightarrow \mathrm{c}\mathrm{o}\mathrm{k}(M_1)$ for independent random matrices $M_1,\dots ,M_k\in {\mathrm{M}\mathrm{a}\mathrm{t}}_n({\mathbb{Z}}_p)$ satisfies the Cohen–Lenstra heuristic. We answer the question affirmatively when $M_1,\dots ,M_k$ 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.